Linear Functions and Trend Analytics

Slope as Rate of Change

In 9th Grade maths, slope is one of the most important ideas in linear functions. Slope tells us how fast one quantity changes compared to another. It connects tables, graphs, words, and equations.

You may already know slope as the “steepness” of a line. That is true, but slope also has a meaning in real situations. It tells us the rate of change: how much the output changes when the input changes.

For example, if a plant grows 2 centimeters every week, the slope is 2. If a car travels 60 miles every hour, the slope is 60. In both cases, slope compares a vertical change to a horizontal change.

1. What slope means

On a graph, slope measures how much a line goes up or down compared to how much it goes right. This is often called rise over run.

The formula for slope between two points \\((x_1, y_1)\\) and \\((x_2, y_2)\\) is:

Here:

• \\(y_2-y_1\\) is the vertical change
• \\(x_2-x_1\\) is the horizontal change
• \\(m\\) stands for slope

This means slope is the ratio:

When we talk about slope as a rate of change, we are saying:

• “For every 1 unit increase in \\((x)\\), how much does \\((y)\\) change?”

2. Interpreting positive, negative, zero, and undefined slope

The sign of the slope tells us how the quantities are related.

• Positive slope: as \\((x)\\) increases, \\((y)\\) increases.
• Negative slope: as \\((x)\\) increases, \\((y)\\) decreases.
• Zero slope: \\((y)\\) stays the same as \\((x)\\) changes.
• Undefined slope: the horizontal change is 0, so the slope cannot be found.

Examples:

• A savings account balance increasing each month has a positive slope.
• The amount of water in a tank draining over time has a negative slope.
• A flat road has zero slope.
• A vertical line has undefined slope.

3. Slope from a graph

To find slope from a graph, choose two clear points on the line. Then count:

1. How far the line rises or falls
2. How far it moves to the right
3. Write the slope as rise over run

Suppose a line goes through \\((1,2)\\) and \\((4,8)\\). The rise is \\((8-2=6)\\) and the run is \\((4-1=3)\\).

This means the line goes up 2 units for every 1 unit it moves right. As a rate of change, \\((y)\\) increases by 2 for every 1 increase in \\((x)\\).

4. Slope from a table

If a relationship is linear, the slope can also be found from a table. Look at how \\((y)\\) changes compared to how \\((x)\\) changes.

Example table:

• \\((x=1, y=5)\\)
• \\((x=2, y=8)\\)
• \\((x=3, y=11)\\)
• \\((x=4, y=14)\\)

Each time \\((x)\\) increases by 1, \\((y)\\) increases by 3. So the slope is:

This means the rate of change is 3 units of \\((y)\\) for every 1 unit of \\((x)\\).

If the changes are not by 1, the idea is the same. For example, if \\((x)\\) increases by 2 while \\((y)\\) increases by 10, then:

5. Slope from an equation

Many linear equations are written in slope-intercept form:

In this form:

• \\((m)\\) is the slope
• \\((b)\\) is the y-intercept

For example, in the equation

the slope is 4. That means for every increase of 1 in \\((x)\\), \\((y)\\) increases by 4.

In the equation

the slope is \\((-2)\\). That means for every increase of 1 in \\((x)\\), \\((y)\\) decreases by 2.

6. Slope in real-world situations

Slope is not just a graph idea. It describes real change between two quantities.

• Distance and time: slope can represent speed.
• Cost and number of items: slope can represent cost per item.
• Temperature and time: slope can represent heating or cooling rate.
• Height and age: slope can represent growth rate.

When you interpret slope, always include units. A slope of 5 by itself is incomplete. A slope of 5 dollars per ticket or 5 miles per hour has a clear meaning.

7. Worked Examples

Example 1: Find slope from two points

Find the slope of the line through \\((2,3)\\) and \\((6,11)\\).

Step 1: Use the slope formula.

Step 2: Substitute the values.

Step 3: Simplify.

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

Interpretation: \\((y)\\) increases by 2 for every increase of 1 in \\((x)\\).

Example 2: Find slope from a table

A table shows the number of hours studied and test score increase.

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

Step 1: Look at the change in \\((x)\\). It increases by 1.

Step 2: Look at the change in \\((y)\\). It increases by 4.

Answer: The slope is \\((4)\\).

Interpretation: The test score increases by 4 points for each extra hour studied.

