Question

Determine the function f(x), where f(x) is a quadratic function with zeros f(2)=f(-4)=0 and y intercept of 6.

Answer 1

start with \(f(x)=a(x-2)(x+4)\) because that will have the appropriate zeros
then solve for \(a\) by saying \(f(0)=6=a(-2)(4)\)

Answer 2

why does that function have the appropriate zeros? I don't really understand this...

Answer 3

ok you know if you want to find the zeros of some quadratic the first step would be to try and factor right?

Answer 4

right, i understand that

Answer 5

like if the zeros of a function were 3 and 2, wouldn't f(x) = (x-3)(x+2)?

Answer 6

no it would be \((x-3)(x-2)\)

Answer 7

because if you replace \(x\) by 3 you get zero for the first factor, and if you replace x by 2 you get zero for the second factor

Answer 8

i meant to say x-2, sorry, i mistyped that. that part i do understand.

Answer 9

so this is the same idea, but in reverse
instead of factoring to find the zeros, you are given the zeros and so you know how to factor

Answer 10

it is called the "factor theorem" if \(r\) is a zero of a polynomial \(p(x)\) then you can factor as
\[p(x)=(x-r)q(x)\] for some polynomial \(q\) in other words it works both ways
factor, find the zeros
have the zeros, factor

Answer 11

but why does it say f(2)=f(-4)? how can f(2) equal f(-4)?

Answer 12

oh i think you are confused by the wording of the question
you are TOLD that \(f(2)=0\) and that \(f(-4)=0\)

Answer 13

so they could have said
"the zeros are at \(2\) and \(-4\)"
or
"the \(x\) intercepts are at \(2\) and \(-4\)"
or
"\(f(2)=0\) and also \(f(-4)=0\)" but to be succinct they just wrote
\[f(2)=f(-4)=0\]

Answer 14

I'm so sorry I disappeared, my internet went down and I had to restart my computer to fix it.

Answer 15

I was confused byt he wording, thank you. If you're still there, I have another question...using your equation, I got a = -3/4
The answer provided to me says y = -3/4x^2 - 3/2x + 6

Answer 16

and I am a bit confused on how they got the 3/2 part

Answer 17

i think they multiplied out

Answer 18

you find \(a(-2)(4)=6\) so \(a=-\frac{3}{4}\) which looks good to me

Answer 19

then
\[-\frac{3}{4}(x-2)(x+4)=-\frac{3}{4}(x^2+2x-8)=-\frac{3}{4}x^2-\frac{3}{2}x+6\]

Answer 20

thank you! that was very helpful. again, i'm sorry i abandoned you for a while!

Answer 21

np, i went away too