Functions and Linear Systems

Definition of a Function

In math, we often look at relationships between two things. For example, the number of hours you practice basketball affects how much total practice time you have. The number of notebooks you buy affects the total cost. These relationships can be shown with tables, mapping diagrams, graphs, and equations.

One very important kind of relationship is called a function.

A function is a rule that matches each input with exactly one output.

You can also think of it this way:

• The input is the starting value.
• The output is the result.
• For it to be a function, one input cannot point to two different outputs.

Inputs are often called independent variables, and outputs are often called dependent variables.

The output is called dependent because it depends on the input.

For example, if the cost of one movie ticket is \(\$8\), then the total cost depends on how many tickets you buy.

If \(t\) is the number of tickets, then the total cost is

Each number of tickets gives exactly one total cost, so this relationship is a function.

Main Idea to Remember: A relation is a function if every input has exactly one output.

It is okay for different inputs to have the same output.

For example:

• Input \(1\) gives output \(5\)
• Input \(2\) gives output \(5\)

This is still a function because each input has only one output.

But if input \(3\) gives output \(4\) and output \(7\), then it is not a function.

That is because one input has more than one output.

Ways to Tell if a Relation Is a Function

You can check whether something is a function in several ways:

• Look at a set of ordered pairs
• Look at a table
• Look at a mapping diagram
• Look at a graph

1. Ordered Pairs

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

Example of a function:

Each input appears once and has one output.

Example that is not a function:

The input \(1\) has two different outputs: \(2\) and \(5\).

2. Tables

In a table, check whether any input value repeats with a different output.

If an input repeats but keeps the same output, it is still a function. But if the repeated input has different outputs, it is not a function.

3. Mapping Diagrams

A mapping diagram shows arrows from inputs to outputs.

It is a function if every input has exactly one arrow going out from it.

If one input has two arrows going to different outputs, it is not a function.

4. Graphs and the Vertical Line Test

When a relation is shown on a graph, we can use the vertical line test.

The vertical line test says:

• If any vertical line touches the graph in more than one point, the graph is not a function.
• If every vertical line touches the graph in only one point, the graph is a function.

Why does this work? A vertical line checks one input value, or one \(x\)-value. If that one \(x\)-value has more than one \(y\)-value, then one input has more than one output.

That breaks the definition of a function.

Worked Example 1: Ordered Pairs

Decide whether this relation is a function:

Step 1: Look at the inputs: \(2, 4, 6, 8\).

Step 2: Check whether any input repeats with a different output.

No input repeats.

Answer: Yes, this is a function.

Each input has exactly one output.

Worked Example 2: Table

Decide whether this table shows a function.

Step 1: Look at the inputs: \(3, 5, 3, 7\).

Step 2: Notice that the input \(3\) appears twice.

Step 3: Check the outputs for input \(3\). They are \(10\) and \(14\).

Because one input has two different outputs, this relation is not a function.

Answer: No, it is not a function.

Worked Example 3: Mapping Diagram

Suppose a mapping diagram shows:

• \(1 \to 4\)
• \(2 \to 4\)
• \(3 \to 7\)

Is this a function?

Step 1: Check each input.

• Input \(1\) goes to one output: \(4\)
• Input \(2\) goes to one output: \(4\)
• Input \(3\) goes to one output: \(7\)

Step 2: Notice that inputs \(1\) and \(2\) both go to \(4\).

That is okay. Different inputs can share the same output.

Answer: Yes, it is a function.

Worked Example 4: Vertical Line Test

Imagine a graph of a straight, slanted line like this:

Would this graph be a function?

Step 1: Picture any vertical line crossing the graph.

Step 2: A vertical line will hit the slanted line only once.

So each \(x\)-value has only one \(y\)-value.

Answer: Yes, the graph is a function.

Now imagine a circle graphed on the coordinate plane.

Some vertical lines would hit the circle twice: once on the top half and once on the bottom half.

That means one input would have two outputs.

So a circle is not a function.

Common Mistakes to Avoid

• Mistake 1: Thinking repeated outputs mean it is not a function. Repeated outputs are okay.
• Mistake 2: Forgetting to check whether an input repeats with different outputs.
• Mistake 3: On a graph, using a horizontal line test instead of a vertical line test. For this topic, use the vertical line test.

Quick Check

Decide if each relation is a function.

1. $$\{(0,1), (1,3), (2,5)\}$$
2. $$\{(4,2), (4,6), (7,8)\}$$
3. Inputs \(2, 3, 4\) each have one arrow to outputs \(9, 9, 10\)

Answers:

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

Summary

A function is a relationship where every input has exactly one output.

To decide if something is a function, check for repeated inputs with different outputs. In mapping diagrams, each input should have one arrow out. On graphs, use the vertical line test.

If you remember one rule, remember this: one input cannot have two different outputs.

Domain and Range help us describe the relationship between inputs and outputs in math. When one value goes into a rule, table, or graph, another value comes out. The domain is the set of all possible input values, and the range is the set of all possible output values.

You can think of it like a machine. A number goes in, the machine does something to it, and a new number comes out. The numbers that are allowed to go in are the domain. The numbers that come out are the range.

In many problems, the input is called the independent variable, and the output is called the dependent variable. Usually, in ordered pairs \,\((x, y)\), the \(x\)-values are the domain and the \(y\)-values are the range.

Main Idea:

• Domain = all possible inputs
• Range = all possible outputs

If you are given ordered pairs, a table, or a graph, your job is often to list all the input values and all the output values.

