Lesson 3: Intercepts

New Concept

Intercepts are the specific points where a graph crosses the axes. We find the $y$-intercept by setting $x=0$ and the $x$-intercept by setting $y=0$. These points reveal key starting values or endpoints in real-world scenarios.

What’s next

Soon, you'll work through interactive examples of finding intercepts. Then, you'll apply this skill to graph lines and solve challenge problems.

Property

The points at which a graph crosses the axes are called the intercepts of the graph.

To find the intercepts of a graph:

1. To find the $x$-intercept, we set $y = 0$ and solve for $x$.
2. To find the $y$-intercept, we set $x = 0$ and solve for $y$.

Examples

• To find the intercepts of $3x + 2y = 12$, first set $y=0$ to get $3x = 12$, so the $x$-intercept is $(4, 0)$. Then set $x=0$ to get $2y=12$, so the $y$-intercept is $(0, 6)$.

Property

The graphs of the equations we have seen so far are all portions of straight lines. For this reason such equations are called linear equations.

The general form for a linear equation is

(where $A$ and $B$ cannot both be 0).

Examples

• A bakery sells muffins for 3 dollars and croissants for 4 dollars. To make 600 dollars in revenue, the equation is $3x + 4y = 600$, where $x$ is muffins and $y$ is croissants.

Property

To Graph a Linear Equation by the Intercept Method:
a. Find the horizontal and vertical intercepts.
b. Plot the intercepts, and draw the line through the two points.

Examples

• To graph $4x + 3y = 24$, find the intercepts. Set $y=0$ to get the $x$-intercept $(6, 0)$. Set $x=0$ to get the $y$-intercept $(0, 8)$. Plot these two points and draw a line through them.

• Let's graph $x - 2y = 4$. The $x$-intercept is $(4, 0)$ and the $y$-intercept is $(0, -2)$. Plot these two points on the coordinate plane and draw the line that connects them.

Property

We have now seen two forms for linear equations: the general linear form,

and the form for a linear model,

Caution: Do not confuse solving for $y$ in terms of $x$ with finding the $y$-intercept.

Examples

• To convert $8x - 2y = 14$ into the linear model form, we solve for $y$. This gives $-2y = -8x + 14$, which simplifies to $y = 4x - 7$.

• To convert $y = -\frac{1}{4}x + 2$ to general form, first multiply by 4 to clear the fraction: $4y = -x + 8$. Then, add $x$ to both sides to get $x + 4y = 8$.