Determine, as exact values, the x and y-intercepts of the graph of the following functions
a) f(x) = -3(x+2)^2+3
b) f(x) = 2(x-6)^2-6
c) f(x) = -1/4(x+1)^2+5
For the y-intercept let x = 0 in each of the given functions, so that is easy
I will do a)
f(x) = -3(x+2)^2+3
f(0) = -3(0+2)^2 + 3 = -9 , so the y-intercept is -9
For the x-intercept, let f(x) = 0 or y = 0 if written in that form
a)
-3(x+2)^2+3 = 0
-3(x+2)^2 = -3
(x+2)^2 = 1
x+2 = ± √1 = ± 1
x = -1 or x = -3
verification:
www.wolframalpha.com/input/?i=plot+f(x)+%3D+-3(x%2B2)%5E2%2B3+from+-4+to+1
To find the x-intercepts of a function, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y.
a) For f(x) = -3(x+2)^2+3:
To find the x-intercept, set y = 0:
0 = -3(x+2)^2 + 3
Divide both sides by -3:
0 = (x+2)^2 - 1
(x+2)^2 = 1
Take the square root of both sides:
x+2 = ±1
For x+2 = 1, solve for x:
x = 1 - 2
x = -1
For x+2 = -1, solve for x:
x = -1 - 2
x = -3
To find the y-intercept, set x = 0:
f(0) = -3(0+2)^2 + 3
f(0) = -3(2)^2 + 3
f(0) = -12 + 3
f(0) = -9
Therefore, the x-intercepts are x = -3 and x = -1, and the y-intercept is y = -9.
b) For f(x) = 2(x-6)^2-6:
To find the x-intercept, set y = 0:
0 = 2(x-6)^2 - 6
Divide both sides by 2:
0 = (x-6)^2 - 3
(x-6)^2 = 3
Take the square root of both sides:
x-6 = ±√3
For x-6 = √3, solve for x:
x = 6 + √3
For x-6 = -√3, solve for x:
x = 6 - √3
To find the y-intercept, set x = 0:
f(0) = 2(0-6)^2 - 6
f(0) = 2(36) - 6
f(0) = 72 - 6
f(0) = 66
Therefore, the x-intercepts are x = 6 + √3 and x = 6 - √3, and the y-intercept is y = 66.
c) For f(x) = -1/4(x+1)^2+5:
To find the x-intercept, set y = 0:
0 = -1/4(x+1)^2 + 5
1/4(x+1)^2 = 5
Multiply both sides by 4:
(x+1)^2 = 20
Take the square root of both sides:
x+1 = ±√20
For x+1 = √20, solve for x:
x = -1 + √20
For x+1 = -√20, solve for x:
x = -1 - √20
To find the y-intercept, set x = 0:
f(0) = -1/4(0+1)^2 + 5
f(0) = -1/4(1)^2 + 5
f(0) = -1/4 + 5/1
f(0) = -1/4 + 20/4
f(0) = 19/4
Therefore, the x-intercepts are x = -1 + √20 and x = -1 - √20, and the y-intercept is y = 19/4.