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x and y-intercepts

Module by: Pradnya Bhawalkar, Kim Johnston. E-mail the authors

Summary: Finding x & y intercepts

A rational function is a function of the form R⁢x=p⁢xq⁢x R x p x q x , where p and q are polynomial functions and q≠0 q 0 .

The domain is all real numbers except for numbers that make the denominator = 0.

x-intercepts are the points at which the graph crosses the x-axis. They are also known as roots, zeros, or solutions.

To find x-intercepts, let y (or f(x)) = 0 and solve for x. In rational functions, this means that you are multiplying by 0 so to find the x-intercept, just set the numerator (the top of the fraction) equal to 0 and solve for x.

Remember: x-intercepts are points that look like (x,0)

Example 1

For y=x−1x−2 y x 1 x 2 find the x-intercept

The x-intercept is (1,0) since x−1=0 x 1 0 , x=1 x 1

The y-intercept is the point where the graph crosses the y-axis. If the graph is a function, there is only one y-intercept (and it only has ONE name)

To find the y-intercept (this is easier than the x-intercept), let x = 0. Plug in 0 for x in the equation and simplify.

Remember: y-intercepts are points that look like (0,y)

Example 2

For y=x+1x−2 y x 1 x 2 find the y-intercept

The y-intercept is (0, -12 -1 2 ) since 0+10−2=-12 01 02 -1 2

Find the x- and y-intercepts of the following:

Exercise 1

y=1x+2 y 1 x 2

Solution

x-intercept: None since 1≠0 1 0

y-intercept: (0, 1212) since 10+2=12 1 0 2 1 2

Exercise 2

y=1−3⁢x1−x y 1 3 x 1 x

Solution

x-intercept: (1313,0) since 1−3⁢x=0 1 3 x 0 , -3⁢x=-1 -3 x -1 , x=13 x 1 3

y-intercept: (0,1) since 1−3×01−0=1 1 3 0 1 0 1

Exercise 3

y=x2x2+9 y x 2 x 2 9

Solution

x-intercept: (0,0) since x2=0 x 2 0 , x=0 x 0

y-intercept: (0,0) since 0202+9=09=0 0 2 0 2 9 0 9 0 or because the x-intercept is (0,0)

Exercise 4

y=x+1x−22 y x 1 x 2 2

Solution

x-intercept: (-1,0) since x+1=0 x 1 0 , x+1=0 x 1 0 , x=−1 x 1

y-intercept: (0, 1414 ) since 0+10−22=1−22=14 0 1 0 2 2 1 2 2 1 4

Exercise 5

y=3⁢xx2−x−2 y 3 x x 2 x 2

Solution

x-intercept: (0,0) since 3⁢x=0 3 x 0 , x=0 x 0

y-intercept: (0,0) since the x-intercept is (0,0)

Exercise 6

y=1x−3+1 y 1 x 3 1

Solution

x-intercept: (2,0) since 1x−3+1=0 1 x 3 1 0 , 1x−3=−1 1 x 3 1 , −x+3=1 x 3 1 , −x=-2 x -2 , x=2 x 2

y-intercept: (0, 23 2 3 ) since 10−3+1=-13+1=23 1 0 3 1 -1 3 1 2 3

Exercise 7

y=x2−4x+1 y x 2 4 x 1

Solution

x-intercepts: (-2,0), (2,0) since x2−4=0 x 2 4 0 , x2=4 x 2 4 , x=-2 x -2 , x=2 x 2

y-intercept: (0, -4) since 02−40+1=−41=-4 0 2 4 0 1 4 1 -4

Exercise 8

y=4+5x2+2 y 4 5 x 2 2

Solution

x-intercept: None since 4+5x2+2=0 4 5 x 2 2 0 , 5x2+2=-4 5 x 2 2 -4 , -4⁢x2−8=5 -4 x 2 8 5 , -4⁢x2=13 -4 x 2 13 , x2=134 x 2 13 4 , a number squared will never be a negative number, so there is no x-intercept

y-intercept: (0, 132 13 2 ) since 4+502+2=4+52=132 4 5 0 2 2 4 5 2 13 2

Exercise 9

y=5⁢x−2x−3 y 5 x 2 x 3

Solution

x-intercept: ( 25 2 5 , 0) since 5⁢x−2=0 5 x 2 0 , 5⁢x−2=0 5 x 2 0 , 5⁢x=2 5 x 2 , x=25 x 2 5

y-intercept: None since y=5×0−20−3 y 5 0 2 0 3 takes the square root of a negative number.

Exercise 10

y=x3−8x2+1 y x 3 8 x 2 1

Solution

x-intercept: (2,0) since x3−8=0 x 3 8 0 , (x−2)⁢(x2+2⁢x+4)=0 x 2 x 2 2 x 4 0 , x=2 x 2

y-intercept: (0,-8) since 03−802+1=−81=-8 0 3 8 0 2 1 8 1 -8

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