How to find domain and range from ordered pairs

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

To find the domain and range:

1. Look at all the first numbers. These make the domain.
2. Look at all the second numbers. These make the range.
3. Do not repeat numbers if the same value appears more than once.

Worked Example 1: Ordered Pairs

Find the domain and range of the relation:

\((2, 5), (4, 7), (6, 9), (8, 11)\)

Step 1: List the first numbers: \(2, 4, 6, 8\)

So the domain is:

$$\{2, 4, 6, 8\}$$

Step 2: List the second numbers: \(5, 7, 9, 11\)

So the range is:

$$\{5, 7, 9, 11\}$$

Answer:

• Domain: \(\{2, 4, 6, 8\}\)
• Range: \(\{5, 7, 9, 11\}\)

Worked Example 2: Repeated Values

Find the domain and range of:

\((1, 3), (2, 3), (3, 5), (4, 5), (5, 7)\)

Step 1: The first numbers are \(1, 2, 3, 4, 5\)

Domain:

$$\{1, 2, 3, 4, 5\}$$

Step 2: The second numbers are \(3, 3, 5, 5, 7\)

Do not repeat values in the range. So the range is:

$$\{3, 5, 7\}$$

Answer:

• Domain: \(\{1, 2, 3, 4, 5\}\)
• Range: \(\{3, 5, 7\}\)

How to find domain and range from a table

In a table, one column usually shows the input values and the other column shows the output values. The input column gives the domain. The output column gives the range.

Worked Example 3: Table

Suppose a table shows:

\[ \begin{array}{c|c} x & y \\ \hline -2 & 4 \\ 0 & 1 \\ 3 & 7 \\ 5 & 1 \end{array} \]

Step 1: Read the \(x\)-values: \(-2, 0, 3, 5\)

Domain:

$$\{-2, 0, 3, 5\}$$

Step 2: Read the \(y\)-values: \(4, 1, 7, 1\)

Do not repeat \(1\). So the range is:

$$\{4, 1, 7\}$$

Answer:

• Domain: \(\{-2, 0, 3, 5\}\)
• Range: \(\{4, 1, 7\}\)

How to find domain and range from a graph

On a graph, the domain is all the \(x\)-values used by the points on the graph. The range is all the \(y\)-values used by the points on the graph.

A good way to remember this is:

• Domain: read the graph from left to right
• Range: read the graph from bottom to top

If the graph shows only a few points, list the \(x\)-values and \(y\)-values from those points. If values repeat, only list them once.

Worked Example 4: Graph Points

A graph has points at:

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

Step 1: Find the domain by listing the \(x\)-values:

$$\{-3, 0, 2, 5\}$$

Step 2: Find the range by listing the \(y\)-values:

\(2, 4, 2, -1\)

Do not repeat \(2\), so the range is:

$$\{2, 4, -1\}$$

Answer:

• Domain: \(\{-3, 0, 2, 5\}\)
• Range: \(\{2, 4, -1\}\)

How to find domain and range from a rule

Sometimes you are given a rule, such as:

$$y = x + 2$$

This rule tells how the output depends on the input. To find domain and range, first ask: Which input values are allowed?

In 7th Grade problems, there are often two common situations:

• You are given a rule and a list of allowed input values.
• You are expected to use values from a table or a graph made from the rule.

For example, if the rule is \(y = x + 2\) and the allowed inputs are \(\{1, 2, 3, 4\}\), then:

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

So:

• Domain: \(\{1, 2, 3, 4\}\)
• Range: \(\{3, 4, 5, 6\}\)

Important Things to Remember

• The domain comes from the inputs.
• The range comes from the outputs.
• In ordered pairs, domain is the set of first numbers and range is the set of second numbers.
• Do not repeat numbers when writing a set.
• On a graph, domain is read across and range is read up and down.

Common Mistakes

• Mixing up \(x\) and \(y\): Remember, \(x\) is usually the input and \(y\) is usually the output.
• Repeating values: If a number appears more than once, list it only once in the domain or range.
• Reading a graph the wrong way: Domain is left to right. Range is bottom to top.

Quick Check

Find the domain and range of:

\((0, 6), (2, 8), (4, 10), (2, 12)\)

Solution:

The first numbers are \(0, 2, 4, 2\), so the domain is:

$$\{0, 2, 4\}$$

The second numbers are \(6, 8, 10, 12\), so the range is:

$$\{6, 8, 10, 12\}$$

Brief Summary

Domain and range tell us about the inputs and outputs in a relationship. The domain is the set of all input values, and the range is the set of all output values. You can find them from ordered pairs, tables, graphs, and rules by carefully separating the \(x\)-values from the \(y\)-values.

Slope as Rate of Change tells us how fast one quantity changes compared to another. In 7th grade, you can think of slope as the rise over run of a line on a graph.

If a line goes up as you move to the right, the slope is positive. If a line goes down as you move to the right, the slope is negative. Slope helps us describe patterns in tables, graphs, and real-life situations.

Another way to say this is that slope is a rate of change. A rate of change compares how much the dependent variable changes when the independent variable changes.

For example, if the number of miles traveled depends on the number of hours driven, then:

• Independent variable: hours
• Dependent variable: miles

The slope tells us how many miles change for each 1 hour change. That is why slope and rate of change mean the same idea in linear relationships.

On a coordinate plane, we usually use points written as \((x, y)\).

• \(x\) is the horizontal value, or input
• \(y\) is the vertical value, or output

To find slope between two points, use this rule:

In symbols, for points \((x_1, y_1)\) and \((x_2, y_2)\):

Here, \(m\) stands for slope.

