Lesson 4: Graphing Linear Equations in Slope-Intercept Form

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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).

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Before comparing two lines, you must write their equations in the slope-intercept form:

In this form, m represents the slope (rate of change), and b represents the y-coordinate of the y-intercept (0, b). By using algebraic manipulation to isolate y on one side, you can rewrite any linear equation into this format.

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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.