Example 3: Interpret slope in a real-world situation

A taxi company charges based on miles traveled. The total cost is given by:

Here, \\((y)\\) is the total cost in dollars and \\((x)\\) is the number of miles.

Step 1: Identify the slope. In \\((y = mx + b)\\), the slope is \\((m)\\).

Answer: The slope is \\((3)\\).

Interpretation: The cost increases by 3 dollars for every 1 mile traveled. The rate of change is 3 dollars per mile.

Example 4: Negative slope

A candle burns down at a steady rate. At hour 1, its height is 18 cm. At hour 5, its height is 10 cm. Find the slope.

The points are \\((1,18)\\) and \\((5,10)\\).

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

Interpretation: The candle’s height decreases by 2 cm each hour. The negative sign shows that the height is going down over time.

8. Common mistakes to avoid

• Mixing up the order: If you subtract the x-values in one order, subtract the y-values in the same order.
• Forgetting the sign: A line that goes down from left to right has a negative slope.
• Using the y-intercept as slope: In \\((y=mx+b)\\), \\((m)\\) is slope, not \\((b)\\).
• Ignoring units: In real situations, slope should be described with units, like miles per hour or dollars per item.
• Dividing the wrong way: Slope is \\((\text{change in } y)/(\text{change in } x)\\), not the other way around.

9. Quick check for understanding

Ask yourself these questions:

• Can I find slope from two points?
• Can I find slope from a table by comparing changes?
• Can I identify slope in an equation like \\((y=mx+b)\\)?
• Can I explain what the slope means in words and units?

If you can do all four, then you understand slope as a rate of change.

Summary

Slope is the ratio of vertical change to horizontal change, or change in \\((y)\\) divided by change in \\((x)\\). It tells how fast one quantity changes compared to another.

For a linear function, the slope stays constant. You can find it from a graph, a table, two points, or an equation. Most importantly, you should be able to interpret slope in context, such as dollars per mile, points per hour, or centimeters per week.

Slope-Intercept Form is one of the most useful ways to write the equation of a line. It helps you quickly understand two important things about a line: how steep it is and where it crosses the vertical axis.

The slope-intercept form of a linear equation is:

$$y = mx + b$$

In this form:

• (m) is the slope, which tells the rate of change of the line.
• (b) is the y-intercept, which tells where the line crosses the y-axis.
• (x) and (y) are the coordinates of points on the line.

This lesson will show you how to recognize slope-intercept form, how to identify the slope and y-intercept, and how to use them to graph and understand a line.

1. What does slope mean?

The slope tells how much the line goes up or down as you move from left to right. Slope is often described as:

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

Rise means the change in the vertical direction, and run means the change in the horizontal direction.

For example:

• If the slope is positive, the line goes up from left to right.
• If the slope is negative, the line goes down from left to right.
• If the slope is 0, the line is flat, or horizontal.

If the slope is a whole number like 3, you can think of it as \(\frac{3}{1}\). That means go up 3 and right 1.

If the slope is negative, like \(-2\), you can think of it as \(\frac{-2}{1}\). That means go down 2 and right 1.

2. What does the y-intercept mean?

The y-intercept is the point where the line crosses the y-axis. On the y-axis, the x-value is always 0.

So if a line has y-intercept \(b\), then the intercept point is:

$$ (0,b) $$

For example:

• If \(b = 4\), the line crosses at \((0,4)\).
• If \(b = -2\), the line crosses at \((0,-2)\).

3. Reading slope and y-intercept from an equation

When a line is written in slope-intercept form,

$$y = mx + b$$

you can read the slope and y-intercept directly.

Example: In the equation

$$y = 2x + 5$$

• The slope is \(m = 2\).
• The y-intercept is \(b = 5\).

This means the line crosses the y-axis at \((0,5)\), and from that point, it rises 2 units for every 1 unit it moves to the right.

Another example:

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

• The slope is \(m = -\frac{1}{2}\).
• The y-intercept is \(b = 3\).

This means the line crosses at \((0,3)\), then goes down 1 and right 2.

4. How to graph a line using slope-intercept form

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

1. Identify the slope \(m\) and y-intercept \(b\).
2. Plot the y-intercept \((0,b)\).
3. Use the slope to find another point.
4. Draw a straight line through the points.

Worked Example 1: Graphing a simple line

Graph the equation:

$$y = x + 2$$

Step 1: Identify slope and y-intercept.

• \(m = 1\)
• \(b = 2\)

Step 2: Plot the y-intercept.

The y-intercept is \((0,2)\).

