Functions and Their Representations

Definition of a Function

In math, we often study how one quantity depends on another. For example, the total cost depends on how many items you buy, and the distance traveled depends on how long you move.

A function is a special kind of rule. It takes an input and gives an output. The most important part is this: each input must match with exactly one output.

This idea is the definition of a function:

A function is a rule that assigns exactly one output to each input.

You can think of a function like a machine. You put something in, and the machine gives one result back. If the same input could give two different outputs, then it is not a function.

Important words

• Input: the value you start with
• Output: the value you get from the rule
• Rule: what the function does to the input

We often use letters like \(x\) for the input and \(y\) for the output. For example, in the rule \(y = 2x + 1\), you choose a value for \(x\), and then the rule tells you the value of \(y\).

What makes something a function?

To decide whether something is a function, ask this question:

Does every input have exactly one output?

• If yes, it is a function.
• If no, it is not a function.

There are two common ways something can fail to be a function:

• An input has more than one output.
• An input is missing an output when it should have one in the situation.

Functions can be shown in different ways

In 8th grade, you may see functions represented as:

• words
• tables
• ordered pairs
• equations
• graphs

No matter how the function is shown, the definition stays the same: one input, one output.

1. Functions in words

A rule written in words can describe a function. For example:

“Multiply the input by 3.”

If the input is \(4\), the output is \(12\). If the input is \(7\), the output is \(21\). Each input leads to one output, so this is a function.

2. Functions in a table

A table shows inputs and outputs in pairs.

Example table:

\[ \begin{array}{c|c} x & y \\ \hline 1 & 4 \\ 2 & 7 \\ 3 & 10 \\ 4 & 13 \end{array} \]

Each input value of \(x\) appears once and has one output value of \(y\). So this table represents a function.

If a table showed the same input with two different outputs, it would not be a function.

For example:

\[ \begin{array}{c|c} x & y \\ \hline 2 & 5 \\ 2 & 8 \\ 3 & 7 \end{array} \]

Here, the input \(2\) has outputs \(5\) and \(8\). Since one input gives two outputs, this is not a function.

3. Functions as ordered pairs

An ordered pair looks like \((x, y)\). The first number is the input, and the second number is the output.

Example: \((1, 2), (2, 4), (3, 6), (4, 8)\)

Each input is different and has one output, so this is a function.

Example: \((2, 3), (2, 6), (5, 9)\)

The input \(2\) is paired with both \(3\) and \(6\). That means it is not a function.

4. Functions as equations

An equation can represent a function if each input gives exactly one output.

Example: \(y = x + 4\)

If you choose any value for \(x\), there is only one value of \(y\).

• If \(x = 1\), then \(y = 5\)
• If \(x = 3\), then \(y = 7\)
• If \(x = 10\), then \(y = 14\)

So \(y = x + 4\) is a function.

5. Functions on a graph

On a graph, a function means that each \(x\)-value matches only one \(y\)-value.

A helpful test is the vertical line test:

• If a vertical line touches the graph at only one point each time, the graph is a function.
• If a vertical line touches the graph at more than one point, the graph is not a function.

Why does this work? A vertical line has one fixed \(x\)-value. If it hits the graph in two places, then that same input \(x\) has two outputs \(y\), so it cannot be a function.

Functions and real-life situations

Many real-life rules are functions.

• A number of movie tickets bought and the total cost
• Hours worked and total pay
• Number of miles traveled at a constant rate and total distance

In each case, one input gives one output.

But not every real-life matching is a function.

For example, “student to favorite sports” may not be a function if one student likes basketball and soccer. That one input would have more than one output.

Worked Example 1: Using a rule

Rule: “Multiply the input by 2, then add 1.”

Is this a function?

Step 1: Check what happens to an input.

If the input is \(3\):

$$y = 2(3) + 1 = 7$$

If the input is \(5\):

$$y = 2(5) + 1 = 11$$

Step 2: Ask whether each input gets exactly one output.

Yes. Every input is changed by the same rule and gives one answer.

Answer: Yes, this is a function.

Worked Example 2: Looking at ordered pairs

Decide whether the set \((1, 4), (2, 6), (3, 8), (2, 6)\) is a function.

Step 1: Look at the inputs: \(1, 2, 3, 2\).

The input \(2\) appears twice.

Step 2: Check whether it has different outputs.

Both times, \(2\) is paired with \(6\).

Important idea: Repeating the same ordered pair does not break the function rule. The problem happens only when one input has different outputs.

Answer: Yes, this is still a function.

Worked Example 3: Looking at a table

Decide whether this table represents a function.

\[ \begin{array}{c|c} x & y \\ \hline 4 & 9 \\ 5 & 11 \\ 6 & 13 \\ 5 & 12 \end{array} \]

Step 1: Look for repeated inputs.

The input \(5\) appears twice.

Step 2: Compare the outputs for that input.

One time \(5\) gives \(11\), and another time \(5\) gives \(12\).

Step 3: Use the definition.

One input has two different outputs.

Answer: No, this is not a function.

Worked Example 4: Looking at an equation

Does the equation \(y = 3x - 2\) represent a function?

Step 1: Try some inputs.

• If \(x = 0\), then \(y = 3(0) - 2 = -2\)
• If \(x = 2\), then \(y = 3(2) - 2 = 4\)
• If \(x = -1\), then \(y = 3(-1) - 2 = -5\)

Step 2: Decide if each input gives one output.

Yes. Every value of \(x\) leads to exactly one value of \(y\).

Answer: Yes, \(y = 3x - 2\) is a function.

How this connects to linear and nonlinear patterns

Some functions are linear, and some are nonlinear. Both can still be functions.

A linear function has a graph that is a straight line, such as \(y = 2x + 1\).

A nonlinear function does not have a straight-line graph, such as \(y = x^2\).

The key point is this: being a function is not the same as being linear.

• Some functions are linear.
• Some functions are nonlinear.
• As long as each input has exactly one output, it is a function.

Common mistakes to avoid

• Mistake 1: Thinking all equations are functions. Some are not. Always check whether one input can give more than one output.
• Mistake 2: Looking only at outputs. Outputs are allowed to repeat. Different inputs can have the same output.
• Mistake 3: Forgetting that repeated inputs are okay only if they repeat the same output.

For example, the set \((1, 5), (2, 5), (3, 5)\) is a function. Different inputs all lead to the same output, and that is allowed.

Quick check questions

1. Is \((1, 2), (2, 3), (3, 4)\) a function?
2. Is \((4, 7), (4, 9), (5, 10)\) a function?
3. Does the rule \(y = x - 8\) describe a function?
4. If a graph fails the vertical line test, is it a function?