Important idea: subtract in the same order for the top and bottom. If you do \(y_2 - y_1\), then you must also do \(x_2 - x_1\).

Let’s connect this to rise and run:

• Rise = how far up or down you move
• Run = how far left or right you move

If you go up 3 and right 2, the slope is \(\frac{3}{2}\). If you go down 4 and right 1, the slope is \(-4\).

Here is what different slopes mean:

• Positive slope: line rises from left to right
• Negative slope: line falls from left to right
• Zero slope: flat horizontal line

In this lesson, we will focus on finding and understanding slope as a rate of change in linear relationships.

Worked Example 1: Find slope from a graph idea

Suppose a line passes through the points \((1, 2)\) and \((4, 5)\).

Step 1: Find the change in \(y\).

$$5 - 2 = 3$$

Step 2: Find the change in \(x\).

$$4 - 1 = 3$$

Step 3: Divide rise by run.

$$m = \frac{3}{3} = 1$$

Answer: The slope is \(1\).

This means the line goes up 1 unit for every 1 unit it moves to the right. As a rate of change, \(y\) increases by 1 when \(x\) increases by 1.

Worked Example 2: Find slope from two points

Find the slope of the line through \((2, 7)\) and \((6, 15)\).

Use the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Substitute the values:

$$m = \frac{15 - 7}{6 - 2} = \frac{8}{4} = 2$$

Answer: The slope is \(2\).

This means for every 1 unit increase in \(x\), the value of \(y\) increases by 2. The rate of change is 2.

Worked Example 3: Negative slope

Find the slope of the line through \((3, 10)\) and \((7, 2)\).

Step 1: Find the change in \(y\).

$$2 - 10 = -8$$

Step 2: Find the change in \(x\).

$$7 - 3 = 4$$

Step 3: Divide.

$$m = \frac{-8}{4} = -2$$

Answer: The slope is \(-2\).

This means when \(x\) increases by 1, \(y\) decreases by 2. The line goes downward from left to right.

Worked Example 4: Slope as a real-world rate of change

A bike rider travels 6 miles in 1 hour, 12 miles in 2 hours, and 18 miles in 3 hours.

We can write the points as:

• \((1, 6)\)
• \((2, 12)\)
• \((3, 18)\)

Find the slope using the first two points:

$$m = \frac{12 - 6}{2 - 1} = \frac{6}{1} = 6$$

Answer: The slope is \(6\).

This means the rider travels 6 miles per hour. The rate of change is 6 miles for each 1 hour.

This example shows why slope is useful. It helps us describe real situations, like speed, cost, earnings, or distance.

How to find slope step by step

1. Pick two points on the line.
2. Find the change in the \(y\)-values.
3. Find the change in the \(x\)-values.
4. Write the fraction \(\frac{\text{change in } y}{\text{change in } x}\).
5. Simplify if needed.

Example of careful subtraction

Use points \((1, 4)\) and \((5, 12)\).

You could write:

$$m = \frac{12 - 4}{5 - 1} = \frac{8}{4} = 2$$

Or you could switch the order for both:

$$m = \frac{4 - 12}{1 - 5} = \frac{-8}{-4} = 2$$

Both ways give the same answer because the order stayed consistent.

What if the slope is 0?

Look at the points \((2, 5)\) and \((7, 5)\).

The \(y\)-values stay the same, so the rise is 0.

$$m = \frac{5 - 5}{7 - 2} = \frac{0}{5} = 0$$

A slope of 0 means the line is horizontal. There is no change in \(y\) as \(x\) changes.

How slope connects to tables

You do not always need a graph. You can also find slope from a table if the relationship is linear.

Example table:

• \(x = 1, y = 3\)
• \(x = 2, y = 5\)
• \(x = 3, y = 7\)
• \(x = 4, y = 9\)

Each time \(x\) increases by 1, \(y\) increases by 2.

So the slope is:

$$\frac{2}{1} = 2$$

This tells us the rate of change is constant, so the relationship is linear.

Common mistakes to avoid

• Mixing up rise and run. Remember: slope is \(\frac{\text{change in } y}{\text{change in } x}\).
• Subtracting in different orders. Keep the order the same in the top and bottom.
• Forgetting that a negative slope means the line goes down from left to right.
• Using points that are not on the same line.

Quick check questions

• What does a slope of \(3\) mean? It means \(y\) goes up 3 when \(x\) goes up 1.
• What does a slope of \(-1\) mean? It means \(y\) goes down 1 when \(x\) goes up 1.
• What does a slope of \(\frac{1}{2}\) mean? It means \(y\) goes up 1 when \(x\) goes up 2.

Summary

Slope is the rate of change in a linear relationship. It shows how much the output changes compared to how much the input changes.

You can find slope using:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Remember that slope is also called rise over run. Positive slopes go up, negative slopes go down, and a slope of 0 is flat.

When you understand slope as rate of change, you can better read graphs, tables, and real-world situations.

Lesson: Understanding Slope-Intercept Form

In 7th grade math, you will often work with linear equations. A linear equation makes a straight line when you graph it.

One of the most useful ways to write a linear equation is called slope-intercept form:

$$y = mx + b$$

This form helps you quickly understand two important things about a line:

• 4slope 014 how steep the line is
• 4y-intercept 014 where the line starts on the y-axis

Once you know the slope and y-intercept, you can graph the line and describe how the two variables are related.

What do the letters mean?

• 4y: the output or dependent variable
• 4x: the input or independent variable
• 4m: the slope, or rate of change
• 4b: the y-intercept, or starting value

So in the equation $$y = mx + b,$$ the line changes by the slope 4m, and it begins at 4b on the y-axis.