Step 3: Use the slope.

A slope of 1 means \(\frac{1}{1}\), so go up 1 and right 1.

Starting at \((0,2)\):

• Go up 1, right 1 to get \((1,3)\).
• Again, go up 1, right 1 to get \((2,4)\).

Step 4: Draw the line.

Draw a straight line through \((0,2)\), \((1,3)\), and \((2,4)\).

Worked Example 2: Graphing with a negative slope

Graph the equation:

$$y = -2x + 1$$

Step 1: Identify slope and y-intercept.

• \(m = -2\)
• \(b = 1\)

Step 2: Plot the y-intercept.

The line crosses the y-axis at \((0,1)\).

Step 3: Use the slope.

\(-2\) can be written as \(\frac{-2}{1}\). So from \((0,1)\), go down 2 and right 1.

• From \((0,1)\), go to \((1,-1)\).
• Again, go down 2 and right 1 to get \((2,-3)\).

Step 4: Draw the line.

Connect the points with a straight line.

5. When the equation is not already in slope-intercept form

Sometimes a linear equation is not written as \(y = mx + b\). If that happens, solve for \(y\) so you can rewrite it in slope-intercept form.

Worked Example 3: Rewriting an equation

Rewrite the equation in slope-intercept form and identify the slope and y-intercept:

$$2y + 4x = 6$$

Step 1: Solve for \(y\).

Subtract \(4x\) from both sides:

$$2y = -4x + 6$$

Divide both sides by 2:

$$y = -2x + 3$$

Step 2: Identify the slope and y-intercept.

• Slope: \(m = -2\)
• y-intercept: \(b = 3\)

So the line goes through \((0,3)\) and falls 2 units for every 1 unit to the right.

6. Connecting slope-intercept form to real situations

Slope-intercept form is also useful in real life because it shows an initial value and a rate of change.

In \(y = mx + b\):

• \(b\) is the starting amount, or initial value.
• \(m\) is how much the value changes each time \(x\) increases by 1.

Worked Example 4: Interpreting a real-world equation

A taxi ride costs a starting fee of $4 plus $2 for each mile. Write and interpret the equation.

Let:

• \(x\) = number of miles
• \(y\) = total cost

The equation is:

$$y = 2x + 4$$

Now interpret the parts:

• The slope is \(2\), so the cost increases by $2 for each mile.
• The y-intercept is \(4\), so the ride starts at $4 before any miles are traveled.

If you travel 3 miles, then:

$$y = 2(3) + 4 = 6 + 4 = 10$$

The total cost is $10.

7. Common mistakes to avoid

• Mixing up slope and y-intercept. In \(y = mx + b\), the number with \(x\) is the slope, and the constant number is the y-intercept.
• Forgetting the sign. In \(y = -3x + 2\), the slope is \(-3\), not 3.
• Missing the coefficient of \(x\). In \(y = x - 4\), the slope is 1 because \(x = 1x\).
• Thinking the y-intercept is \((b,0)\). The y-intercept is always \((0,b)\).

8. Quick check

Try identifying the slope and y-intercept in each equation:

• \(y = 4x - 1\)
• \(y = -\frac{3}{2}x + 6\)
• \(y = 7\)

Answers:

• For \(y = 4x - 1\): slope \(= 4\), y-intercept \(= -1\)
• For \(y = -\frac{3}{2}x + 6\): slope \(= -\frac{3}{2}\), y-intercept \(= 6\)
• For \(y = 7\): slope \(= 0\), y-intercept \(= 7\)

Summary

Slope-intercept form is written as $$y = mx + b$$. The slope, \(m\), tells how fast the line rises or falls, and the y-intercept, \(b\), tells where the line crosses the y-axis.

When you understand this form, you can quickly read information from an equation, graph a line, and explain what the line means in a real-world situation. This makes slope-intercept form a powerful tool for working with linear functions.

Point-Slope Form is a way to write the equation of a line when you know one point on the line and the slope.

This form is very useful because it lets you build a linear equation quickly. Instead of finding many values first, you can start right away with the information you already have.

The point-slope form of a line is:

In this equation:

• m is the slope of the line.
• (x_1, y_1) is a point on the line.
• x and y represent any other point on the same line.

So if you know a slope and one point, you can substitute those values into the formula and get the equation of the line.

Why does this form make sense?

Remember that slope tells how much the change in y compares to the change in x. The slope formula is:

If we multiply both sides by \((x - x_1)\), we get:

That is exactly the point-slope form.