Answers

1. Yes, each input has one output.
2. No, input \(4\) has two outputs.
3. Yes, each input gives exactly one output.
4. No, it is not a function.

Summary

A function is a rule that assigns exactly one output to each input. You can recognize functions in words, tables, ordered pairs, equations, and graphs. To check, always ask: Does each input have only one output? If yes, it is a function. If no, it is not.

The Vertical Line Test

When we look at a graph, we often want to know whether it represents a function. A function is a rule that gives exactly one output for each input.

In graphing, the input is usually the 34x-value and the output is the y-value. So if a graph is a function, each x-value can match with only one y-value.

The Vertical Line Test is a quick way to check this on a graph.

What is the Vertical Line Test?

Imagine drawing a vertical line anywhere on the graph. A vertical line goes straight up and down.

If that vertical line touches the graph in more than one point, then the graph is not a function.

If every vertical line touches the graph at only one point, then the graph is a function.

Why does this work?

A vertical line shows one fixed x-value. If that same x-value hits the graph in two places, then that one input has two different outputs.

That breaks the rule of a function, because a function must give only one output for each input.

Key idea: A graph is a function only if no x-value is used more than once with different y-values.

How to use the Vertical Line Test

1. Look at the graph.

2. Imagine moving a vertical line from left to right across the graph.

3. Check how many times the line touches the graph at each spot.

4. If it ever touches twice or more, the graph is not a function.

5. If it always touches once, the graph is a function.

What kinds of graphs usually pass the test?

• A straight line that is not vertical

• A parabola opening up or down

• Many curves that go through only one y-value for each x-value

What kinds of graphs usually fail the test?

• A vertical line

• A circle

• Sideways curves that give two y-values for one x-value

Worked Example 1: A slanted line

Suppose a graph is a straight line like $$y = 2x + 1$$

If you draw vertical lines across this graph, each vertical line hits the line only once.

That means each x-value has exactly one y-value.

Conclusion: This graph is a function.

Worked Example 2: A circle

Now think about a circle, such as a graph centered at the origin.

If you draw a vertical line through the middle of the circle, it will hit the circle in two places: one point on the top half and one point on the bottom half.

That means the same x-value has two different y-values.

Conclusion: A circle is not a function.

Worked Example 3: A parabola opening upward

Look at the graph of $$y = x^2$$

This is a U-shaped graph. Even though it curves, any vertical line touches it only once.

For example:

• If \(x = 0\), then \(y = 0\)

• If \(x = 2\), then \(y = 4\)

• If \(x = -2\), then \(y = 4\)

Notice that different x-values can share the same y-value. That is okay.

What is not okay is one x-value having two different y-values.

Conclusion: $$y = x^2$$ is a function.

Worked Example 4: A sideways parabola

Now look at a sideways parabola, such as $$x = y^2$$

This graph opens to the right. If you draw a vertical line through part of the graph, it will often hit the graph twice.

For example, when \(x = 4\), the graph includes:

So:

The same x-value, \(4\), gives two different y-values.

Conclusion: This graph is not a function.

Important notes to remember

• The test checks whether a graph is a function by looking at x-values.

• It is okay for two points to have the same y-value.

• It is not okay for one x-value to have more than one y-value.

• A graph can curve and still be a function.

• A graph does not have to be a straight line to be a function.

Common mistake

Some students think that if a graph crosses a horizontal line more than once, it is not a function. That is not true.

The Vertical Line Test uses vertical lines, not horizontal ones.

For example, the graph of $$y = x^2$$ is a function, even though many horizontal lines cross it twice.

Quick checklist

• Does any vertical line hit the graph more than once?

• If yes, it is not a function.

• If no, it is a function.

Summary

The Vertical Line Test helps you decide whether a graph represents a function. If every vertical line touches the graph at only one point, the graph is a function.

If even one vertical line touches the graph more than once, then the graph is not a function. Always remember: one x-value must go with only one y-value.

Domain and Range in Context

When we talk about a function, we are talking about a rule that takes an input and gives exactly one output. In real-life situations, not every input makes sense, and not every output is possible. That is why we study domain and range.

The domain is the set of all possible input values. The range is the set of all possible output values.

For example, if a machine takes the number of hours you work and gives your pay, then the hours worked are the input and the pay is the output. But you cannot work a negative number of hours, so the context helps limit the domain.

Why “in context” matters

In math class, a function might allow many values. But in a real situation, the story often gives limits. These limits help us decide which inputs and outputs are reasonable.

For example, a formula may work for all numbers, but if it represents the number of movie tickets bought, then the input should not be negative. Also, you usually cannot buy half a ticket, so the input may need to be whole numbers only.

Main idea: To find domain and range in context, ask two questions:

• What inputs make sense in this situation?
• What outputs come from those sensible inputs?

How to find the domain

1. Identify the input.
2. Think about what values the input can have.
3. Use the context to rule out impossible values.

Some common context limits are:

• No negative values, such as time, distance, age, or number of items.
• Whole numbers only, such as people, books, or tickets.
• A maximum value, such as the number of seats in a bus or the hours in a day.

How to find the range

1. Identify the output.
2. Use the possible inputs from the domain.
3. Find all output values that can happen.

The range depends on the domain. If the set of inputs changes, the set of outputs may change too.

Worked Example 1: Movie tickets

A movie theater charges \(\$12\) per ticket. Let \(C = 12t\), where \(t\) is the number of tickets and \(C\) is the total cost.

Step 1: Find the domain.

The input is \(t\), the number of tickets. You cannot buy a negative number of tickets, and usually tickets are counted in whole numbers.

So the domain is:

all whole numbers greater than or equal to \(0\)

If we write some possible values, the domain could look like:

\(0, 1, 2, 3, 4, \dots\)

Step 2: Find the range.

Use the rule \(C = 12t\).

• If \(t = 0\), then \(C = 0\)
• If \(t = 1\), then \(C = 12\)
• If \(t = 2\), then \(C = 24\)

So the range is:

\(0, 12, 24, 36, 48, \dots\)

These are all nonnegative multiples of \(12\).

Worked Example 2: Filling a water tank

A water tank starts empty and fills at a rate of \(5\) liters per minute. The amount of water is given by \(W = 5m\), where \(m\) is the number of minutes.

Suppose the tank fills for up to \(20\) minutes.

Step 1: Find the domain.

The input is time in minutes. Time cannot be negative. Since the tank fills for up to \(20\) minutes, the possible input values are from \(0\) to \(20\).

So the domain is:

$$0 \le m \le 20$$

Here, any time between \(0\) and \(20\) makes sense, including decimals like \(3.5\) minutes.

