how to find the intercepts of a graph

Graphs

70 Graphing with Intercepts

Learning Objectives

By the terminate of this section, you volition exist able to:

Identify the intercepts on a graph
Find the intercepts from an equation of a line
Graph a line using the intercepts
Choose the most convenient method to graph a line

Place the Intercepts on a Graph

Every linear equation has a unique line that represents all the solutions of the equation. When graphing a line past plotting points, each person who graphs the line tin cull any 3 points, so two people graphing the line might use dissimilar sets of points.

At first glance, their 2 lines might appear unlike since they would have dissimilar points labeled. Just if all the work was done correctly, the lines will be exactly the same line. 1 way to recognize that they are indeed the aforementioned line is to focus on where the line crosses the axes. Each of these points is chosen an intercept of the line.

Intercepts of a Line

Each of the points at which a line crosses the $x\text{-axis}$ and the $y\text{-axis}$ is called an intercept of the line.

Let's wait at the graph of the lines shown in (Figure).

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through two labeled points,

First, notice where each of these lines crosses the x– axis:

Figure:	The line crosses the ten-centrality at:	Ordered pair of this point
42	3	(3,0)
43	4	(4,0)
44	5	(five,0)
45	0	(0,0)

Practice you see a pattern?

For each row, the y- coordinate of the point where the line crosses the 10- centrality is cypher. The signal where the line crosses the 10- centrality has the form $\left(a,0\right)$ ; and is chosen the 10-intercept of the line. The x- intercept occurs when y is zero.

Now, let'south await at the points where these lines cross the y-axis.

Figure:	The line crosses the y-axis at:	Ordered pair for this bespeak
42	6	(0,6)
43	-3	(0,-3)
44	-v	(0,-5)
45	0	(0,0)

Solution

ⓐ
The graph crosses the ten-axis at the point (4, 0).	The x-intercept is (4, 0).
The graph crosses the y-axis at the betoken (0, 2).	The 10-intercept is (0, 2).

ⓑ
The graph crosses the ten-axis at the signal (2, 0).	The x-intercept is (ii, 0)
The graph crosses the y-axis at the point (0, −6).	The y-intercept is (0, −6).

ⓒ
The graph crosses the x-centrality at the betoken (−5, 0).	The x-intercept is (−5, 0).
The graph crosses the y-axis at the point (0, −v).	The y-intercept is (0, −5).

Find the $x\text{-}$ and $y\text{-intercepts}$ of the graph: $x-y=2.$

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points

x-intercept (2,0): y-intercept (0,−2)

Find the $x\text{-}$ and $y\text{-intercepts}$ of the graph: $2x+3y=6.$

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points

x-intercept (3,0); y-intercept (0,2)

Find the Intercepts from an Equation of a Line

Recognizing that the $x\text{-intercept}$ occurs when $y$ is zero and that the $y\text{-intercept}$ occurs when $x$ is zero gives us a method to find the intercepts of a line from its equation. To observe the $x\text{-intercept,}$ allow $y=0$ and solve for $x.$ To find the $y\text{-intercept},$ let $x=0$ and solve for $y.$

Discover the ten and y from the Equation of a Line

Use the equation to discover:

ten	y
	0
0

Find the intercepts of $2x+y=6$

Find the intercepts of $4x-3y=12.$

Detect the intercepts of the line: $3x-4y=12.$

x-intercept (4,0); y-intercept: (0,−3)

Discover the intercepts of the line: $2x-4y=8.$

x-intercept (4,0); y-intercept: (0,−2)

Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you can use the intercepts as ii of your iii points. Detect the two intercepts, and and then a third point to ensure accuracy, and depict the line. This method is often the quickest mode to graph a line.

Graph $-x+2y=6$ using intercepts.

Solution

First, observe the $x\text{-intercept}.$ Let $y=0,$

$\begin{array}{}\\ \phantom{\rule{0.7em}{0ex}}-x+2y=6\\ -x+2\left(0\right)=6\\ \phantom{\rule{2.8em}{0ex}}-x=6\\ \phantom{\rule{4.3em}{0ex}}x=-6\end{array}$

The $x\text{-intercept}$ is $\left(-6,0\right).$

At present discover the $y\text{-intercept}.$ Let $x=0.$

$\begin{array}{}\\ -x+2y=6\\ -0+2y=6\\ \\ \\ \phantom{\rule{2.4em}{0ex}}2y=6\\ \phantom{\rule{3em}{0ex}}y=3\end{array}$

The $y\text{-intercept}$ is $\left(0,3\right).$

Notice a third point. We'll use $x=2,$

$\begin{array}{}\\ -x+2y=6\\ -2+2y=6\\ \\ \\ \phantom{\rule{2.4em}{0ex}}2y=8\\ \phantom{\rule{3em}{0ex}}y=4\end{array}$

A third solution to the equation is $\left(2,4\right).$

Summarize the three points in a table and then plot them on a graph.