How to use point-slope form

1. Find the slope \(m\).
2. Choose a point \((x_1, y_1)\) on the line.
3. Substitute the values into \(y - y_1 = m(x - x_1)\).
4. Simplify if needed.

Be careful with signs. If the point has a negative number, put it in parentheses when substituting.

For example, if \((x_1, y_1) = (4, -2)\), then:

This simplifies to:

Worked Example 1: Write an equation from a slope and a point

Write the equation of the line with slope \(3\) that passes through \((2, 5)\).

Step 1: Identify the slope and point.

• \(m = 3\)
• \((x_1, y_1) = (2, 5)\)

Step 2: Substitute into point-slope form.

This is a correct equation of the line in point-slope form.

If you want, you can also simplify it into slope-intercept form.

So the line can be written as:

• Point-slope form: \(y - 5 = 3(x - 2)\)
• Slope-intercept form: \(y = 3x - 1\)

Worked Example 2: Use a negative slope

Write the equation of the line with slope \(-2\) through the point \((1, 4)\).

Step 1: Identify the values.

• \(m = -2\)
• \((x_1, y_1) = (1, 4)\)

Step 2: Substitute into the formula.

This is the equation in point-slope form.

Now simplify.

So the equation is:

• Point-slope form: \(y - 4 = -2(x - 1)\)
• Simplified form: \(y = -2x + 6\)

Worked Example 3: Use a point with a negative coordinate

Write the equation of the line with slope \(\frac{1}{2}\) passing through \((-3, 2)\).

Step 1: Identify the values.

• \(m = \frac{1}{2}\)
• \((x_1, y_1) = (-3, 2)\)

Step 2: Substitute carefully.

Subtracting a negative becomes adding, so:

This is the equation in point-slope form.

Now simplify.

Worked Example 4: Find the slope from two points, then use point-slope form

Write the equation of the line through \((2, 1)\) and \((6, 9)\).

First, find the slope.

Now use one of the points. We will use \((2, 1)\).

This is the equation in point-slope form.

Now simplify.

You could also use the other point, \((6, 9)\).

This looks different at first, but it represents the same line. If you simplify it, you still get:

Important ideas to remember

• Point-slope form needs one point and the slope.
• The formula is $$y - y_1 = m(x - x_1)$$
• Always substitute the point carefully.
• Watch for negative signs, especially with negative coordinates.
• Different points on the same line can give different-looking point-slope equations, but they can still describe the same line.

Common mistakes

• Mixing up the coordinates: In \((x_1, y_1)\), the first number is always \(x_1\) and the second is always \(y_1\).
• Forgetting parentheses: If a coordinate is negative, use parentheses when substituting.
• Wrong sign when subtracting: For example, \(x - (-4)\) becomes \(x + 4\).
• Using the wrong slope: If you are given two points, calculate slope first before writing the equation.

Quick check

What is the equation of a line with slope \(4\) through \((3, -1)\)?

Substitute into the formula:

Simplify the left side:

That is the equation in point-slope form.

Summary

Point-slope form is a fast way to write a line when you know the slope and one point. The formula is $$y - y_1 = m(x - x_1)$$. Substitute the slope and point carefully, especially when negative numbers are involved. You can leave the equation in point-slope form or simplify it into another form if needed.

Standard Form Analysis is about working with linear equations written in the form

$$Ax + By = C$$

where A, B, and C are numbers, and x and y are variables. This is called standard form.

In 9th Grade Maths, standard form is important because it helps you quickly find the x-intercept and y-intercept of a line. These intercepts are useful for graphing and for understanding what the equation tells us.

This lesson will show you how to recognize standard form, how to find intercepts, and how to analyze a line from its equation.

1. What is standard form?

A linear equation is in standard form when it looks like

$$Ax + By = C$$

Examples of equations in standard form are:

• \(2x + 3y = 12\)
• \(5x - y = 10\)
• \(-4x + 2y = 8\)

The numbers \(A\), \(B\), and \(C\) can be positive or negative, but \(A\) and \(B\) should not both be zero.

2. What are intercepts?

The x-intercept is the point where the line crosses the x-axis. At this point, the y-value is 0.

So, to find the x-intercept, substitute \(y = 0\) into the equation.

The y-intercept is the point where the line crosses the y-axis. At this point, the x-value is 0.

So, to find the y-intercept, substitute \(x = 0\) into the equation.