Step 2: Find the range.

Use \(W = 5m\).

• At \(m = 0\), \(W = 0\)
• At \(m = 20\), \(W = 100\)

So the range is:

$$0 \le W \le 100$$

The water amount can be any value from \(0\) liters to \(100\) liters.

Notice the difference from Example 1: time can be measured with parts of a minute, so decimals are allowed in the domain.

Worked Example 3: Height on a ladder

A person climbs a ladder at a steady rate. Their height above the ground is given by \(h = 2t + 1\), where \(t\) is time in seconds and \(h\) is height in feet. The person climbs for \(6\) seconds.

Step 1: Find the domain.

The input is time. Since the climb starts at \(0\) seconds and lasts \(6\) seconds, the domain is:

$$0 \le t \le 6$$

Step 2: Find the range.

Find the smallest and largest output.

• At \(t = 0\), \(h = 2(0) + 1 = 1\)
• At \(t = 6\), \(h = 2(6) + 1 = 13\)

So the range is:

$$1 \le h \le 13$$

This means the person’s height above the ground goes from \(1\) foot to \(13\) feet.

Worked Example 4: Renting bikes

A bike shop rents bikes for \(\$8\) each. There are only \(15\) bikes available. Let \(R = 8b\), where \(b\) is the number of bikes rented and \(R\) is the total cost.

Step 1: Find the domain.

The input is the number of bikes rented. You cannot rent a negative number of bikes. You also cannot rent more than \(15\) bikes, and the number of bikes must be a whole number.

So the domain is:

\(0, 1, 2, 3, \dots, 15\)

Step 2: Find the range.

Use \(R = 8b\).

• If \(b = 0\), then \(R = 0\)
• If \(b = 1\), then \(R = 8\)
• If \(b = 15\), then \(R = 120\)

So the range is:

\(0, 8, 16, 24, \dots, 120\)

Important idea: Even though the rule \(R = 8b\) could keep going forever in pure math, the real-world situation limits the domain to at most \(15\) bikes.

How domain and range may look in different representations

You may see a function as a table, graph, equation, or word problem. No matter how it is shown, domain means inputs and range means outputs.

• In a table: domain is the set of input values; range is the set of output values.
• On a graph: domain is the set of \(x\)-values; range is the set of \(y\)-values.
• In an equation: think about what the variable represents in the context.
• In a story: use common sense and the details in the situation.

Example from a table

Suppose a table shows the cost to mail packages.

Inputs: number of packages mailed

Outputs: total cost

• \(1 \to 4\)
• \(2 \to 8\)
• \(3 \to 12\)
• \(4 \to 16\)

The domain is \(\{1, 2, 3, 4\}\).

The range is \(\{4, 8, 12, 16\}\).

Common mistakes to avoid

• Mixing up domain and range. Domain is input; range is output.
• Ignoring the context. A formula may allow many numbers, but the situation may not.
• Forgetting about whole numbers. Some inputs, like people or tickets, cannot be decimals.
• Forgetting about limits. A maximum number of seats, hours, or items can shorten the domain and range.

Quick check questions

1. A taxi charges \(\$3\) per mile. If a ride can be from \(0\) to \(25\) miles, what is the domain?

The domain is all mile values from \(0\) to \(25\). If distance can include parts of a mile, then:

$$0 \le m \le 25$$

2. Using the same taxi ride, what is the range for cost \(C = 3m\)?

At \(m = 0\), cost is \(0\). At \(m = 25\), cost is \(75\). So:

$$0 \le C \le 75$$

3. A classroom has \(30\) desks. If \(s\) is the number of students seated, what kind of numbers are in the domain?

The domain uses whole numbers from \(0\) to \(30\), because you cannot have a negative number of students or part of a student.

Summary

Domain is the set of possible inputs. Range is the set of possible outputs. In real-world problems, you must use the context to decide what values make sense.

Always ask:

• What does the input represent?
• Can it be negative?
• Must it be a whole number?
• Is there a greatest or smallest possible value?

Then use those input values to find the output values. That is how you find domain and range in context.

Multiple Representations of Functions

In math, the same function can be shown in different ways. You might see it as a word description, an equation, a table, or a graph. Learning to move between these forms helps you understand what the function is doing.

A function is a rule that matches each input with exactly one output. The input is often called the independent variable, and the output is often called the dependent variable because it depends on the input.

For example, if you put in a number of hours worked and get out an amount of money earned, the money depends on the hours. That relationship can be described with words, shown in a table, written as an equation, or drawn on a graph.

The 4 common representations of a function

• Verbal description: tells the rule in words
• Algebraic rule: shows the rule using variables and numbers
• Table of values: lists input-output pairs
• Graph: shows the pairs as points on a coordinate plane

These forms all describe the same relationship. Your job is to recognize how they connect.

1. Verbal descriptions

A verbal description explains how the output changes when you know the input. Here are some examples:

• "The cost is \(3\) dollars for each notebook."
• "Start with \(5\), then add \(2\) for each hour."
• "The area of a square is the side length times itself."

When reading a verbal description, ask:

• What is the input?
• What is the output?
• Does the output increase by the same amount each time?
• Can I write a rule for it?

2. Algebraic rules

An algebraic rule uses variables to show the function. For example:

• \(y = 3x\)
• \(y = 2x + 5\)
• \(y = x^2\)

In these rules, \(x\) is the input and \(y\) is the output. To find an output, substitute a value for \(x\).

For example, in \(y = 2x + 5\), if \(x = 4\), then

So the input-output pair is \((4, 13)\).

3. Tables of values

A table shows several inputs and their matching outputs. Tables are useful because they help you see patterns.

For the rule \(y = 2x + 1\), a table could be:

As \(x\) increases by \(1\), \(y\) increases by \(2\). That constant change is an important clue that this function is linear.

4. Graphs

A graph shows the input-output pairs as points \((x, y)\) on the coordinate plane. The horizontal axis is the x-axis, and the vertical axis is the y-axis.

To graph a function from a table:

1. Take each row as an ordered pair.
2. Plot the points.
3. Look for a pattern in the points.

If the points lie on a straight line, the function is linear. If they make a curve or another shape, the function is nonlinear.

How the representations connect

Suppose a function says, "You earn \(4\) dollars for each hour worked."

• Verbal: You earn \(4\) dollars per hour.
• Equation: \(y = 4x\)
• Table: inputs are hours, outputs are dollars earned
• Graph: points like \((1,4)\), \((2,8)\), \((3,12)\)

Each form is saying the same thing in a different way.

