Lesson 4-6: Graph Linear Equations

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

When graphing a line using the slope $m = \frac{\text{rise}}{\text{run}}$, always move to the right for the run and use the sign of the slope to determine the direction of the rise:

• Positive slope ($m > 0$): Move up and to the right.
• Negative slope ($m < 0$): Move down and to the right. Assign the negative sign to the numerator: $-\frac{a}{b} = \frac{-a}{b}$.

Examples

Property

A horizontal line is the graph of an equation of the form $y = b$. The line passes through the $y$-axis at $(0, b)$.
In this type of equation, the value of $y$ is always equal to $b$, no matter the value of $x$.

Examples

• The graph of the equation $y = 4$ is a horizontal line where every point has a y-coordinate of 4, such as $(0, 4)$, $(2, 4)$, and $(-1, 4)$.
• The equation $y = -1$ represents a horizontal line that passes through the y-axis at the point $(0, -1)$.
• A horizontal line that passes through the point $(5, -6)$ has the equation $y = -6$ because the y-coordinate must always be -6.

Explanation

When an equation only has a $y$ variable, like $y = 3$, it means $y$ is always fixed at that number. No matter how far left or right you move along the x-axis, the result is a perfectly flat, horizontal line.