5.3 x- and y-Intercepts

Leanne Thompson

5.3 x- and y-Intercepts

When analyzing a graph, we are sometimes interested in what we call the x-intercept or the y-intercept. The x-intercept is the point where the graph intersects the x-axis. The y-intercept is the point where the graph intersects the y-axis.

Example 1

Identify the x-intercept and y-intercept of the graph below.

The x- axis of the planes runs from negative 6 to 6. The y- axis of the planes runs from negative 6 to 6. Figure a shows a straight line crossing the x- axis at the point (3, 0) and crossing the y- axis at the point (0, 6).

Table 5.3.1
Steps	Solution
Identify the x-intercept.	The graph crosses the x-axis at 3, so the x-intercept is the point (3, 0).
Identify the y-intercept.	The graph crosses the y-axis at 6, so the y-intercept is the point (0, 6).

Practice 1

Identify the x-intercept(s) and y-intercept(s) of the graphs below.

a)

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

b)

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).

c)

Parabola shaped (u-shape) graph opening up with x-int at (-3,0) and (3,0) and y-int at (0,-9)

Use your answers from Practice 1 to help you fill in the following blanks.

The y-coordinate of the x-intercept is always equal to ; therefore, x-intercepts will always be in the form (x, 0).

The x-coordinate of the y-intercept is always equal to ; therefore, y-intercepts will always be in the form (0, y).

Recognizing that the x-intercept occurs when y is zero, and that the y-intercept occurs when x is zero, gives us a method for finding the intercepts of a line from its equation.

Table 5.3.2
To find the x-intercepts and y-intercepts from the equation of a line:
x-intercepts Step 1: Set y = 0. Step 2: Solve for x. y-intercepts Step 1: Set x = 0. Step 2: Solve for y.

Example 2

Algebraically determine the intercepts of the relation $2x+y=6$ .

Table 5.3.3
Steps	Solution
x-intercept Set y = 0, and solve for x.	$2x+y=6$ $2x+0=6$ $2x=6$ $x=3$ The x-intercept is (3, 0).
y-intercept Set x = 0, and solve for y.	$2x+y=6$ $2(0)+y=6$ $y=6$ The y-intercept is (0, 6).

Practice 2

Algebraically determine the intercepts of the following equations.

a) $4x-3y=12$

b) $y=x^2-25$

Practice 3

Kimiwan is driving from Thunder Bay to Montreal, a distance of 1 000 miles. The equation $D=\frac{-200}{3}t+1 000$ represents the relationship between the time in hours since Kimiwan left Thunder Bay, t, and the distance he has left to drive, D, in miles.
a) Fill in the table of values.
b) Label each axis and give the graph a title.
c) Plot the points, then draw a line to connect the points.

Time (hr)	Distance from Thunder Bay (mi)
0
6
12
15

Quadrant 1 of a blank coordinate plane. The x-axis has a minimum of 0 and maximum of 16 and a scale of 0.5. The y-axis has a minimum of 0 and maximum of 1200 and a scale of 50.

d) What is the t-intercept of this graph? What does it represent?

e) What is the D-intercept of this graph? What does it represent?

f) How many more miles did Kimiwan still need to drive after 3.5 hr of driving? Round to the nearest whole mile.

g) How many miles had Kimiwan driven after 9 hr of driving?

h) How many hours did it take to get to the midpoint between Thunder Bay and Montreal?

Homework

Identify the x-intercept(s) and y-intercept(s) of the graphs below.

a)
b)
c)

d)
e)
f)
Algebraically determine the intercepts of the equations.

a) $3x+y=12$

b) $3x-4y=12$

c) $x+y=4$

d) $x+y=-2$

e) $x-y=-3$

f) $x+2y=8$

g) $3x+y=6$

h) $x-3y=12$

i) $4x-y=8$

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

k) $y=-4x$

l) $y=\frac{1}{3}x+1$

m) $y=2x$

n) $y=x^2-49$

o) $x^2-36=y$

p) $y=2x^2-50$

q) $4x^2-324=y$

r) $x^2-100=y$
Lila filled up the gas tank of her truck and headed out on a road trip. The equation $g=\frac{-4}{75}d+16$ represents the relationship between the number of miles Lila has driven since filling up, d, and the number of gallons of gas in the truck’s gas tank, g.
a) Fill in the table of values.
b) Label each axis and give the graph a title.
c) Plot the points, draw a line to connect the points.

Distance (mi) Gas in Gas Tank (gal)

0

150

300

d) What is the d-intercept of this graph? What does it represent?

e) What is the g-intercept of this graph? What does it represent?

f) How much gas was left in the tank at the midpoint of the trip?

g) How much gas was left in the tank after driving 20 miles? Round to the nearest tenth.

h) If Lila does not want the amount of gas in the tank to go below 3 gallons, after how many miles is the latest she should fill up the tank?

i) If Lila has 10 gallons in the tank, how many miles has she driven?

Answers

1.

a) (3, 0), (0, 3)	b) (0, 0), (0, 0)	c) (–1, 0), (1, 0), (0, 1)
d) (6, 0), (0, 3)	e) (0, 0), (0, 0), (0, 1)	f) (0, 0), (0, 0)

2.

a) (4, 0), (0, 12)	b) (4, 0), (0, –3)	c) (4, 0), (0, 4)
d) (–2, 0), (0, –2)	e) (–3, 0), (0, 3)	f) (8, 0), (0, 4)
g) (2, 0), (0, 6)	h) (12, 0), (0, –4)	i) (2, 0), (0,–8)
j) (–10, 0), (0, 2)	k) (0, 0), (0,0 )	l) (–3, 0), (0, 1)
m) (0, 0), (0, 0)	n) (–7, 0), (7, 0), (0, –49)	o) (–6, 0), (6, 0), (0, –36)
p) (–5, 0), (5, 0), (0, –50)	q) (–9, 0), (9, 0), (0, –324)	r) (–10, 0), (10, 0), (0, –100)

3.

a)

Distance (mi)	Gas in Gas Tank (gal)	(x, y)
0	16	(0, 16)
150	8	(150, 8)
300	0	(300, 0)

d) (300, 0). There is no gas left in the tank, and Lila has driven 300 miles.
e) (0, 16). Lila has driven 0 miles, and there are 16 gallons left in the tank.
f) 8 gal
g) 14.9 gal
h) 243.75 mi
i) 112.5 mi

b) and c)

Quadrant 1 of a blank coordinate plane. The x-axis has a minimum of 0 and maximum of 350 and a scale of 12.5. The y-axis has a minimum of 0 and maximum of 18 and a scale of 0.5. With equation g=-4/75d+16 graphed.

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resource:

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

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