Linear and nonlinear functions

A linear function changes at a constant rate. This means when the input increases by the same amount each time, the output also changes by the same amount each time.

Examples of linear rules:

• \(y = 3x\)
• \(y = x + 7\)
• \(y = 5 - 2x\)

On a graph, a linear function makes a straight line.

A nonlinear function does not change at a constant rate.

Examples of nonlinear rules:

• \(y = x^2\)
• \(y = 2^x\) is beyond this course, so focus mainly on examples like \(y = x^2\)

For 8th Grade, a common nonlinear example is \(y = x^2\). Its graph is not a straight line, and the outputs do not increase by the same amount each time.

How to tell if a table is linear

Check how the output changes when the input increases evenly.

Example table:

The outputs go up by \(3\) each time. Since the change is constant, the function is linear.

Now look at this table:

The outputs increase by \(3\), then \(5\), then \(7\). The change is not constant, so the function is nonlinear.

Worked Example 1: From words to equation to table

Problem: A taxi charges \(2\) dollars for each mile traveled. Show this function as a verbal rule, an equation, and a table.

Step 1: Identify the variables.

• Input: number of miles, \(x\)
• Output: total cost, \(y\)

Step 2: Write the verbal rule.

The cost is \(2\) dollars for each mile.

Step 3: Write the equation.

Since the cost is \(2\) times the number of miles,

Step 4: Make a table.

Step 5: Describe the graph.

Plot the points \((0,0)\), \((1,2)\), \((2,4)\), and \((3,6)\). They form a straight line, so the function is linear.

Worked Example 2: From equation to table to graph idea

Problem: Represent the function \(y = x + 3\) using a table and describe its graph.

Step 1: Choose some input values.

Let \(x = 0, 1, 2, 3\).

Step 2: Find the outputs.

• If \(x = 0\), then \(y = 0 + 3 = 3\)
• If \(x = 1\), then \(y = 1 + 3 = 4\)
• If \(x = 2\), then \(y = 2 + 3 = 5\)
• If \(x = 3\), then \(y = 3 + 3 = 6\)

Step 3: Write the table.

Step 4: Describe the graph.

The points are \((0,3)\), \((1,4)\), \((2,5)\), and \((3,6)\). The graph is a straight line. Each time \(x\) increases by \(1\), \(y\) increases by \(1\), so the function is linear.

Worked Example 3: From table to rule

Problem: Write a rule for this table and decide whether the function is linear.

Step 1: Look for the pattern.

As \(x\) goes up by \(1\), \(y\) goes up by \(3\). So this suggests a linear rule of the form:

Step 2: Use a point from the table.

Use \((1,5)\):

So the rule is

Step 3: Check the rule.

If \(x = 4\), then

This matches the table, so the rule is correct. The function is linear because the output changes by a constant amount.

Worked Example 4: Recognizing a nonlinear function

Problem: Compare the verbal rule, equation, and table for the function where the output is the square of the input.

Verbal rule: The output is the input multiplied by itself.

Equation:

Table:

Step 1: Look at the changes in output.

The outputs change by \(1\), then \(3\), then \(5\), then \(7\). The changes are not the same each time.

Step 2: Decide if it is linear.

No. The function is nonlinear.

Step 3: Describe the graph.

The graph does not make a straight line. It curves upward.

Tips for translating between representations

• From words to equation: look for phrases like “for each,” “times,” “plus,” or “starts at.”
• From equation to table: choose input values and substitute them into the rule.
• From table to graph: turn each row into a point.
• From table to equation: look for a constant change in the output.
• From graph to table: read the coordinates of points on the graph.

Common mistakes to avoid

• Mixing up input and output
• Writing points as \((y, x)\) instead of \((x, y)\)
• Thinking every table is linear
• Forgetting to check whether the output changes by a constant amount
• Using the wrong variable in an equation

Quick check questions

1. If a function is described as “add \(6\) to the input,” what equation could represent it?
2. If the equation is \(y = 5x\), what is the output when \(x = 3\)?
3. If a table’s outputs increase by \(4\) each time the inputs increase by \(1\), is it linear or nonlinear?
4. What does the graph of a linear function look like?

Answers:

1. \(y = x + 6\)
2. \(y = 15\)
3. Linear
4. A straight line

Summary

A function matches each input to exactly one output. The same function can be represented with words, an equation, a table, and a graph. These forms all show the same relationship in different ways.

When working with multiple representations, look for patterns. Linear functions have a constant rate of change and make straight lines on graphs. Nonlinear functions do not have a constant rate of change and do not graph as straight lines.

Comparing Functions means looking at two different functions and deciding how they are alike or different.

Sometimes one function is shown with an equation, while another is shown in a table, graph, or word description. Your job is to compare important features, such as the rate of change and the starting value.

This skill helps you answer questions like:

• Which function grows faster?
• Which function starts at a higher value?
• Are both functions linear?
• How can I compare them if they are shown in different ways?

In this lesson, you will learn how to compare functions clearly and step by step.

1. What is a function?

A function is a rule that gives exactly one output for each input.

Functions can be shown in different representations:

• Equation, such as \(y = 3x + 2\)
• Table of values
• Graph
• Word description

Even if two functions are shown differently, you can still compare them.

2. Important features to compare

When comparing two functions, pay close attention to these features:

• Rate of change: how much the output changes when the input increases by 1
• Starting value: the output when the input is 0
• Linear or nonlinear: whether the function changes at a constant rate

Rate of change tells how steeply a function rises or falls.

For a linear function written as \(y = mx + b\):

• \(m\) is the rate of change
• \(b\) is the starting value

For example, in \(y = 4x + 1\):

• Rate of change = \(4\)
• Starting value = \(1\)

3. How to find these features in different representations

From an equation

If the equation is in the form \(y = mx + b\), then:

• Rate of change = \(m\)
• Starting value = \(b\)

Example: in \(y = 2x + 5\), the rate of change is \(2\) and the starting value is \(5\).

From a table

Look at how the output changes when the input increases by 1.

If the change is always the same, the function is linear.

Example table:

\[\begin{array}{c|c}x & y \\ \hline 0 & 3 \\ 1 & 6 \\ 2 & 9 \\ 3 & 12\end{array}\]

As \(x\) increases by 1, \(y\) increases by 3 each time. So:

• Rate of change = \(3\)
• Starting value = \(3\) because when \(x=0\), \(y=3\)

From a graph

To find the starting value, look for where the graph crosses the y-axis. That happens when \(x=0\).

To find the rate of change, choose two points on the graph and see how much the function goes up or down compared to how much it moves right.

If the graph is a straight line, the function is linear.