1. What is slope?

Slope tells how much a line goes up or down compared to how much it moves to the right.

You can think of slope as:

$$\text{slope} = \frac{\text{rise}}{\text{run}}$$

Rise means how far the line goes up or down.

Run means how far the line goes left or right.

For example, if a line goes up 2 and right 1, the slope is:

$$m = \frac{2}{1} = 2$$

If a line goes down 3 and right 2, the slope is:

$$m = \frac{-3}{2}$$

Here is how to understand slope values:

• If 4m > 0, the line rises from left to right.
• If 4m < 0, the line falls from left to right.
• If 4m = 0, the line is flat.

2. What is the y-intercept?

The y-intercept is the point where the line crosses the y-axis.

In slope-intercept form, the y-intercept is the number 4b.

Because the y-axis is where 4x = 0, the y-intercept is always the point:

$$ (0, b) $$

For example:

• In $$y = 3x + 4,$$ the y-intercept is 4, so the point is 4(0,4)4.
• In $$y = -2x - 1,$$ the y-intercept is -1, so the point is 4(0,-1)4.

3. How to read an equation in slope-intercept form

When an equation is already written as $$y = mx + b,$$ you can find the slope and y-intercept right away.

Example:

$$y = 5x + 2$$

• Slope: 4m = 5
• Y-intercept: 4b = 2

This means the line starts at 4(0,2)4 and goes up 5 and right 1.

Another example:

$$y = -\frac{1}{2}x + 3$$

• Slope: 4m = -\frac{1}{2}
• Y-intercept: 4b = 3

This line starts at 4(0,3)4 and goes down 1 and right 2.

4. How to graph using slope-intercept form

To graph a line from slope-intercept form, follow these steps:

1. Find the y-intercept 4b4 and plot the point 4(0,b)4.
2. Use the slope 4m4 to move from that point.
3. Plot another point.
4. Draw a straight line through the points.

If the slope is a whole number like 434, you can write it as:

$$3 = \frac{3}{1}$$

So you would go up 3 and right 1.

If the slope is negative, the line goes downward as you move to the right.

Worked Example 1: Identify slope and y-intercept

Find the slope and y-intercept of:

$$y = 2x + 5$$

Step 1: Compare the equation to $$y = mx + b.$$

Here, 4m = 24 and 4b = 54.

Answer:

• Slope: 2
• Y-intercept: 5
• Y-intercept point: 4(0,5)4

Worked Example 2: Graph a line from an equation

Graph:

$$y = -x + 1$$

Step 1: Find the slope and y-intercept.

• 4m = -14
• 4b = 14

Step 2: Plot the y-intercept.

Plot the point 4(0,1)4.

Step 3: Use the slope.

A slope of 4-14 means:

$$-1 = \frac{-1}{1}$$

Go down 1 and right 1.

Starting from 4(0,1)4, that gives another point: 4(1,0)4.

You can do it again to get 4(2,-1)4.

Step 4: Draw the line through the points.

Worked Example 3: Write an equation from slope and y-intercept

Write the equation of a line with slope 434 and y-intercept 4-24.

Step 1: Start with slope-intercept form.

$$y = mx + b$$

Step 2: Replace 4m4 with 434 and 4b4 with 4-24.

$$y = 3x - 2$$

Answer: $$y = 3x - 2$$

Worked Example 4: Write an equation from a graph description

A line crosses the y-axis at 4(0,4)4 and has slope 4\frac{1}{2}4. Write the equation.

Step 1: Identify 4b4.

The line crosses the y-axis at 4(0,4)4, so 4b = 44.

Step 2: Identify 4m4.

The slope is 4\frac{1}{2}4, so 4m = \frac{1}{2}4.

Step 3: Substitute into $$y = mx + b.$$

$$y = \frac{1}{2}x + 4$$

Answer: $$y = \frac{1}{2}x + 4$$

5. How slope-intercept form shows a real-world relationship

Slope-intercept form is useful because it can describe real-life situations.

For example, imagine you earn \$2 each time you wash a car, and you already have \$5 saved.

If 4x4 is the number of cars washed, and 4y4 is the total amount of money, the equation is:

$$y = 2x + 5$$

• The slope 424 means you earn \$2 for each car.
• The y-intercept 454 means you started with \$5.

This is why slope is called the rate of change, and the y-intercept is called the starting value.

6. Common mistakes to avoid

• Mixing up slope and y-intercept: In $$y = mx + b,$$ the number with 4x4 is the slope. The number by itself is the y-intercept.
• Forgetting negative signs: In $$y = -2x + 3,$$ the slope is 4-24, not 424.
• Not using the y-intercept first when graphing: Always start by plotting 4(0,b)4.
• Reading slope backward: Slope is rise over run, not run over rise.

7. Quick check

Try answering these on your own:

1. In $$y = 4x - 6,$$ what are 4m4 and 4b4?
2. What is the y-intercept point of $$y = -3x + 2$$?
3. Write an equation with slope 4-24 and y-intercept 474.
4. If a line has slope 4\frac{2}{3}4 and y-intercept 4-14, what is its equation?

Answers:

1. 4m = 44, 4b = -64
2. 4(0,2)4
3. $$y = -2x + 7$$
4. $$y = \frac{2}{3}x - 1$$

Summary

Slope-intercept form is written as $$y = mx + b.$$ The slope, 4m4, tells how steep the line is and how it changes. The y-intercept, 4b4, tells where the line crosses the y-axis.

