# 11.3 Graphing with intercepts

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Graph using the intercepts: $y=3x.$

Graph using the intercepts: $y=-x.$

## Choose the most convenient method to graph a line

While we could graph any linear equation    by plotting points, it may not always be the most convenient method. This table shows six of equations we’ve graphed in this chapter, and the methods we used to graph them.

Equation Method
#1 $y=2x+1$ Plotting points
#2 $y=\frac{1}{2}x+3$ Plotting points
#3 $x=-7$ Vertical line
#4 $y=4$ Horizontal line
#5 $2x+y=6$ Intercepts
#6 $4x-3y=12$ Intercepts

What is it about the form of equation that can help us choose the most convenient method to graph its line?

Notice that in equations #1 and #2, y is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for x on the right side of the equation and then simplifying to get the corresponding y- values.

Equations #3 and #4 each have just one variable. Remember, in this kind of equation the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.

In equations #5 and #6, both x and y are on the same side of the equation. These two equations are of the form $Ax+By=C$ . We substituted $y=0$ and $x=0$ to find the x- and y- intercepts, and then found a third point by choosing a value for x or y .

This leads to the following strategy for choosing the most convenient method to graph a line.

## Choose the most convenient method to graph a line.

1. If the equation has only one variable. It is a vertical or horizontal line.
• $x=a$ is a vertical line passing through the $x\text{-axis}$ at $a$
• $y=b$ is a horizontal line passing through the $y\text{-axis}$ at $b.$
2. If $y$ is isolated on one side of the equation. Graph by plotting points.
• Choose any three values for $x$ and then solve for the corresponding $y\text{-}$ values.
3. If the equation is of the form $Ax+By=C,$ find the intercepts.
• Find the $x\text{-}$ and $y\text{-}$ intercepts and then a third point.

Identify the most convenient method to graph each line:

1. $\phantom{\rule{0.2em}{0ex}}y=-3\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}4x-6y=12\phantom{\rule{0.2em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}x=2\phantom{\rule{0.2em}{0ex}}$
4. $\phantom{\rule{0.2em}{0ex}}y=\frac{2}{5}x-1$

## Solution

$\phantom{\rule{0.2em}{0ex}}y=-3$

This equation has only one variable, $y.$ Its graph is a horizontal line crossing the $y\text{-axis}$ at $-3.$

$\phantom{\rule{0.2em}{0ex}}4x-6y=12$

This equation is of the form $Ax+By=C.$ Find the intercepts and one more point.

$\phantom{\rule{0.2em}{0ex}}x=2$

There is only one variable, $x.$ The graph is a vertical line crossing the $x\text{-axis}$ at $2.$

$\phantom{\rule{0.2em}{0ex}}y=\frac{2}{5}x-1$

Since $y$ is isolated on the left side of the equation, it will be easiest to graph this line by plotting three points.

Identify the most convenient method to graph each line:

1. $\phantom{\rule{0.2em}{0ex}}3x+2y=12$
2. $\phantom{\rule{0.2em}{0ex}}y=4$
3. $\phantom{\rule{0.2em}{0ex}}y=\frac{1}{5}x-4$
4. $\phantom{\rule{0.2em}{0ex}}x=-7$
1. intercepts
2. horizontal line
3. plotting points
4. vertical line

Identify the most convenient method to graph each line:

1. $\phantom{\rule{0.2em}{0ex}}x=6$
2. $\phantom{\rule{0.2em}{0ex}}y=-\frac{3}{4}x+1$
3. $\phantom{\rule{0.2em}{0ex}}y=-8$
4. $\phantom{\rule{0.2em}{0ex}}4x-3y=-1$

1. vertical line
2. plotting points
3. horizontal line
4. intercepts

## Key concepts

• Intercepts
• The x- intercept is the point, $\left(a,0\right)$ , where the graph crosses the x- axis. The x- intercept occurs when y is zero.
• The y- intercept is the point, $\left(0,b\right)$ , where the graph crosses the y- axis. The y- intercept occurs when y is zero.
• The x- intercept occurs when y is zero.
• The y- intercept occurs when x is zero.
• Find the x and y intercepts from the equation of a line
• To find the x- intercept of the line, let $y=0$ and solve for x .
• To find the y- intercept of the line, let $x=0$ and solve for y .
x y
0
0
• Graph a line using the intercepts
1. Find the x- and y- intercepts of the line.
• Let $y=0$ and solve for x.
• Let $x=0$ and solve for y.
2. Find a third solution to the equation.
3. Plot the three points and then check that they line up.
4. Draw the line.
• Choose the most convenient method to graph a line
1. Determine if the equation has only one variable. Then it is a vertical or horizontal line.
$x=a$ is a vertical line passing through the x- axis at a .
$y=b$ is a vertical line passing through the y- axis at b .
2. Determine if y is isolated on one side of the equation. The graph by plotting points.
Choose any three values for x and then solve for the corresponding y- values.
3. Determine if the equation is of the form $Ax+By=C$ , find the intercepts.
Find the x- and y- intercepts and then a third point.