From a word description

Look for:

• A value the function starts with
• How much it changes each time

Example: “A bike rental costs \(\$5\) to start and \(\$2\) for each hour.”

This means:

• Starting value = \(5\)
• Rate of change = \(2\)

The equation would be \(y = 2x + 5\).

4. Steps for comparing functions

1. Identify how each function is represented.
2. Find the rate of change for each function.
3. Find the starting value for each function.
4. Decide which function has the greater rate of change.
5. Decide which function has the greater starting value.
6. Check whether each function is linear or nonlinear.

Now let’s work through some examples.

Worked Example 1: Equation vs. Table

Compare these two functions.

Function A: \(y = 3x + 4\)

Function B:

\[\begin{array}{c|c}x & y \\ \hline 0 & 2 \\ 1 & 7 \\ 2 & 12 \\ 3 & 17\end{array}\]

Step 1: Find the rate of change and starting value of Function A.

In \(y = 3x + 4\):

• Rate of change = \(3\)
• Starting value = \(4\)

Step 2: Find the rate of change and starting value of Function B.

In the table, as \(x\) increases by 1, the outputs go:

\(2, 7, 12, 17\)

The output increases by \(5\) each time.

• Rate of change = \(5\)
• Starting value = \(2\)

Step 3: Compare.

• Function B has the greater rate of change because \(5 > 3\).
• Function A has the greater starting value because \(4 > 2\).

Both functions are linear because both have a constant rate of change.

Worked Example 2: Word Description vs. Equation

Function C: “A streaming service charges \(\$8\) each month plus a one-time sign-up fee of \(\$12\).”

Function D: \(y = 10x + 5\)

Step 1: Write the information for Function C.

The sign-up fee is the starting value, so \(12\).

The monthly charge is the rate of change, so \(8\).

Function C can be written as:

$$y = 8x + 12$$

Step 2: Read Function D.

For \(y = 10x + 5\):

• Rate of change = \(10\)
• Starting value = \(5\)

Step 3: Compare.

• Function D has the greater rate of change because \(10 > 8\).
• Function C has the greater starting value because \(12 > 5\).

Worked Example 3: Table vs. Nonlinear Table

Compare these two functions.

Function E:

\[\begin{array}{c|c}x & y \\ \hline 0 & 1 \\ 1 & 4 \\ 2 & 7 \\ 3 & 10\end{array}\]

Function F:

\[\begin{array}{c|c}x & y \\ \hline 0 & 1 \\ 1 & 2 \\ 2 & 4 \\ 3 & 7\end{array}\]

Step 1: Look at Function E.

The outputs increase by:

\(4-1=3\), \(7-4=3\), \(10-7=3\)

The change is always \(3\), so Function E is linear.

• Rate of change = \(3\)
• Starting value = \(1\)

Step 2: Look at Function F.

The outputs increase by:

\(2-1=1\), \(4-2=2\), \(7-4=3\)

The change is not constant, so Function F is nonlinear.

• It does not have one constant rate of change.
• Starting value = \(1\)

Step 3: Compare.

• Function E is linear, but Function F is nonlinear.
• Both functions have the same starting value, \(1\).
• Only Function E has a constant rate of change, which is \(3\).

This is important: if a function does not change at a constant rate, you should not compare it as if it were linear.

Worked Example 4: Graph Idea Compared to an Equation

Suppose a graph shows a line crossing the y-axis at \(6\). From one point to another, the line goes up \(2\) units when it moves right \(1\) unit.

Compare this graph to Function G: \(y = x + 9\).

Step 1: Find the features of the graph.

• Starting value = \(6\)
• Rate of change = \(2\)

So the graph represents a line like:

$$y = 2x + 6$$

Step 2: Find the features of Function G.

For \(y = x + 9\):

• Rate of change = \(1\)
• Starting value = \(9\)

Step 3: Compare.

• The graphed function has the greater rate of change because \(2 > 1\).
• Function G has the greater starting value because \(9 > 6\).

5. Helpful comparison language

When answering comparison questions, you can use sentence starters like these:

• “Function ___ has a greater rate of change than Function ___.”
• “Function ___ has a greater starting value.”
• “Both functions are linear because they have constant rates of change.”
• “Function ___ is nonlinear because its rate of change is not constant.”

6. Common mistakes to avoid

• Mixing up rate of change and starting value
  In \(y = mx + b\), \(m\) is the rate of change and \(b\) is the starting value.
• Not checking for constant change in a table
  A table is only linear if the output changes by the same amount each time the input increases equally.
• Reading the wrong value as the starting value
  The starting value is the output when \(x=0\).
• Comparing nonlinear functions as if they were linear
  If the rate of change is not constant, say that the function is nonlinear.

7. Quick check

Try these on your own:

1. Function H: \(y = 4x + 2\)
   Function I starts at \(5\) and increases by \(3\) each step.
   Which has the greater rate of change? Which has the greater starting value?
2. A table has values \((0,7), (1,11), (2,15), (3,19)\).
   What is the rate of change? What is the starting value? Is it linear?
3. A function has starting value \(2\) and rate of change \(6\). Another has equation \(y = 5x + 4\).
   Which grows faster? Which starts higher?

Answers:

1. Function H has greater rate of change because \(4 > 3\). Function I has greater starting value because \(5 > 2\).
2. Rate of change = \(4\), starting value = \(7\), and it is linear.
3. The first function grows faster because \(6 > 5\). The second function starts higher because \(4 > 2\).

Summary

To compare functions, find the rate of change and the starting value for each function, even if they are shown in different ways.

In an equation \(y = mx + b\), \(m\) is the rate of change and \(b\) is the starting value.

In a table, look for constant change in the outputs. In a graph, use the y-axis crossing for the starting value and the steepness of the line for the rate of change.

Always check whether each function is linear or nonlinear before comparing them.

Identifying Nonlinear Functions means deciding whether a function does not have a constant rate of change.

In 8th Grade, a linear function is a function whose graph is a straight line and whose rate of change stays the same. A nonlinear function is a function where the rate of change changes.

This lesson will help you recognize nonlinear functions in tables, graphs, and equations by checking whether the change is constant.

First, remember what a function is. A function is a rule that gives each input exactly one output. For example, if every value of \(x\) matches with only one value of \(y\), then the relationship is a function.

The big idea: To prove a function is nonlinear, show that its rate of change is not constant.

For a linear function, equal changes in \(x\) give equal changes in \(y\). For a nonlinear function, equal changes in \(x\) do not give equal changes in \(y\).