When you see an equation in this form, you can quickly identify the slope and starting point, graph the line, and describe what the equation means in a real-world situation.

Linear vs. Non-Linear Functions

In math, a function shows a relationship between two variables. Usually, one variable is the independent variable, which is the input, and the other is the dependent variable, which is the output.

For example, if the number of hours you practice piano affects how many songs you can learn, then the hours practiced is the input and the number of songs learned is the output.

In this lesson, you will learn how to tell the difference between linear and non-linear functions by looking at tables, graphs, and patterns.

1. What is a linear function?

A linear function has a constant rate of change. This means that when the input increases by the same amount each time, the output changes by the same amount each time too.

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

A linear function can often be written in the form:

$$y = mx + b$$

Here:

• m is the rate of change, also called the slope.
• b is the starting value, also called the y-intercept.

Example: In the equation \(y = 3x + 2\), the output goes up by 3 every time the input goes up by 1. That constant change of 3 means the function is linear.

2. What is a non-linear function?

A non-linear function does not have a constant rate of change. When the input increases by the same amount, the output does not keep changing by the same amount.

On a graph, a non-linear function does not make a straight line. It may curve or change direction.

Some non-linear patterns include:

• Quadratic patterns, where the graph curves
• Exponential-like patterns, where values grow faster and faster

For example, in the function \(y = x^2\), the outputs do not increase by the same amount each time. So it is non-linear.

3. How to tell from a table

One of the easiest ways to check if a function is linear is to look at how the outputs change.

If the input values increase by equal amounts, then:

• If the output changes by the same amount each time, the function is linear.
• If the output changes by different amounts, the function is non-linear.

Worked Example 1: A linear table

Look at this table:

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

Step 1: Check how the input changes.

The input increases by 1 each time.

Step 2: Check how the output changes.

\(7 - 4 = 3\)
\(10 - 7 = 3\)
\(13 - 10 = 3\)

The output increases by 3 each time. The rate of change is constant, so this function is linear.

Worked Example 2: A non-linear table

Look at this table:

\[ \begin{array}{c|c} x & y \\ \hline 1 & 1 \\ 2 & 4 \\ 3 & 9 \\ 4 & 16 \end{array} \]

Step 1: The input still increases by 1 each time.

Step 2: Check the output changes.

\(4 - 1 = 3\)
\(9 - 4 = 5\)
\(16 - 9 = 7\)

The output does not increase by the same amount. Since the rate of change is not constant, this function is non-linear.

4. How to tell from a graph

You can also tell whether a function is linear by looking at its graph.

• A linear function graphs as a straight line.
• A non-linear function graphs as a curve or a graph that is not a straight line.

If you plotted points from a table and they all lie on one straight line, the function is linear. If the points bend into a curve, it is non-linear.

5. How to tell from an equation

You can also look at the equation itself.

A function is usually linear if the variable has an exponent of 1 and the equation looks like:

$$y = mx + b$$

Examples of linear equations:

• \(y = 2x + 5\)
• \(y = -x + 1\)
• \(y = 7\)

Notice that \(y = 7\) is still linear. Its graph is a horizontal straight line, and the rate of change is 0.

A function is usually non-linear if:

• the variable is squared, like \(x^2\)
• the variable is multiplied by itself
• the pattern does not have a constant rate of change

Examples of non-linear equations:

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

6. Comparing linear and non-linear functions

• Linear: constant rate of change, straight-line graph, often written as \(y = mx + b\)
• Non-linear: changing rate of change, graph is not a straight line

Here is a quick way to remember it:

• Linear = line
• Non-linear = not a line

Worked Example 3: Decide from an equation

Is \(y = 5x - 2\) linear or non-linear?

Step 1: Look at the equation form.

It matches \(y = mx + b\), where \(m = 5\) and \(b = -2\).

Step 2: Check the variable.

The variable \(x\) has an exponent of 1.

So this function is linear.

Worked Example 4: Decide from a pattern

A ball drops and the distance it falls after each second is shown below:

\[ \begin{array}{c|c} t & d \\ \hline 1 & 5 \\ 2 & 20 \\ 3 & 45 \\ 4 & 80 \end{array} \]

Step 1: The time increases by 1 each time.

Step 2: Find how the distance changes.

\(20 - 5 = 15\)
\(45 - 20 = 25\)
\(80 - 45 = 35\)

The changes are 15, 25, and 35. These are not the same.

So the rate of change is not constant, and the function is non-linear.

7. Common mistakes to avoid

• Mistake 1: Looking only at whether the numbers increase. Both linear and non-linear functions can increase. What matters is whether they increase by the same amount.
• Mistake 2: Thinking every equation with an \(x\) is linear. Some equations, like \(y = x^2\), are non-linear.
• Mistake 3: Forgetting to check that the input changes equally in a table before comparing output changes.

8. Quick check questions

1. Does a linear function have a constant or changing rate of change?
2. If a graph is a curve, is it linear or non-linear?
3. Is \(y = 4x + 1\) linear or non-linear?
4. If outputs change by 2, then 2, then 2, is the function linear or non-linear?
5. If outputs change by 3, then 5, then 7, is the function linear or non-linear?

Answers:

1. Constant rate of change
2. Non-linear
3. Linear
4. Linear
5. Non-linear

Summary

A linear function has a constant rate of change and makes a straight-line graph. A non-linear function does not have a constant rate of change and does not graph as a straight line.

To decide whether a function is linear or non-linear, you can check a table, a graph, or an equation. If the output changes by the same amount for equal input changes, it is linear. If not, it is non-linear.