## Practice makes perfect

Identify the Intercepts on a Graph

In the following exercises, find the $x\text{-}$ and $y\text{-}$ intercepts.

(3,0),(0,3)

(5,0),(0,−5)

(−2,0),(0,−2)

(−1,0),(0,1)

(0,0)

Find the $x$ and $y$ Intercepts from an Equation of a Line

In the following exercises, find the intercepts.

$x+y=4$

(4,0),(0,4)

$x+y=3$

$x+y=-2$

(−2,0),(0,−2)

$x+y=-5$

$x-y=5$

(5,0),(0,−5)

$x-y=1$

$x-y=-3$

(−3,0),(0,3)

$x-y=-4$

$x+2y=8$

(8,0),(0,4)

$x+2y=10$

$3x+y=6$

(2,0),(0,6)

$3x+y=9$

$x-3y=12$

(12,0),(0,−4)

$x-2y=8$

$4x-y=8$

(2,0),(0,−8)

$5x-y=5$

$2x+5y=10$

(5,0),(0,2)

$2x+3y=6$

$3x-2y=12$

(4,0),(0,−6)

$3x-5y=30$

$y=\frac{1}{3}x-1$

(3,0),(0,−1)

$y=\frac{1}{4}x-1$

$y=\frac{1}{5}x+2$

(−10,0),(0,2)

$y=\frac{1}{3}x+4$

$y=3x$

(0,0)

$y=-2x$

$y=-4x$

(0,0)

$y=5x$

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

$-x+5y=10$

$-x+4y=8$

$x+2y=4$

$x+2y=6$

$x+y=2$

$x+y=5$

$x+y=-3$

$x+y=-1$

$x-y=1$

$x-y=2$

$x-y=-4$

$x-y=-3$

$4x+y=4$

$3x+y=3$

$3x-y=-6$

$2x-y=-8$

$2x+4y=12$

$3x+2y=12$

$3x-2y=6$

$5x-2y=10$

$2x-5y=-20$

$3x-4y=-12$

$y=-2x$

$y=-4x$

$y=x$

$y=3x$

Choose the Most Convenient Method to Graph a Line

In the following exercises, identify the most convenient method to graph each line.

$x=2$

vertical line

$y=4$

$y=5$

horizontal line

$x=-3$

$y=-3x+4$

plotting points

$y=-5x+2$

$x-y=5$

intercepts

$x-y=1$

$y=\frac{2}{3}x-1$

plotting points

$y=\frac{4}{5}x-3$

$y=-3$

horizontal line

$y=-1$

$3x-2y=-12$

intercepts

$2x-5y=-10$

$y=-\frac{1}{4}x+3$

plotting points

$y=-\frac{1}{3}x+5$

## Everyday math

Road trip Damien is driving from Chicago to Denver, a distance of $1,000$ miles. The $x\text{-axis}$ on the graph below shows the time in hours since Damien left Chicago. The $y\text{-axis}$ represents the distance he has left to drive.

Find the $x\text{-}$ and $y\text{-}$ intercepts

Explain what the $x\text{-}$ and $y\text{-}$ intercepts mean for Damien.

(0,1,000),(15,0). At (0,1,000) he left Chicago 0 hours ago and has 1,000 miles left to drive. At (15,0) he left Chicago 15 hours ago and has 0 miles left to drive.

Road trip Ozzie filled up the gas tank of his truck and went on a road trip. The $x\text{-axis}$ on the graph shows the number of miles Ozzie drove since filling up. The $y\text{-axis}$ represents the number of gallons of gas in the truck’s gas tank.

Find the $x\text{-}$ and $y\text{-}$ intercepts.

Explain what the $x\text{-}$ and $y\text{-}$ intercepts mean for Ozzie.

## Writing exercises

How do you find the $x\text{-intercept}$ of the graph of $3x-2y=6?$

How do you find the $y\text{-intercept}$ of the graph of $5x-y=10?$

Do you prefer to graph the equation $4x+y=-4$ by plotting points or intercepts? Why?

Do you prefer to graph the equation $y=\frac{2}{3}x-2$ by plotting points or intercepts? Why?

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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