For example, if \(x\) increases by 1 each time, a linear function might have \(y\) increase by 3 each time. But if \(y\) changes by 2, then 5, then 9, the rate of change is not constant, so the function is nonlinear.

How to check a table:

• Look at how \(x\) changes.
• If the changes in \(x\) are equal, compare the changes in \(y\).
• If the changes in \(y\) are not equal, the function is nonlinear.

If the changes in \(x\) are not equal, you need to compare the rate of change over each interval. That means checking how much \(y\) changes compared to how much \(x\) changes.

The rate of change between two points is:

If those rates are different for different intervals, the function is nonlinear.

How to check a graph:

• A linear function makes a straight line.
• A nonlinear function does not make a straight line.
• You can also check whether the steepness changes from one part of the graph to another.

If the graph bends or curves, the rate of change is changing, so the function is nonlinear.

How to check an equation:

• Linear equations in 8th Grade usually look like \(y = mx + b\).
• If the equation does not fit that form, it may be nonlinear.

For example, \(y = 4x + 1\) is linear because it is in the form \(y = mx + b\).

But equations like \(y = x^2\) or \(y = 2x^2 + 3\) are nonlinear. Their graphs are not straight lines, and their rates of change are not constant.

Worked Example 1: Table with equal \(x\)-steps

Decide whether this function is linear or nonlinear.

Step 1: Check the change in \(x\). Each time, \(x\) increases by 1.

Step 2: Check the change in \(y\).

• From 4 to 7: change of \(+3\)
• From 7 to 10: change of \(+3\)
• From 10 to 13: change of \(+3\)

The change in \(y\) is constant, so the rate of change is constant.

Conclusion: This function is linear, not nonlinear.

Worked Example 2: Table showing a nonlinear function

Decide whether this function is linear or nonlinear.

Step 1: The changes in \(x\) are all \(+1\).

Step 2: Find the changes in \(y\).

• From 1 to 2: \(+1\)
• From 2 to 5: \(+3\)
• From 5 to 10: \(+5\)

The changes in \(y\) are not the same.

That means the rate of change is not constant.

Conclusion: This function is nonlinear.

Worked Example 3: Unequal \(x\)-steps

Decide whether this function is linear or nonlinear.

Here, the changes in \(x\) are not all 1, so we should compare rates of change.

Between \((1,2)\) and \((3,8)\):

Between \((3,8)\) and \((6,17)\):

The rate of change is 3 in both intervals.

Conclusion: This function is linear.

Now compare that to this table:

Between \((1,2)\) and \((3,8)\):

Between \((3,8)\) and \((6,20)\):

The rates are different, so the rate of change is not constant.

Conclusion: This function is nonlinear.

Worked Example 4: Using an equation

Decide whether \(y = x^2 + 1\) is linear or nonlinear.

This equation is not in the form \(y = mx + b\). It has \(x^2\), so that is a clue that it is nonlinear.

We can also test it with values.

Now check the changes in \(y\):

• \(+1\)
• \(+3\)
• \(+5\)

The changes are not constant.

Conclusion: \(y = x^2 + 1\) is nonlinear.

Common mistakes to avoid

• Looking only at the first change: You must compare more than one interval.
• Forgetting to check \(x\)-changes: If \(x\) does not change by the same amount, use the rate of change formula.
• Thinking every function is linear: A relationship can be a function and still be nonlinear.
• Using only the shape of the points without checking: When possible, calculate to be sure.

Quick checklist for identifying a nonlinear function

1. Make sure the relationship is a function.
2. Check whether the rate of change stays the same.
3. If the rate of change changes, the function is nonlinear.
4. On a graph, if it is not a straight line, it is nonlinear.
5. In an equation, if it is not of the form \(y = mx + b\), check carefully because it may be nonlinear.

Summary

A function is nonlinear when its rate of change is not constant. In a table, equal changes in \(x\) should give equal changes in \(y\) for a linear function. If the changes or rates are different, the function is nonlinear.

On a graph, linear functions make straight lines, while nonlinear functions curve or change steepness. In equations, linear functions look like \(y = mx + b\), but equations such as \(y = x^2\) are nonlinear.

Graphing Nonlinear Functions

In earlier lessons, you may have graphed linear functions. A linear function makes a straight line on a graph because the rate of change stays the same.

In this lesson, you will learn about nonlinear functions. These functions do not make a straight line when graphed. Instead, the graph may curve or change direction.

You will learn how to:

• recognize a nonlinear function,
• make a table of values,
• plot points on a coordinate plane, and
• graph simple quadratic and absolute value functions.

1. What is a nonlinear function?

A function is a rule that gives exactly one output for each input. The input is usually called \(x\), and the output is usually called \(y\).

A function is nonlinear if its graph is not a straight line. This usually happens when the pattern in the outputs does not change by the same amount each time.

For example, in a linear function, if \(x\) goes up by 1 each time, then \(y\) changes by the same amount each time. In a nonlinear function, that is not true.

Here are two common nonlinear functions in 8th grade:

• Quadratic functions, such as \(y = x^2\)
• Absolute value functions, such as \(y = |x|\)

2. How to graph a nonlinear function

To graph a nonlinear function, follow these steps:

1. Choose several \(x\)-values.
2. Substitute each \(x\)-value into the rule.
3. Find the matching \(y\)-values.
4. Make a table of ordered pairs.
5. Plot the points on the coordinate plane.
6. Connect the points smoothly if the graph curves.

When graphing nonlinear functions, it is important to plot enough points. Since the graph is not a straight line, two points are usually not enough to show the shape.

3. Looking for patterns in a table

A table can help you tell whether a function is linear or nonlinear.

If the \(x\)-values increase by the same amount, check how the \(y\)-values change:

• If the \(y\)-values change by the same amount, the function is linear.
• If the \(y\)-values do not change by the same amount, the function is nonlinear.

Example of a nonlinear pattern:

If the \(y\)-values are \(0, 1, 4, 9\), the changes are \(+1, +3, +5\). Since the changes are not the same, the function is nonlinear.

4. Graphing a quadratic function

A simple quadratic function is:

$$y = x^2$$

This means you square each \(x\)-value to get \(y\).

The graph of \(y = x^2\) is a curved shape called a parabola. For 8th grade, it is enough to notice that it forms a smooth U-shape.

Worked Example 1: Graph \(y = x^2\)

Step 1: Choose some \(x\)-values.

A good choice is values around 0:

$$x = -3, -2, -1, 0, 1, 2, 3$$

Step 2: Find the \(y\)-values.