Translating Multiple Representations means showing the same relationship in different ways.

In 7th Grade math, you will often see a pattern or function written as:

• a description in words,
• an input-output table,
• a graph, or
• an equation.

Your job is to recognize that all four can tell the same story.

When you can move from one form to another, you understand the relationship more deeply. This helps with graphing, slope, and solving real-world problems.

Important idea: A function matches each input with exactly one output.

Usually, the independent variable is the input, often written as \(x\). The dependent variable is the output, often written as \(y\), because its value depends on the input.

For example, if the rule is “multiply the input by 3 and add 1,” then the output depends on the input. If \(x=2\), then \(y=7\).

We can write that rule as an equation:

$$y=3x+1$$

Now let’s learn how each representation works.

1. Words

A word description tells the relationship as a rule or a situation.

Examples:

• “The output is 4 more than twice the input.”
• “A taxi ride costs \(\$5\) plus \(\$2\) for each mile.”

When reading words, look for clues:

• more than often means add
• less than often means subtract
• times means multiply
• per often shows a rate
• starts with or initial fee often means a starting amount

2. Input-output table

A table shows pairs of values. The first column is the input, and the second column is the output.

Example:

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

To understand a table, ask:

• How does \(y\) change when \(x\) goes up by 1?
• What is the output when the input is 0?

In this table, the output increases by 3 each time, and when \(x=0\), \(y=1\). That matches the equation \(y=3x+1\).

3. Graph

A graph shows the ordered pairs \((x,y)\) on a coordinate plane.

Each point on the graph must satisfy the same rule.

If the relationship is linear, the points lie on a straight line.

For example, the equation \(y=3x+1\) includes points like:

• \((0,1)\)
• \((1,4)\)
• \((2,7)\)

These points form a line.

4. Equation

An equation is a math rule that connects input and output.

Many 7th Grade linear relationships are written in the form:

$$y=mx+b$$

Here:

• \(m\) is the amount \(y\) changes when \(x\) increases by 1. This is the slope or rate of change.
• \(b\) is the value of \(y\) when \(x=0\). This is the starting value.

So if a rule says “start at 2 and add 5 each time,” the equation is:

$$y=5x+2$$

How the four representations connect

All four forms should match the same pattern.

Here is one example of the same relationship in all four forms:

• Words: The output is 2 more than 4 times the input.
• Equation: \(y=4x+2\)
• Table:

$$\begin{array}{c|c}x&y\\ \hline 0&2\\1&6\\2&10\\3&14\end{array}$$

• Graph: Plot \((0,2)\), \((1,6)\), \((2,10)\), and \((3,14)\), then draw the line through them.

If one form does not match the others, something is wrong.

Strategy for translating between representations

1. Find the input and output.
2. Look for the rate of change: how much the output changes when the input changes by 1.
3. Find the starting value: the output when the input is 0.
4. Write or check the equation using \(y=mx+b\).
5. Use the equation to make a table or graph points.
6. Make sure the words, table, graph, and equation all agree.

Now let’s work through some examples.

Worked Example 1: Words to equation and table

A rule says: “The output is 3 more than twice the input.”

Step 1: Write the equation.

“Twice the input” means \(2x\).

“3 more than” means add 3.

So the equation is:

$$y=2x+3$$

Step 2: Make a table.

Substitute some input values.

If \(x=0\), then:

$$y=2(0)+3=3$$

If \(x=1\), then:

$$y=2(1)+3=5$$

If \(x=2\), then:

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

If \(x=3\), then:

$$y=2(3)+3=9$$

The table is:

$$\begin{array}{c|c}x&y\\ \hline 0&3\\1&5\\2&7\\3&9\end{array}$$

Step 3: Describe the graph.

Plot the points \((0,3)\), \((1,5)\), \((2,7)\), and \((3,9)\).

They make a straight line.

The line goes up 2 units for every 1 unit to the right, and it crosses the \(y\)-axis at 3.

Worked Example 2: Table to words and equation

Look at this table:

$$\begin{array}{c|c}x&y\\ \hline 0&-1\\1&2\\2&5\\3&8\end{array}$$

Step 1: Find the rate of change.

As \(x\) increases by 1, the outputs go:

$$-1, 2, 5, 8$$

They increase by 3 each time.

So the slope is 3.

Step 2: Find the starting value.

When \(x=0\), \(y=-1\).

So the starting value is \(-1\).

Step 3: Write the equation.

$$y=3x-1$$

Step 4: Write the rule in words.

The output is 1 less than 3 times the input.

Or you could say: Start at \(-1\) and add 3 for each increase of 1 in the input.

Worked Example 3: Equation to graph and words

Given the equation:

$$y=-2x+4$$

Step 1: Identify the slope and starting value.

The slope is \(-2\), so the line goes down 2 units for every 1 unit to the right.

The starting value is 4, so when \(x=0\), \(y=4\).

Step 2: Make a table.

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

Step 3: Graph the points.

Plot \((0,4)\), \((1,2)\), \((2,0)\), and \((3,-2)\).

Draw a straight line through them.

Step 4: Write the relationship in words.

The output is 4 minus 2 times the input.

Or: Start at 4 and subtract 2 each time the input increases by 1.

Worked Example 4: Real-world situation to all representations

A movie theater charges \(\$6\) for a ticket plus \(\$2\) for each snack item bought.

Let \(x\) be the number of snack items.

Let \(y\) be the total cost in dollars.

Step 1: Write the equation.

The cost starts at 6 dollars.