$-x+2y=6$
ten	y	(x,y)
$-6$	$0$	$\left(-6,0\right)$
$0$	$3$	$\left(0,3\right)$
$2$	$4$	$\left(2,4\right)$

The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at

Do the points line upward? Aye, so draw line through the points.

The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at

Graph the line using the intercepts: $x-2y=4.$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

Graph the line using the intercepts: $-x+3y=6.$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

Graph a line using the intercepts.

Find the $x-$ and $\text{y-intercepts}$ of the line.
Notice a third solution to the equation.
Plot the iii points then bank check that they line up.
Draw the line.

Graph $4x-3y=12$ using intercepts.

Graph the line using the intercepts: $5x-2y=10.$

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points

Graph the line using the intercepts: $3x-4y=12.$

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points

Graph $y=5x$ using the intercepts.

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

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

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

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

Choose the Nigh Convenient Method to Graph a Line

While nosotros could graph any linear equation past plotting points, it may non always exist the most convenient method. This table shows six of equations nosotros'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
#three	$x=-7$	Vertical line
#iv	$y=4$	Horizontal line
#5	$2x+y=6$	Intercepts
#six	$4x-3y=12$	Intercepts

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

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

Equations #three and #4 each take just one variable. Retrieve, in this kind of equation the value of that one variable is constant; it does non depend on the value of the other variable. Equations of this form accept graphs that are vertical or horizontal lines.

In equations #five and #vi, both x and y are on the aforementioned side of the equation. These two equations are of the grade $Ax+By=C$ . We substituted $y=0$ and $x=0$ to notice the x- and y- intercepts, and then found a tertiary point past choosing a value for ten or y.

This leads to the post-obit strategy for choosing the almost user-friendly method to graph a line.

Choose the nigh convenient method to graph a line.

If the equation has only i variable. It is a vertical or horizontal line.
If $y$ is isolated on one side of the equation. Graph past plotting points.
If the equation is of the form $Ax+By=C,$ discover the intercepts.

Solution

ⓐ $\phantom{\rule{0.2em}{0ex}}y=-3$

This equation has simply 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 course $Ax+By=C.$ Find the intercepts and one more point.

In that location 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, information technology volition be easiest to graph this line by plotting iii points.

ⓐ intercepts
ⓑ horizontal line
ⓒ plotting points
ⓓ vertical line

ⓐ vertical line
ⓑ plotting points
ⓒ horizontal line
ⓓ intercepts

Key Concepts

Practice Makes Perfect

Identify the Intercepts on a Graph

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

The graph shows the x y-coordinate plane. The axes run from -10 to 10. A line passes through the points

(3,0),(0,three)

The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through the points

(5,0),(0,−5)

The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through the points

(−2,0),(0,−2)

The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through the points

(−i,0),(0,one)

The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through the points

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points

(0,0)

The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through the points

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

In the following exercises, find the intercepts.

$x+y=4$

(4,0),(0,iv)

$x+y=3$

$x+y=-2$

(−two,0),(0,−2)

$x+y=-5$

$x-y=5$

(v,0),(0,−5)

$x-y=1$

$x-y=-3$

(−3,0),(0,3)

$x-y=-4$

$x+2y=8$

(8,0),(0,four)

$x+2y=10$

$3x+y=6$

(2,0),(0,6)

$3x+y=9$

$x-3y=12$

(12,0),(0,−iv)

$x-2y=8$

$4x-y=8$

(2,0),(0,−8)

$5x-y=5$

$2x+5y=10$

(5,0),(0,2)

$2x+3y=6$

$3x-2y=12$

(4,0),(0,−six)

$3x-5y=30$

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

(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.

$-x+5y=10$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

$-x+4y=8$

$x+2y=4$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

$x+2y=6$

$x+y=2$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

$x+y=5$

$x+y=-3$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

$x+y=-1$

$x-y=1$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

$x-y=2$

$x-y=-4$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

$x-y=-3$

$4x+y=4$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

$3x+y=3$

$3x-y=-6$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

$2x-y=-8$

$2x+4y=12$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

$3x+2y=12$

$3x-2y=6$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

$5x-2y=10$

$2x-5y=-20$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

$3x-4y=-12$

$y=-2x$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

$y=-4x$

$y=x$

The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points

$y=3x$

Choose the Most Convenient Method to Graph a Line

In the post-obit exercises, identify the near user-friendly 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 altitude 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.

The graph shows the x y-coordinate plane. The x and y-axis each run from - to . A line passes through the labeled points

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

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

ⓐ (0,one,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 fifteen hours ago and has 0 miles left to bulldoze.

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'south gas tank.

The graph shows the x y-coordinate plane. The x and y-axis each run from - to . A line passes through labeled points