• If \(x = -3\), then \(y = (-3)^2 = 9\)
• If \(x = -2\), then \(y = (-2)^2 = 4\)
• If \(x = -1\), then \(y = (-1)^2 = 1\)
• If \(x = 0\), then \(y = 0^2 = 0\)
• If \(x = 1\), then \(y = 1^2 = 1\)
• If \(x = 2\), then \(y = 2^2 = 4\)
• If \(x = 3\), then \(y = 3^2 = 9\)

Step 3: Make a table.

$$ \begin{array}{c|c} x & y \\\hline -3 & 9 \\ -2 & 4 \\ -1 & 1 \\ 0 & 0 \\ 1 & 1 \\ 2 & 4 \\ 3 & 9 \end{array} $$

Step 4: Plot the points.

Plot:

• \((-3, 9)\)
• \((-2, 4)\)
• \((-1, 1)\)
• \((0, 0)\)
• \((1, 1)\)
• \((2, 4)\)
• \((3, 9)\)

Step 5: Connect the points with a smooth curve.

You should see a U-shaped graph. It is not a line, so it is nonlinear.

Important observation: The graph is the same on both sides of the \(y\)-axis. This happens because squaring a positive number and squaring its matching negative number give the same result.

5. Graphing an absolute value function

A simple absolute value function is:

$$y = |x|$$

The absolute value of a number is its distance from 0. Because distance cannot be negative:

• \(|3| = 3\)
• \(|-3| = 3\)
• \(|0| = 0\)

The graph of \(y = |x|\) makes a V-shape.

Worked Example 2: Graph \(y = |x|\)

Step 1: Choose \(x\)-values.

$$x = -3, -2, -1, 0, 1, 2, 3$$

Step 2: Find \(y\)-values.

• If \(x = -3\), then \(y = |-3| = 3\)
• If \(x = -2\), then \(y = |-2| = 2\)
• If \(x = -1\), then \(y = |-1| = 1\)
• If \(x = 0\), then \(y = |0| = 0\)
• If \(x = 1\), then \(y = |1| = 1\)
• If \(x = 2\), then \(y = |2| = 2\)
• If \(x = 3\), then \(y = |3| = 3\)

Step 3: Make a table.

$$ \begin{array}{c|c} x & y \\\hline -3 & 3 \\ -2 & 2 \\ -1 & 1 \\ 0 & 0 \\ 1 & 1 \\ 2 & 2 \\ 3 & 3 \end{array} $$

Step 4: Plot the points and connect them.

The graph forms a V-shape with its lowest point at \((0, 0)\).

Even though the graph has straight pieces, the whole graph is not one straight line, so it is still nonlinear.

6. Comparing linear and nonlinear graphs

Here is a quick comparison:

• Linear: graph is a straight line, and the change in \(y\) is constant.
• Nonlinear: graph is not a straight line, and the change in \(y\) is not constant.

For example:

• \(y = 2x + 1\) is linear.
• \(y = x^2\) is nonlinear.
• \(y = |x|\) is nonlinear.

7. Worked Example 3: Use a table to graph \(y = x^2 + 1\)

This function is similar to \(y = x^2\), but every output is 1 more.

Step 1: Choose \(x\)-values.

$$x = -2, -1, 0, 1, 2$$

Step 2: Calculate each output.

• If \(x = -2\), then \(y = (-2)^2 + 1 = 4 + 1 = 5\)
• If \(x = -1\), then \(y = (-1)^2 + 1 = 1 + 1 = 2\)
• If \(x = 0\), then \(y = 0^2 + 1 = 1\)
• If \(x = 1\), then \(y = 1^2 + 1 = 2\)
• If \(x = 2\), then \(y = 2^2 + 1 = 5\)

Step 3: Write the ordered pairs.

$$(-2,5),\;(-1,2),\;(0,1),\;(1,2),\;(2,5)$$

Step 4: Graph the points.

Connect them with a smooth U-shaped curve.

What do you notice? The graph looks like \(y = x^2\), but it is shifted up by 1 unit because every \(y\)-value is 1 greater.

8. Worked Example 4: Graph \(y = |x| + 2\)

This function is similar to \(y = |x|\), but every output is 2 more.

Step 1: Choose \(x\)-values.

$$x = -3, -2, -1, 0, 1, 2, 3$$

Step 2: Find the outputs.

• If \(x = -3\), then \(y = |-3| + 2 = 3 + 2 = 5\)
• If \(x = -2\), then \(y = |-2| + 2 = 2 + 2 = 4\)
• If \(x = -1\), then \(y = |-1| + 2 = 1 + 2 = 3\)
• If \(x = 0\), then \(y = |0| + 2 = 2\)
• If \(x = 1\), then \(y = |1| + 2 = 1 + 2 = 3\)
• If \(x = 2\), then \(y = |2| + 2 = 2 + 2 = 4\)
• If \(x = 3\), then \(y = |3| + 2 = 3 + 2 = 5\)

Step 3: Plot the points.

$$(-3,5),\;(-2,4),\;(-1,3),\;(0,2),\;(1,3),\;(2,4),\;(3,5)$$

Step 4: Connect the points.

You get a V-shape with the lowest point at \((0, 2)\).

What changed? The graph of \(y = |x|\) moved up 2 units.

9. Common mistakes to avoid

• Forgetting parentheses with negative numbers: \((-2)^2 = 4\), but \(-2^2 = -4\). When squaring a negative number, use parentheses.
• Using too few points: Nonlinear graphs need several points to show the correct shape.
• Connecting curved graphs with straight line segments only: For a quadratic function, draw a smooth curve.
• Mixing up absolute value and negatives: Absolute value is always 0 or greater.

10. Quick check for understanding

Ask yourself these questions when graphing:

• Did I choose enough \(x\)-values?
• Did I substitute each \(x\)-value correctly?
• Did I plot each ordered pair in the right place?
• Does the graph look like a curve or a V-shape instead of one straight line?

11. Summary

A nonlinear function is a function whose graph is not a straight line. To graph one, make a table of values, plot the ordered pairs, and connect the points in a way that matches the shape of the function.

Quadratic functions like \(y = x^2\) make a U-shaped curve. Absolute value functions like \(y = |x|\) make a V-shaped graph.

The more carefully you build your table and plot your points, the easier it is to graph nonlinear functions correctly.

Describing Qualitative Function Graphs

In math, a function is a rule that matches each input with exactly one output. A graph of a function shows how one quantity changes as another quantity changes.

Sometimes, a graph does not need exact numbers. Instead, we describe its shape. This is called a qualitative graph. A qualitative graph shows what is happening overall: whether the graph goes up, goes down, stays flat, changes quickly, or changes slowly.