3. Finding intercepts from standard form

If you have the equation

$$Ax + By = C$$

then:

• To find the x-intercept, let \(y = 0\). Then solve \(Ax = C\), so \(x = \frac{C}{A}\).
• To find the y-intercept, let \(x = 0\). Then solve \(By = C\), so \(y = \frac{C}{B}\).

This gives the intercept points:

$$\left(\frac{C}{A}, 0\right) \quad \text{and} \quad \left(0, \frac{C}{B}\right)$$

This method works as long as \(A \neq 0\) for the x-intercept and \(B \neq 0\) for the y-intercept.

4. Why standard form is useful

Standard form is especially helpful when you want to graph a line quickly using intercepts.

1. Find the x-intercept.
2. Find the y-intercept.
3. Plot both points.
4. Draw a straight line through them.

This is sometimes faster than making a full table of values.

Worked Example 1: Finding intercepts from a simple equation

Find the intercepts of

$$2x + 3y = 12$$

Step 1: Find the x-intercept

Set \(y = 0\):

$$2x + 3(0) = 12$$

$$2x = 12$$

$$x = 6$$

So the x-intercept is

$$ (6, 0) $$

Step 2: Find the y-intercept

Set \(x = 0\):

$$2(0) + 3y = 12$$

$$3y = 12$$

$$y = 4$$

So the y-intercept is

$$ (0, 4) $$

Answer: The intercepts are \((6, 0)\) and \((0, 4)\).

Worked Example 2: Including a negative coefficient

Find the intercepts of

$$5x - y = 10$$

Step 1: Find the x-intercept

Set \(y = 0\):

$$5x - 0 = 10$$

$$5x = 10$$

$$x = 2$$

So the x-intercept is

$$ (2, 0) $$

Step 2: Find the y-intercept

Set \(x = 0\):

$$5(0) - y = 10$$

$$-y = 10$$

$$y = -10$$

So the y-intercept is

$$ (0, -10) $$

Answer: The intercepts are \((2, 0)\) and \((0, -10)\).

5. Checking your work

A good habit is to substitute your intercept points back into the original equation.

For example, in \(2x + 3y = 12\), check \((6, 0)\):

$$2(6) + 3(0) = 12$$

$$12 + 0 = 12$$

This is true, so the point is correct.

Now check \((0, 4)\):

$$2(0) + 3(4) = 12$$

$$0 + 12 = 12$$

This is also true.

6. When an equation is not already in standard form

Sometimes a linear equation is written in another form, such as slope-intercept form:

$$y = mx + b$$

You can rearrange it into standard form if needed.

Worked Example 3: Rearranging into standard form and finding intercepts

Find the intercepts of

$$y = 2x + 6$$

Step 1: Rewrite in standard form

Start with

$$y = 2x + 6$$

Move \(2x\) to the left side:

$$-2x + y = 6$$

This is standard form.

Step 2: Find the x-intercept

Set \(y = 0\):

$$-2x + 0 = 6$$

$$-2x = 6$$

$$x = -3$$

So the x-intercept is

$$(-3, 0)$$

Step 3: Find the y-intercept

Set \(x = 0\):

$$-2(0) + y = 6$$

$$y = 6$$

So the y-intercept is

$$ (0, 6) $$

Answer: The intercepts are \((-3, 0)\) and \((0, 6)\).

7. Understanding what the intercepts mean

Intercepts show where the line crosses each axis.

• The x-intercept tells you where \(y = 0\).
• The y-intercept tells you where \(x = 0\).

These points help you understand the position of the line on the coordinate plane.

If both intercepts are easy to find, they give you two points, and two points are enough to draw a straight line.

Worked Example 4: Analyzing a line from standard form

Analyze the equation

$$3x + 2y = 6$$

Step 1: Find the x-intercept

Set \(y = 0\):

$$3x + 2(0) = 6$$

$$3x = 6$$

$$x = 2$$

So the x-intercept is \((2, 0)\).

Step 2: Find the y-intercept

Set \(x = 0\):

$$3(0) + 2y = 6$$

$$2y = 6$$

$$y = 3$$

So the y-intercept is \((0, 3)\).

Step 3: Describe the graph

The line crosses the x-axis at 2 and the y-axis at 3. If you plot \((2, 0)\) and \((0, 3)\), then draw a straight line through them, you have the graph.

