SOLUTION: Graph f(x)=x^4-4x^2 First find the x-intercepts then evaluate f(-3),f(-1),f(1),f(3)

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Question 113794: Graph f(x)=x^4-4x^2

First find the x-intercepts then evaluate f(-3),f(-1),f(1),f(3)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29=x%5E4-4x%5E2 Start with the given function


f%28x%29=x%5E2%28x%5E2-4%29 Factor out the GCF x%5E2


f%28x%29=x%5E2%28x-2%29%28x%2B2%29 Factor x%5E2-4 using the difference of squares to get %28x-2%29%28x%2B2%29


Now to find the x-intercepts, set f%28x%29 equal to zero and solve for x


x%5E2%28x-2%29%28x%2B2%29=0

Break up the equation using the zero product property
x%5E2=0 , x-2=0 or x%2B2=0


x=0 , x=2 or x=-2 Solve for x in each case


So our solutions are x=0 , x=2 or x=-2 which means the x-intercepts are (0,0), (2,0), and (-2,0)


Notice if we graph, we get

+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E4-4x%5E2%29+

and we can see that the x-intercepts are (0,0), (2,0), and (-2,0)


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Let's evaluate f%28-3%29
f%28x%29=x%5E4-4x%5E2 Start with the given function.


f%28-3%29=%28-3%29%5E4-4%28-3%29%5E2 Plug in x=-3. In other words, replace each x with -3.


f%28-3%29=%2881%29-4%28-3%29%5E2 Evaluate %28-3%29%5E4 to get 81.


f%28-3%29=%2881%29-4%289%29 Evaluate %28-3%29%5E2 to get 9.


f%28-3%29=%2881%29-36 Multiply -4 and 9 to get -36


f%28-3%29=45 Now combine like terms


-----------Now let's evaluate another value---------
Let's evaluate f%28-1%29
f%28x%29=x%5E4-4x%5E2 Start with the given function.


f%28-1%29=%28-1%29%5E4-4%28-1%29%5E2 Plug in x=-1. In other words, replace each x with -1.


f%28-1%29=%281%29-4%28-1%29%5E2 Evaluate %28-1%29%5E4 to get 1.


f%28-1%29=%281%29-4%281%29 Evaluate %28-1%29%5E2 to get 1.


f%28-1%29=%281%29-4 Multiply -4 and 1 to get -4


f%28-1%29=-3 Now combine like terms


-----------Now let's evaluate another value---------
Let's evaluate f%281%29
f%28x%29=x%5E4-4x%5E2 Start with the given function.


f%281%29=%281%29%5E4-4%281%29%5E2 Plug in x=1. In other words, replace each x with 1.


f%281%29=%281%29-4%281%29%5E2 Evaluate %281%29%5E4 to get 1.


f%281%29=%281%29-4%281%29 Evaluate %281%29%5E2 to get 1.


f%281%29=%281%29-4 Multiply -4 and 1 to get -4


f%281%29=-3 Now combine like terms


-----------Now let's evaluate another value---------
Let's evaluate f%283%29
f%28x%29=x%5E4-4x%5E2 Start with the given function.


f%283%29=%283%29%5E4-4%283%29%5E2 Plug in x=3. In other words, replace each x with 3.


f%283%29=%2881%29-4%283%29%5E2 Evaluate %283%29%5E4 to get 81.


f%283%29=%2881%29-4%289%29 Evaluate %283%29%5E2 to get 9.


f%283%29=%2881%29-36 Multiply -4 and 9 to get -36


f%283%29=45 Now combine like terms