Lesson 5-3: Construct Linear Functions

Property

The slope-intercept form for a linear equation is $y = mx + b$, where $m$ is the slope of the line and the point $(0, b)$ is the y-intercept.

The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$.

This formula calculates the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run).

Property

To write an equation from a graph in slope-intercept form $y = mx + b$:

1. Identify the y-intercept $b$ where the line crosses the y-axis
2. Find the slope $m = \frac{\text{rise}}{\text{run}}$ using two clear points on the line
3. Substitute $m$ and $b$ into $y = mx + b$

Examples

Property

To write a linear function in the form $y = mx + b$ from a table of values, first find the slope ($m$) and then determine the y-intercept ($b$).

1. Find the slope ($m$): Use any two points $(x_1, y_1)$ and $(x_2, y_2)$ from the table.

2. Find the y-intercept ($b$): Identify the value of $y$ when $x=0$. If $x=0$ is not in the table, use the slope $m$ and any point $(x, y)$ from the table to solve for $b$ in the equation $y = mx + b$.