8. Common mistakes to avoid

• Mixing up the intercept rules: For the x-intercept, set \(y = 0\). For the y-intercept, set \(x = 0\).
• Forgetting signs: Be careful with negative numbers when solving.
• Writing the point incorrectly: The x-intercept should look like \((x, 0)\), and the y-intercept should look like \((0, y)\).
• Not solving fully: After substituting 0, make sure you solve for the remaining variable.

9. Quick strategy for standard form analysis

Whenever you see an equation in standard form, use this plan:

1. Check that the equation is in the form \(Ax + By = C\).
2. Set \(y = 0\) to find the x-intercept.
3. Set \(x = 0\) to find the y-intercept.
4. Write each answer as a coordinate point.
5. Use the two points to graph or describe the line.

Summary

Standard form is a way to write linear equations as \(Ax + By = C\). To analyze a line in standard form, the most important skill is finding the intercepts.

Set \(y = 0\) to get the x-intercept, and set \(x = 0\) to get the y-intercept. These two points help you graph the line and understand where it crosses the axes.

Parallel and Perpendicular Dynamics is about understanding how the slopes of lines are connected.

When two lines are parallel, they move in the same direction and never meet. When two lines are perpendicular, they cross at a right angle of 490^491.

In linear functions, slope tells us how steep a line is. So to decide whether lines are parallel or perpendicular, we compare their slopes.

This lesson will show you how to recognize these relationships in equations, in graphs, and from points.

1. Review: What is slope?

The slope of a line measures how much the line rises or falls as you move to the right.

If a line passes through points (x_1, y_1) and (x_2, y_2), its slope is

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

We often write linear equations in slope-intercept form:

$$y = mx + b$$

Here:

• m is the slope
• b is the y-intercept

Example: In y = 3x + 2, the slope is 3.

Example: In y = -\frac{1}{2}x + 4, the slope is -\frac{1}{2}.

2. Parallel lines

Two lines are parallel if they have the same slope.

Why? Because they rise and fall at exactly the same rate, so they stay the same distance apart and never intersect.

If two lines are written as

$$y = m_1x + b_1$$

and

$$y = m_2x + b_2$$

then they are parallel when

$$m_1 = m_2$$

Usually, parallel lines have different y-intercepts. If they had the same slope and the same y-intercept, they would actually be the same line.

Parallel line rule:

• Same slope
• Different intercepts if they are distinct lines

Example:

• y = 2x + 1
• y = 2x - 5

Both slopes are 2, so the lines are parallel.

3. Perpendicular lines

Two lines are perpendicular if they meet at a right angle.

For non-vertical and non-horizontal lines, the slopes of perpendicular lines are negative reciprocals.

This means:

1. Flip the fraction
2. Change the sign

If one slope is m, the perpendicular slope is

$$-\frac{1}{m}$$

So if two lines have slopes m_1 and m_2, then they are perpendicular when

$$m_1m_2 = -1$$

Examples of negative reciprocals:

• 2 and -\frac{1}{2}
• \frac{3}{4} and -\frac{4}{3}
• -5 and \frac{1}{5}

Important: A sign change alone is not enough.

For example, the negative of 3 is -3, but the negative reciprocal of 3 is -\frac{1}{3}.

4. Special case: horizontal and vertical lines

A horizontal line has slope 0. Its equation looks like y = 4.

A vertical line has an undefined slope. Its equation looks like x = -2.

Horizontal and vertical lines are perpendicular to each other.

So:

• All horizontal lines are parallel to each other
• All vertical lines are parallel to each other
• Any horizontal line is perpendicular to any vertical line

5. How to tell if lines are parallel or perpendicular

There are several common situations in problems.

Case A: Equations are already in slope-intercept form

Look at the slopes directly.

• Equal slopes  parallel
• Negative reciprocal slopes  perpendicular

Case B: Equations are not in slope-intercept form

Rewrite each equation so that y is by itself.

Then compare slopes.

Case C: You are given points

Find the slope of each line using

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

Then compare the slopes.

6. Worked Example 1: Identifying parallel lines from equations

Are the lines y = 4x + 7 and y = 4x - 3 parallel, perpendicular, or neither?

Step 1: Find the slopes.

• First line slope: 4
• Second line slope: 4

Step 2: Compare the slopes.

The slopes are equal.

Answer: The lines are parallel.

7. Worked Example 2: Identifying perpendicular lines from equations

Are the lines y = \frac{2}{3}x + 1 and y = -\frac{3}{2}x + 5 parallel, perpendicular, or neither?