This skill is useful in real life. For example, you may not know the exact temperature every hour, but you can still describe whether the temperature is rising, falling, or staying the same. That description can be shown with a qualitative graph.

In this lesson, you will learn how to read and sketch qualitative function graphs from words and situations.

1. What is a qualitative graph?

A qualitative graph focuses on the behavior of a function instead of exact coordinates. It answers questions like:

• Is the graph increasing or decreasing?
• Is it flat for part of the time?
• Does it change fast or slowly?
• Does it curve, or is it a straight line?

When sketching a qualitative graph, the exact numbers on the axes are often not important. What matters is the overall pattern.

2. Reading the axes

Before drawing or interpreting a graph, always ask:

• What does the horizontal axis represent?
• What does the vertical axis represent?

Many real-world graphs use time on the horizontal axis. The vertical axis shows the quantity that changes, such as distance, height, temperature, or money.

For example:

• Time and temperature
• Time and distance from home
• Time and amount of water in a tank

3. Important graph behaviors

Here are the main shapes and what they mean.

• Increasing: As you move from left to right, the graph goes up. The output is getting larger.
• Decreasing: As you move from left to right, the graph goes down. The output is getting smaller.
• Constant: The graph is flat. The output stays the same.
• Steep increase: The graph rises quickly. The output is increasing fast.
• Gentle increase: The graph rises slowly. The output is increasing, but not fast.
• Steep decrease: The graph falls quickly. The output is decreasing fast.
• Gentle decrease: The graph falls slowly. The output is decreasing, but not fast.

4. Straight graphs and curved graphs

A straight line means the quantity changes at a steady rate. If the graph rises in a straight line, the output increases by the same amount over equal time intervals.

A curved graph means the rate of change is not steady. The quantity may start changing slowly and then change faster, or the other way around.

This helps us tell the difference between linear and nonlinear behavior:

• Linear: straight-line pattern
• Nonlinear: curved or changing pattern

5. Continuous graphs

In this topic, we often use continuous graphs. A continuous graph can be drawn without lifting your pencil from the paper. It represents situations where change happens smoothly over time.

For example, a person's height above the ground while riding a Ferris wheel changes smoothly. Temperature during the day also changes in a continuous way.

6. Turning words into a graph

When a problem gives a real-world description, follow these steps:

1. Identify the input and output.
2. Decide how the output changes over time.
3. Break the story into parts.
4. Sketch each part of the graph.
5. Make sure the graph matches the story from left to right.

Look for clue words such as:

• increases or rises → graph goes up
• decreases or falls → graph goes down
• stays the same → flat graph
• quickly → steeper graph
• slowly → less steep graph

7. Worked Example 1: Temperature during a day

Situation: Early in the morning, the temperature is cool. By noon, it rises steadily. In the afternoon, it stays about the same for a while. In the evening, it drops.

Step 1: Choose axes.

• Horizontal axis: time of day
• Vertical axis: temperature

Step 2: Break the story into parts.

• Morning to noon: temperature rises
• Afternoon: temperature stays constant
• Evening: temperature falls

Step 3: Describe the graph.

The graph starts low, slopes upward, becomes flat for a while, and then slopes downward.

What this means: The graph is increasing, then constant, then decreasing.

Worked Example 2: Walking away from home

Situation: A student leaves home and walks to a park at a steady speed. They stay at the park for a while. Then they walk back home faster than before.

Step 1: Choose axes.

• Horizontal axis: time
• Vertical axis: distance from home

Step 2: Think about the distance from home.

• Walking to the park: distance from home increases steadily
• Staying at the park: distance stays the same
• Walking home faster: distance decreases quickly

Step 3: Describe the graph.

The graph starts at 0 because the student begins at home. It rises in a straight line, becomes flat, and then drops back to 0 with a steeper line.

Why is the last part steeper? A steeper line shows faster change. Since the student walks home faster, the distance decreases more quickly.

Worked Example 3: Filling a container

Situation: Water is poured into a container. At first, the water level rises slowly. Then it rises faster. Finally, the pouring stops.

Axes:

• Horizontal axis: time
• Vertical axis: water level

Interpret the story:

• At first, the level increases slowly
• Later, the level increases more quickly
• When pouring stops, the level stays the same

Describe the graph: The graph rises gently at first, then rises more steeply, and then becomes flat.

This graph could be made of two line segments and then a flat segment. The important idea is that the graph shows increasing faster over time, followed by constant.

Worked Example 4: Height of a ball thrown upward

Situation: A ball is thrown up into the air. It rises, slows down, reaches its highest point, and then falls back down.

Axes:

• Horizontal axis: time
• Vertical axis: height

Describe the motion:

• The height increases at first
• The increase slows as the ball reaches the top
• Then the height decreases as the ball falls

Describe the graph: The graph rises, curves over to a highest point, and then curves downward.

This is a nonlinear graph because it is curved, not a straight line.

8. How to describe a graph in words

If you are given a graph and need to explain it, move from left to right. Say what happens in each part.

You can use sentence starters like these:

• The graph starts by increasing.
• Then it remains constant.
• Next, it decreases slowly.
• After that, it increases more quickly.
• The graph reaches a highest point.
• The graph returns to its starting value.

9. Common mistakes to avoid

• Ignoring the axes: A graph of distance from home is different from a graph of speed. Always check what each axis means.
• Mixing up height and steepness: A higher graph value means more output, but a steeper graph means faster change.
• Reading the graph out of order: Always read from left to right because that shows how the situation changes over time.
• Forgetting flat sections: If something stays the same, the graph should be horizontal there.
• Using exact numbers when none are given: In a qualitative graph, the overall shape matters more than exact values.

10. Quick check: match the description to the graph shape

Here are some short descriptions and the graph behavior they suggest:

• A plant grows steadily each week. → increasing straight line
• A car stops at a red light. → flat section on a distance graph
• The amount of money in a piggy bank decreases as coins are spent. → decreasing graph
• A runner starts slowly and then speeds up. → graph increases, becoming steeper

11. A helpful way to think about qualitative graphs

You are telling the story of change using a picture. The graph does not need exact scale marks to be useful. It just needs to show the right pattern.

Ask yourself:

• What is changing?
• Is it going up, down, or staying the same?
• Is it changing quickly or slowly?
• Is the change steady or not steady?

Summary

A qualitative function graph shows the overall shape of how one quantity changes with another, usually over time. Instead of focusing on exact numbers, it shows whether the output increases, decreases, stays the same, or changes quickly or slowly.

To sketch or describe one, first identify the axes, then break the situation into parts, and match each part to a graph shape. Straight lines show steady change, while curves show change that is not steady. Reading from left to right helps you explain the story the graph tells.