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Graphing
Linear Functions

Learn to graph lines using \( y = mx + b \), identify intercepts, and understand rate of change.

Pre-Algebra

What Is a Linear Function?

A linear function is a relationship between two variables that creates a straight line when graphed on the coordinate plane. Every linear function can be written in the form:

Here, \( m \) controls the steepness of the line, and \( b \) tells you where the line crosses the y-axis.

Key Vocabulary

Slope (\( m \))
The rate of change: how much \( y \) changes for each unit increase in \( x \). Computed as \( \frac{\text{rise}}{\text{run}} \).
y-intercept (\( b \))
The point where the line crosses the y-axis. It occurs at \( (0, b) \).
x-intercept
The point where the line crosses the x-axis. It occurs where \( y = 0 \).

Visualizing \( y = 2x + 1 \)

x y -4 -3 -2 -1 1 2 3 4 1 2 3 -1 -2 -3 -4 -5 (0, 1) (-0.5, 0) run = 1 rise = 2 (1, 3)

The Slope-Intercept Form

\( m \) : Slope
\( m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{\text{rise}}{\text{run}} \). Positive slopes go up, negative slopes go down.
\( b \) : y-intercept
The value of \( y \) when \( x = 0 \). The line always passes through the point \( (0, b) \).

This form is the quickest way to graph a line: start at the y-intercept, then use the slope to find additional points.

Understanding Slope as Rate of Change

x y Positive slope Negative slope Zero slope

Slope vs. y-intercept

Slope \( (m) \) y-intercept \( (b) \)
What it tells you How steep the line is Where the line starts on the y-axis
How to find it \( \frac{y_2 - y_1}{x_2 - x_1} \) Set \( x = 0 \) and solve for \( y \)
Changes what? The tilt of the line The position (shifts up/down)
Example in \( y = 3x - 2 \) \( m = 3 \) \( b = -2 \)

How to Graph a Linear Function

1
Identify \( m \) and \( b \). Write the equation in the form \( y = mx + b \).
2
Plot the y-intercept. Place a point at \( (0, b) \) on the y-axis.
3
Use the slope. From the y-intercept, move rise units up (or down) and run units right to plot a second point.
4
Draw the line. Connect the points with a straight line extending in both directions.
5
Find the x-intercept (optional). Set \( y = 0 \) and solve: \( x = -\frac{b}{m} \).

Example 1: \( y = \frac{1}{2}x + 3 \)

x y 1 2 3 4 -1 -2 -3 -4 1 2 3 -1 -2 (0, 3) (2, 4) run = 2 rise = 1

Example 2: \( y = -x + 4 \)

x y 1 2 3 4 5 -1 -2 1 2 3 4 -1 -2 (0, 4) (4, 0) (1, 3) run 1 rise -1

Example 3: \( y = -\frac{2}{3}x - 1 \)

x y 1 2 3 4 -1 -2 -3 -4 1 2 3 -1 -2 -3 (0, -1) (-1.5, 0) (3, -3) run = 3 rise = -2

Finding the x-intercept

The x-intercept is the point where the line crosses the x-axis. At this point, \( y = 0 \). To find it, substitute \( y = 0 \) into the equation and solve for \( x \).

1
Start with the equation: \( y = mx + b \)
2
Set \( y = 0 \): \( 0 = mx + b \)
3
Solve for \( x \): \( x = -\dfrac{b}{m} \)

Special Linear Functions

Horizontal Line
\( y = b \) (slope is zero). The line is flat. It has no x-intercept unless \( b = 0 \).
Vertical Line
\( x = a \) (slope is undefined). This is NOT a function. It has no y-intercept unless \( a = 0 \).
Through the Origin
\( y = mx \) (when \( b = 0 \)). The line passes through \( (0, 0) \). Both intercepts are at the origin.

Linear Functions in the Real World

Linear functions model situations with a constant rate of change. The slope tells you the rate, and the y-intercept tells you the starting value.

Earnings
You earn \( \$12\text{/hour} \). After \( x \) hours, your earnings are \( y = 12x \). The slope is the hourly wage and the \(y\)-intercept is \( \$0 \).
Taxi Fare
A \( \$3 \) base fare plus \( \$2\text{/mile} \). The equation is \( y = 2x + 3 \). The slope is the cost per mile and the \(y\)-intercept is the \( \$3 \) base fare.
Water Draining
A pool with 200 gallons loses \( 5~\text{gal/min} \). The equation is \( y = -5x + 200 \). The negative slope means the amount is decreasing.

Common Mistakes

✕ Swapping Rise and Run
Writing \( m = \frac{\text{run}}{\text{rise}} \) instead of \( \frac{\text{rise}}{\text{run}} \). Rise is always the vertical change (numerator).
✕ Forgetting the Sign
Dropping the negative in \( y = -2x + 5 \) and graphing a line that goes up instead of down.
✕ Mixing Up Intercepts
Plotting the y-intercept on the x-axis, or confusing \( (0, b) \) with \( (b, 0) \).

Summary Cheat Sheet

Concept Formula / Rule How to Find
Slope-intercept form \( y = mx + b \) Solve for \( y \)
Slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \) Pick two points
y-intercept Point \( (0, b) \) Set \( x = 0 \)
x-intercept Point \( \left(-\frac{b}{m},\, 0\right) \) Set \( y = 0 \)
Rate of change Same as slope \( m \) Units: \( \frac{\Delta y}{\Delta x} \)

Question 1 of 5

What is the slope and y-intercept of the line \( y = 4x - 7 \)?

Question 2 of 5

A line passes through the points \( (1, 5) \) and \( (3, 11) \). What is the slope?

Question 3 of 5

What is the x-intercept of the line \( y = 2x - 8 \)?

Question 4 of 5

Which equation represents a line with a negative slope that crosses the y-axis at \( 6 \)?

Question 5 of 5

A plumber charges a \( \$50 \) service fee plus \( \$30 \) per hour. If \( y \) is total cost and \( x \) is hours worked, what is the rate of change?