6.3 Graph with Intercepts

Izabela Mazur

CHAPTER 6 Linear Equations and Graphing

6.3 Graph with Intercepts

Learning Objectives

By the end of this section, you will be able to:

Identify the $x$ – and $y$ – intercepts on a graph
Find the $x$ – and $y$ – intercepts from an equation of a line
Graph a line using the intercepts

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.

At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x– axis and the y– axis. These points are called the intercepts of the line.

Intercepts of a line

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

Let’s look at the graphs of the lines in (Figure 1).

Examples of graphs crossing the x-negative axis.

First, notice where each of these lines crosses the $x$ negative axis. See (Figure 1).

Figure

The line crosses the x– axis at:

Ordered pair of this point

Figure (a)

3

$\left(3,0\right)$

Figure (b)

4

$\left(4,0\right)$

Figure (c)

5

$\left(5,0\right)$

Figure (d)

0

$\left(0,0\right)$

Do you see a pattern?

For each row, the y– coordinate of the point where the line crosses the x– axis is zero. The point where the line crosses the x– axis has the form $\left(a,0\right)$ and is called the x– intercept of a line. The x– intercept occurs when $y$ is zero.

Now, let’s look at the points where these lines cross the y– axis. See the table below.

Figure

The line crosses the y-axis at:

Ordered pair for this point

Figure (a)

6

$\left(0,6\right)$

Figure (b)

$-3$

$\left(0,-3\right)$

Figure (c)

$-5$

$\left(0,5\right)$

Figure (d)

0

$\left(0,0\right)$

What is the pattern here?

In each row, the x– coordinate of the point where the line crosses the y– axis is zero. The point where the line crosses the y– axis has the form $\left(0,b\right)$ and is called the y- intercept of the line. The y– intercept occurs when $x$ is zero.

x– intercept and y– intercept of a line

The x– intercept is the point $\left(a,0\right)$ where the line crosses the x– axis.

The y– intercept is the point $\left(0,b\right)$ where the line crosses the y– axis.

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EXAMPLE 1

Find the x– and y– intercepts on each graph.

Solution

The graph crosses the x– axis at the point $\left(4,0\right)$ . The x– intercept is $\left(4,0\right)$ .
The graph crosses the y– axis at the point $\left(0,2\right)$ . The y– intercept is $\left(0,2\right)$ .
The graph crosses the x– axis at the point $\left(2,0\right)$ . The x– intercept is $\left(2,0\right)$
The graph crosses the y– axis at the point $\left(0,-6\right)$ . The y– intercept is $\left(0,-6\right)$ .
The graph crosses the x– axis at the point $\left(-5,0\right)$ . The x– intercept is $\left(-5,0\right)$ .
The graph crosses the y– axis at the point $\left(0,-5\right)$ . The y– intercept is $\left(0,-5\right)$ .

TRY IT 1.1

Find the x– and y– intercepts on the graph.

Graph of the equation y = x − 2. The x-intercept is the point (2, 0) and the y-intercept is the point (0, −2)

Show answer

x– intercept: $\left(2,0\right)$ ; y– intercept: $\left(0,-2\right)$

TRY IT 1.2

Find the x– and y– intercepts on the graph.

Graph of the equation y = − 2 thirds x + 2 and the x-intercept is the point (3, 0) and the y-intercept is the point (0, 2).

Show answer

x– intercept: $\left(3,0\right)$ , y– intercept: $\left(0,2\right)$

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Let y = 0.
Simplify.

The x-intercept is	(3, 0)
To find the y-intercept, let x = 0.

Let x = 0.
Simplify.

The y-intercept is	(0, 6)

To find the x-intercept, let y = 0.

Let y = 0.
Simplify.


The x-intercept is	(3, 0)
To find the y-intercept, let x = 0.

Let x = 0.
Simplify.