Each snack adds 2 dollars.

So:

$$y=2x+6$$

Step 2: Make a table.

$$\begin{array}{c|c}x&y\\ \hline 0&6\\1&8\\2&10\\3&12\end{array}$$

Step 3: Write ordered pairs for the graph.

\((0,6)\), \((1,8)\), \((2,10)\), \((3,12)\)

These points form a line.

Step 4: Describe in words.

The total cost is 6 dollars plus 2 dollars for each snack item.

How to check if representations match

Use these questions:

• Does the table follow the rule in the equation?
• Do the graph points appear in the table?
• Does the word description tell the same starting value and rate of change?
• When \(x=0\), do all forms give the same \(y\)-value?

If the answer is yes, the representations match.

Common mistakes to avoid

• Mixing up input and output. Remember: input is usually \(x\), output is usually \(y\).
• Using the wrong starting value. The starting value is the output when \(x=0\).
• Reading “more than” in the wrong order. “3 more than twice the input” means \(2x+3\), not \(3x+2\).
• Forgetting negative change. If outputs decrease as inputs increase, the slope is negative.
• Graphing points incorrectly. Always plot \((x,y)\) in that order.

Quick practice ideas

If you want to practice on your own, try these steps:

1. Take an equation like \(y=4x-2\).
2. Make a table with at least 4 values.
3. Write the ordered pairs.
4. Describe the rule in words.
5. Sketch the graph.

Then pick a table and work backward to the equation.

Summary

Translating multiple representations means moving between words, tables, graphs, and equations while keeping the same relationship.

For linear functions, the most important ideas are the rate of change and the starting value.

If you can identify how much the output changes and what happens when \(x=0\), you can connect all four forms more easily.

With practice, you will be able to recognize that different-looking representations can describe the exact same function.

Solving Systems of Equations by Graphing

Sometimes in math, you are given two equations at the same time. When we look at both equations together, we call that a system of equations.

A solution to a system is a point that makes both equations true. When we graph the two lines on the same coordinate plane, the solution is the point where the lines intersect, or cross.

In this lesson, you will learn how to graph both equations, find their intersection point, and understand what that point means.

1. What is a system of equations?

A system of equations is a set of two or more equations with the same variables. In 7th grade, you will often work with two linear equations using the variables \(x\) and \(y\).

For example:

These are two different lines. If we graph them on the same coordinate plane, they may cross at one point. That point is the solution to the system.

2. What does the solution mean?

If a point is the solution to a system, then its \(x\)-value and \(y\)-value work in both equations.

For example, if the lines intersect at \((2, 3)\), then:

• \((2, 3)\) is on the first line, and
• \((2, 3)\) is also on the second line.

That means \(x = 2\) and \(y = 3\) make both equations true at the same time.

3. How to solve a system by graphing

To solve a system by graphing, follow these steps:

1. Graph the first equation.
2. Graph the second equation on the same coordinate plane.
3. Look for the point where the two lines cross.
4. Write that intersection point as the solution.

4. Writing equations in graph-friendly form

The easiest equations to graph are often written in the form:

In this form:

• \(m\) is the slope, which tells how steep the line is.
• \(b\) is the y-intercept, which tells where the line crosses the \(y\)-axis.

For example, in \(y = 2x + 1\):

• The slope is \(2\).
• The y-intercept is \(1\), so the line starts at \((0,1)\).

If needed, you can also graph by making a table of values and plotting points.

5. Worked Example 1: A simple intersection

Solve the system by graphing:

Step 1: Graph the first line

For \(y = x + 1\), the y-intercept is \(1\), so one point is \((0,1)\).

The slope is \(1\), which means rise 1 and run 1. Some points on the line are:

• \((0,1)\)
• \((1,2)\)
• \((2,3)\)

Step 2: Graph the second line

For \(y = -x + 5\), the y-intercept is \(5\), so one point is \((0,5)\).

The slope is \(-1\), which means down 1 and right 1. Some points are:

• \((0,5)\)
• \((1,4)\)
• \((2,3)\)

Step 3: Find the intersection

The lines cross at \((2,3)\).

Answer: The solution is \((2,3)\).

Step 4: Check the solution

Substitute \(x=2\) into both equations:

Both equations give \(y=3\), so \((2,3)\) is correct.

6. Worked Example 2: Using tables to graph

Solve the system by graphing:

Step 1: Make a table for \(y = 2x\)

• If \(x=0\), then \(y=0\), so point \((0,0)\)
• If \(x=1\), then \(y=2\), so point \((1,2)\)
• If \(x=2\), then \(y=4\), so point \((2,4)\)

Step 2: Make a table for \(y = x + 2\)

• If \(x=0\), then \(y=2\), so point \((0,2)\)
• If \(x=1\), then \(y=3\), so point \((1,3)\)
• If \(x=2\), then \(y=4\), so point \((2,4)\)

Step 3: Find the intersection

Both lines pass through \((2,4)\).

Answer: The solution is \((2,4)\).

7. Worked Example 3: When the lines do not cross

Solve the system by graphing:

Both lines have the same slope, which is \(1\).

But they have different y-intercepts:

• The first line crosses the y-axis at \(1\).
• The second line crosses the y-axis at \(-3\).

Because the lines have the same slope, they go in the same direction. Because they have different y-intercepts, they never meet.

Answer: There is no solution.

This happens when the lines are parallel.

8. Worked Example 4: When the equations are really the same line

Solve the system by graphing:

First, rewrite the second equation by dividing every term by \(2\):

Now both equations are exactly the same:

That means both equations graph as the same line.

