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Linear Functions
Construct & Interpret

Learn to build linear functions from real-world situations, tables, and graphs. Discover how rate of change and initial value reveal the story behind the numbers.

8th Grade Math

What Is a Linear Function?

A linear function is a relationship between two quantities where the rate of change is constant. When you graph it, you always get a straight line.

x y 1 2 3 1 2 3 0 b = 1 run = 1 rise = 1

Understanding Rate of Change

The rate of change (slope, \( m \)) tells you how much \( y \) changes every time \( x \) increases by 1. It measures the steepness and direction of the line.

Positive Slope
Line rises left to right. As \( x \) grows, \( y \) grows.
Negative Slope
Line falls left to right. As \( x \) grows, \( y \) shrinks.
Zero Slope
Horizontal line. \( y \) stays the same no matter what \( x \) is.

Understanding the Initial Value

The initial value (\( b \)) is the output when the input is zero: \( y = m(0) + b = b \). On a graph, it is the point where the line crosses the y-axis.

x y 1 2 3 1 2 3 0 b = 1 b = 2 y = 0.5x + 1 y = 0.5x + 2

The Core Formula: \( y = mx + b \)

\( m \)
Rate of change (slope): how much \( y \) changes per unit increase in \( x \).
\( b \)
Initial value (y-intercept): the value of \( y \) when \( x = 0 \).
y = m x + b output slope input y-intercept Real-world meaning: m = per-unit cost, speed, growth rate b = starting amount, base fee, initial position

Finding \( m \) and \( b \) from a Table

\( x \) 0 1 2 3 4
\( y \) 5 8 11 14 17
1
Find \( b \): Look at the \( y \)-value when \( x = 0 \). Here, \( b = 5 \).
2
Find \( m \): Pick any two consecutive rows. From \( x = 0 \) to \( x = 1 \): \( m = \dfrac{8 - 5}{1 - 0} = \) 3.
3
Write the equation: \( y = \) 3\( x + \) 5.

Slope vs. Initial Value

Rate of Change (\( m \)) Initial Value (\( b \))
What it is How fast \( y \) changes per unit of \( x \) Where the line starts when \( x = 0 \)
On a graph Steepness & direction of the line Where line crosses the \( y \)-axis
In a table Constant difference between consecutive \( y \)-values \( y \)-value in the row where \( x = 0 \)
Real-world Per-mile charge, monthly growth, speed Base fee, starting balance, initial height

How to Construct a Linear Function

1
Identify two quantities: Which is the input (\( x \)) and which is the output (\( y \))?
2
Find the rate of change (\( m \)): Use \( m = \dfrac{y_2 - y_1}{x_2 - x_1} \) with any two data points.
3
Find the initial value (\( b \)): Read it from \( x = 0 \), or substitute a known point into \( y = mx + b \) and solve for \( b \).
4
Write the equation: Plug \( m \) and \( b \) into \( y = mx + b \).
5
Verify: Check that your equation works for another data point.

Example 1: From a Description

1
Input: miles driven (\( m \)). Output: total cost (\( C \)).
2
Rate of change: $2.50 per mile, so slope \( = 2.5 \).
3
Initial value: $4 base fee (cost when \( m = 0 \)), so \( b = 4 \).
4
Equation: \( C = 2.5m + 4 \).

Example 2: From Two Points

1
Points: \( (2, 6) \) and \( (5, 15) \).
2
Slope: \( m = \dfrac{15 - 6}{5 - 2} = \dfrac{9}{3} = \) 3 cm per week.
3
Find \( b \): Substitute \( (2, 6) \): \( 6 = 3(2) + b \Rightarrow b = \) 0.
4
Equation: \( h = 3w \). The plant grows 3 cm/week from a seed at ground level.

Example 3: From a Graph

hours $ 1 2 3 4 10 20 30 40 50 0 (0, 20) (2, 40) run = 2 rise = 20

Interpreting in Context

Once you have the equation, every number tells a story. The key is connecting math back to meaning.

\( m = 25 \)
You pay $25 each month: the monthly rate.
\( b = 50 \)
There is a $50 sign-up fee: the one-time starting cost.

Reading \( m \) and \( b \) from a Graph

1
Find \( b \): Locate where the line crosses the y-axis. Read the \( y \)-coordinate of that point.
2
Pick two clear points where the line passes exactly through grid intersections.
3
Calculate \( m \): Use \( m = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1} \).
4
Check the sign: Line goes up? \( m \) is positive. Line goes down? \( m \) is negative.

Example 4: Negative Slope

t (hr) h (cm) 3 5 10 15 5 10 15 20 30 (0, 30) (15, 0)

Common Mistakes to Avoid

✕ Swapping Rise and Run
Putting the \( x \)-difference on top instead of the \( y \)-difference. Remember: rise (y) over run (x).
✕ Forgetting the Sign
Ignoring a negative slope. If the line goes down, the rate of change must be negative.
✕ Misreading \( b \)
Using the first \( y \)-value in a table as \( b \) even when \( x \neq 0 \). The initial value is only at \( x = 0 \).

Summary: Three Sources, One Equation

From a Description From a Table From a Graph
Find \( m \) "per unit" phrase Difference in \( y \) / difference in \( x \) Rise over run between two points
Find \( b \) Starting/initial amount \( y \)-value when \( x = 0 \) Where line hits the \( y \)-axis
Write equation \( y = mx + b \)

Question 1 of 5

A phone plan charges a $15 monthly fee plus $0.10 per text. Which function models the monthly cost \( C \) for \( t \) texts?

Question 2 of 5

A line passes through \( (1, 7) \) and \( (4, 22) \). What is the rate of change?

Question 3 of 5

A table shows: when \( x = 0 \), \( y = 12 \); when \( x = 3 \), \( y = 3 \). What is the initial value and what does the slope tell you?

Question 4 of 5

The equation \( y = 4x + 10 \) models a savings account. What does the 10 represent?

Question 5 of 5

A graph shows a line crossing the y-axis at \( (0, 8) \) and passing through \( (6, 2) \). Which equation represents this line?