Section 6.3: Linear Functions

Property

An equation of the form $y = ax + b$, where $a$ and $b$ are constants, is called a linear equation because its graph is a straight line. We can graph a linear equation by evaluating the expression $ax + b$ at several values of $x$ and then plotting the resulting points.

Examples

• A sapling is 5 inches tall and grows 2 inches each week. The height $H$ after $w$ weeks is given by the equation $H = 5 + 2w$. After 3 weeks, the height is $H = 5 + 2(3) = 11$ inches.

• You are 200 miles from home and driving away at 60 miles per hour. Your distance $D$ from home after $h$ hours is $D = 200 + 60h$. After 2 hours, the distance is $D = 200 + 60(2) = 320$ miles.

Property

To graph a line using its slope and intercept:

1. Find the slope-intercept form of the equation, $y = mx + b$.
2. Identify the slope ($m$) and y-intercept ($b$).
3. Plot the y-intercept at point $(0, b)$.
4. Use the slope formula $m = \frac{\text{rise}}{\text{run}}$ to find a second point by counting from the y-intercept.
5. Connect the two points with a straight line.

Examples

• To graph $y = 3x - 1$, start by plotting the y-intercept at $(0, -1)$. The slope $m=3$ means $\frac{\text{rise}}{\text{run}} = \frac{3}{1}$. From $(0, -1)$, go up 3 units and right 1 unit to find the next point, $(1, 2)$. Connect them.

• To graph $y = -\frac{1}{2}x + 3$, plot the y-intercept at $(0, 3)$. The slope $m=-\frac{1}{2}$ means $\frac{\text{rise}}{\text{run}} = \frac{-1}{2}$. From $(0, 3)$, go down 1 unit and right 2 units to find the point $(2, 2)$. Connect them.

Property

To find a linear function from a table:
(1) verify the data shows constant rate of change,
(2) calculate slope using any two points with $m = \frac{y_2 - y_1}{x_2 - x_1}$,
(3) identify the y-intercept when $x = 0$ or use point-slope form,
(4) write the function as $y = mx + b$.

Examples