Every point on the line works for both equations, so there is not just one solution.

Answer: There are infinitely many solutions.

9. The 3 possible results when graphing a system

• One solution: The lines cross once.
• No solution: The lines are parallel and never cross.
• Infinitely many solutions: The lines are the same line.

10. Tips for graphing carefully

• Always label the axes.
• Plot points neatly and accurately.
• Use at least two points for each line.
• Check whether the intersection lands exactly on a grid point.
• If the equations are in \(y=mx+b\) form, start with the y-intercept and use the slope.

11. Common mistakes to avoid

• Mixing up the slope and y-intercept: In \(y = mx + b\), \(m\) is slope and \(b\) is the y-intercept.
• Graphing only one line: A system needs both lines on the same graph.
• Choosing the wrong intersection point: The solution must be where both lines cross.
• Forgetting to check: Plug the point back into both equations if you are unsure.

12. Why graphing is helpful

Graphing helps you see the solution. It shows whether the system has one solution, no solution, or infinitely many solutions.

It also helps you understand that the solution is a point that belongs to both lines at the same time.

Summary

A system of equations is a pair of equations with the same variables. To solve a system by graphing, graph both lines on the same coordinate plane and find where they intersect. The intersection point is the solution because it makes both equations true.

If the lines cross once, there is one solution. If they never cross, there is no solution. If they lie on top of each other, there are infinitely many solutions.

Solving Systems via Substitution

Sometimes in math, you will see two equations with the same two variables. This is called a system of equations.

The goal is to find the one pair of values that makes both equations true at the same time.

One very useful way to solve a system is called substitution.

In this lesson, you will learn what substitution means, when to use it, and how to solve systems step by step.

What is a system of equations?

A system of equations is a set of two equations that use the same variables. For example:

We want to find the values of \(x\) and \(y\) that make both equations true.

What does substitution mean?

Substitution means replacing one expression with another equal expression.

For example, if:

then anywhere we see \(y\), we can replace it with \(x + 2\).

That is the main idea of this method.

When is substitution a good method?

Substitution is especially helpful when:

• one variable is already by itself, like \(y = 3x + 4\)
• or one variable can be isolated easily

Steps for solving by substitution

1. Isolate one variable in one equation if it is not already alone.
2. Substitute that expression into the other equation.
3. Solve the new equation.
4. Substitute back to find the other variable.
5. Check your answer in both original equations.

Let’s go through some examples.

Worked Example 1: A simple substitution problem

Solve the system:

Step 1: Look for a variable that is already isolated.

Both equations have \(y\) by itself, so this is a great system for substitution.

Step 2: Substitute.

Since both expressions equal \(y\), we can set them equal to each other:

Step 3: Solve for \(x\).

So, \(x = 4\).

Step 4: Substitute back to find \(y\).

Use \(y = x + 3\):

Solution: $$\boxed{(4, 7)}$$

Step 5: Check.

Check in the second equation:

The answer is correct.

Worked Example 2: One equation needs substitution into the other

Solve the system:

Step 1: Find the isolated variable.

In the first equation, \(y\) is already isolated:

Step 2: Substitute \(3x + 2\) for \(y\) in the second equation.

Step 3: Solve for \(x\).

Step 4: Substitute back to find \(y\).

Solution: $$\boxed{(3, 11)}$$

Step 5: Check.

The solution works.

Worked Example 3: First isolate a variable

Solve the system:

Step 1: Find the isolated variable.

The second equation already has \(y\) by itself:

Step 2: Substitute into the first equation.

Step 3: Solve for \(x\).

Step 4: Substitute back to find \(y\).

Solution: $$\boxed{(3, 7)}$$

Check:

Worked Example 4: Isolate a variable before substituting

Solve the system:

In this system, no variable is already isolated, so we start by isolating one.

Step 1: Isolate \(y\) in one equation.

Use the second equation:

Now we have:

Step 2: Substitute into the other equation.

Step 3: Solve for \(x\).

Step 4: Substitute back to find \(y\).

Solution: $$\boxed{(5, 2)}$$

Check:

It works.

Important idea: The solution is an ordered pair

When you solve a system, your answer is usually written as an ordered pair:

For example, if \(x = 3\) and \(y = 11\), the solution is:

This pair must make both equations true.

Why substitution works

If two expressions are equal to the same variable, then they are equal to each other.

For example, if:

then \(y\) and \(x + 3\) have the same value. So in another equation, you can replace \(y\) with \(x + 3\).

This turns a two-variable problem into a one-variable problem, which is easier to solve.

Common mistakes to avoid

• Forgetting parentheses when substituting. For example, write \(2x + (3x + 2)\), not \(2x + 3x + 2\) without thinking carefully.
• Solving for only one variable and stopping too early. You need both \(x\) and \(y\).
• Not checking your answer in both original equations.
• Making sign mistakes with positive and negative numbers.

Helpful tips

• Choose the equation where a variable is already alone, if possible.
• If no variable is alone, pick the equation where isolating a variable looks easiest.
• Work one small step at a time.
• Always substitute your answer back in to check.

Let’s review the method

1. Get one variable by itself.
2. Replace that variable in the other equation.
3. Solve the new equation.
4. Plug the value back in.
5. Write the solution as an ordered pair.
6. Check both equations.

Brief Summary

Substitution is a method for solving systems of equations by replacing one variable with an equal expression.

First, isolate a variable. Next, substitute that expression into the other equation. Then solve, substitute back, and check your answer.

If your ordered pair makes both equations true, then you have solved the system correctly.