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Interpret Linear Function
Equations

Understand how the equation \( y = mx + b \) defines a straight-line graph, identify its slope and intercept, and learn to spot functions that are not linear.

8th Grade Math

What Is a Function?

A function is a rule that assigns exactly one output to every input. We write \( y = f(x) \), meaning the output \( y \) depends on the input \( x \).

Input \( x \)
The independent variable: the value you choose or plug in.
Output \( y \)
The dependent variable: the value produced by the function rule.

What Makes a Function Linear?

A function is linear when its equation can be written in the form \( y = mx + b \). The graph of every linear function is a straight line: no curves, no bends.

\( x \) \( y \) 1 2 3 4 1 2 3 y = 0.5x + 1 b = 1

Anatomy of \( y = mx + b \)

\( m \) : Slope
The rate of change: how much \( y \) increases (or decreases) when \( x \) increases by 1. Computed as \( \frac{\text{rise}}{\text{run}} \).
\( b \) : y-intercept
The value of \( y \) when \( x = 0 \). This is where the line crosses the y-axis.

The Core Rule

Two important consequences:

1
Constant rate of change: For any two points on the line, \( \frac{\Delta y}{\Delta x} = m \) (always the same).
2
First-degree equation: The variable \( x \) has exponent 1. No \( x^2 \), no \( \frac{1}{x} \), no \( \sqrt{x} \).

Visualizing Slope

m = 2 (steep up) m = 0 (flat) m = −1 (down) run = 1 rise = 2

Linear vs. Nonlinear

Linear: y = x (0,0) (1,1) (2,2) Nonlinear: y = x² (0,0) (1,1) (2,4)

Spotting Linear vs. Nonlinear

Linear Nonlinear
Equation form \( y = mx + b \) \( y = x^2,\; y = \frac{1}{x},\; y = 2^x \)
Exponent on \( x \) Always 1 Not 1 (e.g., 2, −1, \( \frac{1}{2} \))
Rate of change Constant Changes
Graph shape Straight line Curve (parabola, etc.)
Table pattern Equal \( \Delta y \) for equal \( \Delta x \) Unequal \( \Delta y \)

How to Identify a Linear Function

1
Look at the equation: Is the variable \( x \) only raised to the first power? No \( x^2 \), no \( \sqrt{x} \), no \( x \) in a denominator?
2
Check the form: Can you rewrite it as \( y = mx + b \)? If yes, identify the slope \( m \) and the y-intercept \( b \).
3
Test with a table: Pick at least 3 values of \( x \). Compute \( y \). Check if \( \Delta y \) is the same each time.
4
Graph it: If the points form a straight line, the function is linear. If they curve, it is not.

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

Question: Is \( y = 3x - 2 \) a linear function? If so, what are its slope and y-intercept?

1
The equation is already in the form \( y = mx + b \).
2
Slope: \( m = 3 \). For every 1 unit increase in \( x \), \( y \) increases by 3.
3
y-intercept: \( b = -2 \). The line crosses the y-axis at \( (0, -2) \).

Example 2: \( A = s^2 \)

Question: The area of a square as a function of its side length is \( A = s^2 \). Is this linear?

1
The variable \( s \) is raised to the second power: \( s^2 \). This is not of the form \( y = mx + b \).
2
Build a table: \( s = 1 \to A = 1 \), \( s = 2 \to A = 4 \), \( s = 3 \to A = 9 \).
3
Check differences: \( \Delta A = 3 \), then \( \Delta A = 5 \). The rate of change is not constant.

Example 3: Rearranging to Find the Form

Question: Is \( 2y - 6x = 10 \) a linear function?

1
Solve for \( y \): Add \( 6x \) to both sides: \( 2y = 6x + 10 \).
2
Divide by 2: \( y = 3x + 5 \).
3
Identify: Slope \( m = 3 \), y-intercept \( b = 5 \).

The Table Test for Linearity

If you have a table of values instead of an equation, check whether the first differences in \( y \) are constant.

\( x \) \( y \) \( \Delta y \)
0 4 —
1 7 +3
2 10 +3
3 13 +3

Examples of Nonlinear Functions

\( y = x^2 \)
Quadratic: exponent is 2. Graph is a parabola.
\( y = \frac{1}{x} \)
Reciprocal: \( x \) is in the denominator. Graph is a hyperbola.
\( y = 2^x \)
Exponential: \( x \) is in the exponent. Graph curves steeply upward.

Common Mistakes

✕ Confusing \( m \) and \( b \)
In \( y = 5x + 3 \), students sometimes call 3 the slope. Remember: the number multiplied by \( x \) is the slope.
✕ Thinking all equations are linear
\( y = x^2 + 1 \) has a "\( + 1 \)" like a y-intercept, but the \( x^2 \) makes it nonlinear.
✕ Checking only 2 points
Any 2 points define a line. You need at least 3 points to confirm a pattern is truly linear.

Summary: Linear Function Checklist

✓
Form: Can be written as \( y = mx + b \).
✓
Degree: The highest power of \( x \) is exactly 1.
✓
Rate of change: \( \frac{\Delta y}{\Delta x} \) is constant for all pairs of points.
✓
Graph: Always a straight line.

Question 1 of 5

Which equation defines a linear function?

Question 2 of 5

In the equation \( y = -4x + 9 \), what is the slope and what is the y-intercept?

Question 3 of 5

A table shows: \( x = 1, y = 5 \); \( x = 2, y = 8 \); \( x = 3, y = 11 \); \( x = 4, y = 14 \). Is this function linear?

Question 4 of 5

Which of these is an example of a function that is not linear?

Question 5 of 5

The equation \( 3y + 9 = 6x \) can be rewritten in slope-intercept form. What are \( m \) and \( b \)?