Lesson 4.3: Graph with Intercepts

New Concept

This lesson introduces intercepts—the points where a line crosses the axes. You'll learn to find the $x$-intercept (where $y=0$) and $y$-intercept (where $x=0$) to efficiently graph linear equations.

What’s next

Now, let's apply this. You'll use our interactive examples to find intercepts from equations and then master graphing lines with a series of practice cards.

Property

The points where a line crosses the $x$-axis and the $y$-axis are called the intercepts of a line.

Examples

• A line crosses the x-axis at the point $(4, 0)$. The x-intercept is $(4, 0).

• A line crosses the y-axis at the point $(0, -1)$. The y-intercept is $(0, -1).

Property

The $x$-intercept is the point $(a, 0)$ where the line crosses the $x$-axis.

The $y$-intercept is the point $(0, b)$ where the line crosses the $y$-axis.

The $x$-intercept occurs when $y$ is zero.

Property

Use the equation of the line. To find:
• the $x$-intercept of the line, let $y = 0$ and solve for $x$.
• the $y$-intercept of the line, let $x = 0$ and solve for $y$.

Examples

• For the equation $x + 4y = 8$: set $y=0$ to get $x=8$, so the x-intercept is $(8,0)$. Set $x=0$ to get $4y=8$ or $y=2$, so the y-intercept is $(0,2)$.

• For the equation $3x - 2y = 12$: set $y=0$ to get $3x=12$ or $x=4$, so the x-intercept is $(4,0)$. Set $x=0$ to get $-2y=12$ or $y=-6$, so the y-intercept is $(0,-6)$.

Property

To graph a linear equation using the intercepts:
Step 1. Find the $x$- and $y$-intercepts of the line.
• Let $y = 0$ and solve for $x$
• Let $x = 0$ and solve for $y$.
Step 2. Find a third solution to the equation.
Step 3. Plot the three points and check that they line up.
Step 4. Draw the line.

Examples

• Graph $x+2y=4$. The x-intercept is $(4,0)$ and y-intercept is $(0,2)$. For a third point, let $x=2$, then $2+2y=4$, so $y=1$. Plot $(4,0), (0,2),$ and $(2,1)$.

• Graph $5x-y=5$. The x-intercept is $(1,0)$ and y-intercept is $(0,-5)$. For a third point, let $x=2$, then $10-y=5$, so $y=5$. Plot $(1,0), (0,-5),$ and $(2,5)$.