Step 1: Find the slopes.

• First line slope: \frac{2}{3}
• Second line slope: -\frac{3}{2}

Step 2: Check for negative reciprocals.

The reciprocal of \frac{2}{3} is \frac{3}{2}. Changing the sign gives -\frac{3}{2}.

That matches the second slope.

Answer: The lines are perpendicular.

8. Worked Example 3: Rewriting equations first

Determine the relationship between

$$2x + y = 6$$

and

$$4x - 2y = 8$$

Step 1: Rewrite the first equation in slope-intercept form.

$$2x + y = 6$$

Subtract 2x from both sides:

$$y = -2x + 6$$

So the slope is -2.

Step 2: Rewrite the second equation in slope-intercept form.

$$4x - 2y = 8$$

Subtract 4x from both sides:

$$-2y = -4x + 8$$

Divide by -2:

$$y = 2x - 4$$

So the slope is 2.

Step 3: Compare slopes.

• They are not equal, so the lines are not parallel.
• The negative reciprocal of -2 is \frac{1}{2}, not 2.

Answer: The lines are neither parallel nor perpendicular.

9. Worked Example 4: Writing an equation of a parallel or perpendicular line

Write an equation of the line that passes through (3, -1) and is perpendicular to

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

Step 1: Find the given slope.

The given line has slope \frac{1}{4}.

Step 2: Find the perpendicular slope.

The negative reciprocal of \frac{1}{4} is -4.

So the new line has slope m = -4.

Step 3: Use point-slope idea with the point (3,-1).

Substitute into y = mx + b:

$$-1 = -4(3) + b$$

$$-1 = -12 + b$$

$$b = 11$$

Step 4: Write the equation.

$$y = -4x + 11$$

Answer: The equation is

$$y = -4x + 11$$

10. Finding slopes from points

Sometimes you are not given equations. Instead, you may get two pairs of points.

Suppose one line goes through (1,2) and (5,10).

Its slope is

$$m = \frac{10-2}{5-1} = \frac{8}{4} = 2$$

Suppose another line goes through (0,3) and (2,2).

Its slope is

$$m = \frac{2-3}{2-0} = \frac{-1}{2}$$

Now compare:

• The slopes are not equal, so the lines are not parallel.
• -\frac{1}{2} is the negative reciprocal of 2.

So the lines are perpendicular.

11. Common mistakes to avoid

• Forgetting to rewrite equations first. If the equation is not in the form y = mx + b, the slope may not be easy to see.
• Confusing opposite numbers with negative reciprocals. The perpendicular slope of 3 is not -3. It is -\frac{1}{3}.
• Ignoring special cases. Horizontal and vertical lines do not follow the usual negative reciprocal pattern in the same simple way. Just remember that they are perpendicular to each other.
• Using the same intercept for parallel lines without noticing. If two equations have the same slope and same intercept, they are the same line, not two different parallel lines.

12. Quick strategy checklist

1. Find each slope.
2. If needed, rewrite equations into y = mx + b.
3. Compare the slopes:

• Same slope  parallel
• Negative reciprocals  perpendicular
• Otherwise  neither

13. Summary

In linear functions, slope controls the relationship between lines.

Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals.

If one slope is m, then a perpendicular slope is -\frac{1}{m}. Also remember the special case: horizontal and vertical lines are perpendicular.

Once you can find and compare slopes, you can decide whether lines are parallel, perpendicular, or neither, and you can also write equations for new lines with those relationships.

Secant Lines and Average Rate of Change help us describe how a quantity changes between two points.

Even when a graph is not a straight line, we can still measure how fast it changes over an interval. This is called the average rate of change.

A secant line is a straight line that passes through two points on a graph. The slope of that secant line tells us the average rate of change between those two points.

This idea is useful when working with curves, tables, real-world data, and non-linear functions. It gives us a way to estimate how quickly something is increasing or decreasing over a stretch of time or distance.

1. Review: Rate of Change and Slope

For a straight line, the rate of change is constant. We find it using slope:

This means:

• change in output divided by change in input
• or rise over run

For example, if a line goes through \((2,5)\) and \((6,13)\), its slope is:

So the output increases by 2 for every increase of 1 in the input.

2. What Is a Secant Line?

A secant line is a line drawn through two points on a graph. If the graph is curved, the secant line does not match the graph everywhere, but it does connect the two chosen points.

The secant line shows the overall change between those two points, not the change at every single moment in between.