The y-intercept is	(0, −4)

$4x-3y=12$
$x$	$y$	$\left(x,y\right)$
3	0	$\left(3,0\right)$
0	$-4$	$\left(0,-4\right)$
6	4	$\left(6,4\right)$

$y=5x$
$x$	$y$	$\left(x,y\right)$
0	0	$\left(0,0\right)$
1	5	$\left(1,5\right)$
$-1$	$-5$	$\left(-1,-5\right)$

1.	2.
3.	4.
5.	6.
7.	8.
9.	10.
11.	12.

6.3 Graph with Intercepts

Identify the x– and y– Intercepts on a Graph

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

Graph a Line Using the Intercepts

Everyday Math

Writing Exercises

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13. $x+y=4$	14. $x+y=3$
15. $x+y=-2$	17. $x-y=5$
18. $x-y=1$	19. $x-y=-3$
20. $x-y=-4$	21. $x+2y=8$
22. $x+2y=10$	23. $3x+y=6$
24. $3x+y=9$	25. $x-3y=12$
25. $x-3y=12$	27. $4x-y=8$
28. $5x-y=5$	28. $5x-y=5$
30. $2x+3y=6$	31. $3x-2y=12$
32. $3x-5y=30$	33. $y=\frac{1}{3}x+1$
34. $y=\frac{1}{4}x-1$	35. $y=\frac{1}{5}x+2$
36. $y=\frac{1}{3}x+4$	37. $y=3x$
38. $y=-2x$	39. $y=-4x$
40. $y=5x$

41. $-x+5y=10$	42. $-x+4y=8$
43. $x+2y=4$	44. $x+2y=6$
45. $x+y=2$	46. $x+y=5$
47. $x+y=-3$	48. $x+y=-1$
49. $x-y=1$	49. $x-y=1$
51. $x-y=-4$	52. $x-y=-3$
53. $4x+y=4$	54. $3x+y=3$
55. $2x+4y=12$	56. $3x+2y=12$
57. $3x-2y=6$	58. $5x-2y=10$
59. $2x-5y=-20$	60. $3x-4y=-12$
61. $3x-y=-6$	62. $2x-y=-8$
63. $y=\frac{3}{2}x$	64. $y=-4x$
65. $y=x$	66. $y=3x$

69. How do you find the x– intercept of the graph of $3x-2y=6$ ?	70. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $4x+y=-4$ ? Why?
71. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $y=\frac{2}{3}x-2$ ? Why?	72. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $y=6$ ? Why?

1. $\left(3,0\right),\left(0,3\right)$	3. $\left(5,0\right),\left(0,-5\right)$
5. $\left(-2,0\right),\left(0,-2\right)$	7. $\left(-1,0\right),\left(0,1\right)$
9. $\left(6,0\right),\left(0,3\right)$	11. $\left(0,0\right)$
13. $\left(4,0\right),\left(0,4\right)$	15. $\left(-2,0\right),\left(0,-2\right)$
17. $\left(5,0\right),\left(0,-5\right)$	19. $\left(-3,0\right),\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}\left(0,3\right)$
21. $\left(8,0\right),\left(0,4\right)$	23. $\left(2,0\right),\left(0,6\right)$
25. $\left(12,0\right),\left(0,-4\right)$	27. $\left(2,0\right),\left(0,-8\right)$
29. $\left(5,0\right),\left(0,2\right)$	31. $\left(4,0\right),\left(0,-6\right)$
33. $\left(-3,0\right),\left(0,1\right)$	35. $\left(-10,0\right),\left(0,2\right)$
37. $\left(0,0\right)$	39. $\left(0,0\right)$
41.	43.
45.	47.
49.	51.
53.	55.
57.	59.
61.	63.
65.	67. a) $\left(0,1000\right),\left(15,0\right)$ b) At $\left(0,1000\right)$ , he has been gone 0 hours and has 1000 miles left. At $\left(15,0\right)$ , he has been gone 15 hours and has 0 miles left to go.
69. Answers will vary.	71. Answers will vary.

Identify the x– and y– Intercepts on a Graph

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

Graph a Line Using the Intercepts

Key Concepts

Glossary

Practice Makes Perfect

Identify the x– and y– Intercepts on a Graph

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

Graph a Line Using the Intercepts

Everyday Math

Writing Exercises

Answers

Attributions

License

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