Connexions

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x and y-intercepts

Module by: Pradnya Bhawalkar, Kim Johnston

Summary: Finding x & y intercepts

A rational function is a function of the form Rx=pxqx R x p x q x , where p and q are polynomial functions and q0 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:
Problem 1
y=1x+2 y 1 x 2
[ Click for Solution 1 ]
Solution 1
x-intercept: None since 10 1 0
y-intercept: (0, 1212) since 10+2=12 1 0 2 1 2
[ Hide Solution 1 ]
Problem 2
y=1-3x1-x y 1 3 x 1 x
[ Click for Solution 2 ]
Solution 2
x-intercept: (1313,0) since 1-3x=0 1 3 x 0 , -3x=-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
[ Hide Solution 2 ]
Problem 3
y=x2x2+9 y x 2 x 2 9
[ Click for Solution 3 ]
Solution 3
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)
[ Hide Solution 3 ]
Problem 4
y=x+1x-22 y x 1 x 2 2
[ Click for Solution 4 ]
Solution 4
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
[ Hide Solution 4 ]
Problem 5
y=3xx2-x-2 y 3 x x 2 x 2
[ Click for Solution 5 ]
Solution 5
x-intercept: (0,0) since 3x=0 3 x 0 , x=0 x 0
y-intercept: (0,0) since the x-intercept is (0,0)
[ Hide Solution 5 ]
Problem 6
y=1x-3+1 y 1 x 3 1
[ Click for Solution 6 ]
Solution 6
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
[ Hide Solution 6 ]
Problem 7
y=x2-4x+1 y x 2 4 x 1
[ Click for Solution 7 ]
Solution 7
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
[ Hide Solution 7 ]
Problem 8
y=4+5x2+2 y 4 5 x 2 2
[ Click for Solution 8 ]
Solution 8
x-intercept: None since 4+5x2+2=0 4 5 x 2 2 0 , 5x2+2=-4 5 x 2 2 -4 , -4x2-8=5 -4 x 2 8 5 , -4x2=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
[ Hide Solution 8 ]
Problem 9
y=5x-2x-3 y 5 x 2 x 3
[ Click for Solution 9 ]
Solution 9
x-intercept: ( 25 2 5 , 0) since 5x-2=0 5 x 2 0 , 5x-2=0 5 x 2 0 , 5x=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.
[ Hide Solution 9 ]
Problem 10
y=x3-8x2+1 y x 3 8 x 2 1
[ Click for Solution 10 ]
Solution 10
x-intercept: (2,0) since x3-8=0 x 3 8 0 , x-2x2+2x+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
[ Hide Solution 10 ]

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