If a function is written as \(f(x)\), and we choose two inputs \(a\) and \(b\), then the two points are:

The slope of the secant line is:

This is also the average rate of change of the function from \(x=a\) to \(x=b\).

3. Average Rate of Change

The average rate of change tells us how much the output changes, on average, for each 1-unit increase in the input over a given interval.

Formula:

This looks exactly like the slope formula because the average rate of change is the slope of the secant line.

Important idea:

• For a linear function, the rate of change is constant, so the average rate of change is the same on every interval.
• For a non-linear function, the average rate of change can be different on different intervals.

4. How to Find Average Rate of Change

1. Choose two input values.
2. Find the corresponding output values.
3. Subtract the outputs: \(f(x_2)-f(x_1)\).
4. Subtract the inputs: \(x_2-x_1\).
5. Divide.

5. Worked Example 1: From a Table

A plant's height is recorded in the table below.

\(x\): days after planting
\(f(x)\): height in cm

• Day 2: 11 cm
• Day 6: 19 cm

Find the average rate of change from day 2 to day 6.

Step 1: Write the two points.

Step 2: Use the formula.

Answer: The average rate of change is \(2\) cm per day.

Meaning: Between day 2 and day 6, the plant grew an average of 2 cm each day.

6. Worked Example 2: From a Function Rule

Find the average rate of change of \(f(x)=x^2\) from \(x=1\) to \(x=5\).

Step 1: Find the output values.

Step 2: Use the formula.

Answer: The average rate of change is \(6\).

Meaning: Over the interval from 1 to 5, the output increases by an average of 6 for every 1-unit increase in \(x\).

Notice that \(f(x)=x^2\) is not a straight line. Its graph is curved, so its rate of change is not constant. That is why we say average rate of change.

7. Worked Example 3: Comparing Intervals on a Non-Linear Function

Use \(f(x)=x^2\) again, but now compare two different intervals.

Interval A: from \(x=1\) to \(x=3\)

Average rate of change on Interval A is \(4\).

Interval B: from \(x=3\) to \(x=5\)

Average rate of change on Interval B is \(8\).

What do we learn?

The average rate of change is different on different intervals. This shows that the function is non-linear.

As \(x\) gets larger, \(x^2\) increases faster, so the secant lines on later intervals are steeper.

8. Worked Example 4: A Negative Average Rate of Change

Suppose \(g(x)=20-x^2\). Find the average rate of change from \(x=2\) to \(x=4\).

Step 1: Find the outputs.

Step 2: Use the formula.

Answer: The average rate of change is \(-6\).

Meaning: On average, the output decreases by 6 units for each increase of 1 in \(x\).

A negative average rate of change means the graph is going downward over that interval.

9. How Secant Lines Look on a Graph

Imagine a curved graph. Pick two points on that curve and draw a straight line through them. That line is the secant line.

• If the secant line slopes upward from left to right, the average rate of change is positive.
• If it slopes downward from left to right, the average rate of change is negative.
• If it is horizontal, the average rate of change is 0.

The steeper the secant line, the greater the size of the average rate of change.

10. Real-World Meaning

Average rate of change is useful in many situations:

• Distance and time: average speed over a trip
• Money and time: average increase in savings each month
• Height and age: average growth over a period of years
• Temperature and time: average change in temperature during a day

For example, if a car travels 180 miles in 3 hours, its average rate of change in distance is:

So the average speed is 60 miles per hour.

11. Common Mistakes to Avoid

• Mixing up the order: If you subtract \(y\)-values in one order, subtract \(x\)-values in the same order.
• Using the wrong inputs: Make sure the output values match the chosen input values.
• Forgetting units: In real-world problems, include units like miles per hour or cm per day.
• Assuming the rate is constant: For non-linear functions, the average rate of change only describes that specific interval.

For example, this is correct:

But this would be wrong:

The numerator and denominator must follow the same order.

12. Quick Check

Find the average rate of change of \(h(x)=2x^2+1\) from \(x=0\) to \(x=3\).

Step 1: Find the outputs.

Step 2: Use the formula.

So the average rate of change is \(6\).

13. Summary

A secant line connects two points on a graph. Its slope gives the average rate of change over that interval.

To find average rate of change, use:

For linear functions, this value stays the same on every interval. For non-linear functions, it can change depending on which interval you choose.

When you see a table, graph, or function rule, think about how much the output changes compared to how much the input changes. That is the key idea behind secant lines and average rate of change.