Lesson 1: Intercepts

New Concept

Intercepts are the points where a line's graph crosses the axes. We'll find them algebraically by setting $x=0$ or $y=0$ in an equation. This powerful skill lets you quickly graph lines and understand their real-world meaning.

What’s next

This is just the foundation. Next, you’ll master this concept through interactive examples, practice graphing with the intercept method, and solve challenge problems.

Property

The intercepts of a line are the points where the graph crosses the axes. Because the $y$-intercept of a graph lies on the $y$-axis, its $x$-coordinate must be zero. And because the $x$-intercept lies on the $x$-axis, its $y$-coordinate must be zero.

Examples

• A line crosses the x-axis at $(5, 0)$ and the y-axis at $(0, -2)$. The x-intercept is $(5, 0)$ and the y-intercept is $(0, -2).

• For the line $y = x + 3$, the graph intersects the y-axis at $(0, 3)$ and the x-axis at $(-3, 0)$. These are its intercepts.

Property

To find the $x$-intercept of a graph: Substitute $0$ for $y$ in the equation and solve for $x$. To find the $y$-intercept of a graph: Substitute $0$ for $x$ in the equation and solve for $y$.

Examples

• To find the intercepts of $y = 4x - 8$: for the y-intercept, set $x=0$ to get $y = -8$, so it's $(0, -8)$. For the x-intercept, set $y=0$ to get $0 = 4x - 8$, which means $x=2$, so it's $(2, 0)$.

• To find the intercepts of $3x + 5y = 15$: for the y-intercept, set $x=0$ to get $5y = 15$, which means $y=3$, so it's $(0, 3)$. For the x-intercept, set $y=0$ to get $3x = 15$, which means $x=5$, so it's $(5, 0)$.

Property

To Graph a Linear Equation Using the Intercept Method.

1. Find the $x$- and $y$-intercepts of the graph.
2. Draw the line through the two intercepts.
3. Find a third point on the graph as a check. (Choose any convenient value for $x$ and solve for $y$.)

Examples

• To graph $x - 2y = 6$, find the intercepts. Setting $x=0$ gives $y=-3$, so $(0, -3)$. Setting $y=0$ gives $x=6$, so $(6, 0)$. Plot these two points and draw a line through them.

• To graph $4x + 3y = 12$, find the intercepts. Setting $x=0$ gives $y=4$, so $(0, 4)$. Setting $y=0$ gives $x=3$, so $(3, 0)$. Plot $(0, 4)$ and $(3, 0)$ and connect them with a line.

Property

The intercepts of a graph give us valuable information about a problem. They often represent the starting or ending values for a particular variable.

Examples

• A phone's battery percentage is modeled by $P = 100 - 5h$. The P-intercept $(0, 100)$ means the battery starts at 100%. The h-intercept $(20, 0)$ means the battery will be dead after 20 hours.

• The value of a car is $V = 20000 - 2500y$. The V-intercept $(0, 20000)$ is the initial price of the car (20000 dollars). The y-intercept $(8, 0)$ means the car will be worth 0 dollars after 8 years.