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iwanttogotostanford · Answer

@Cardinal_Carlo

anonymous · Answer

A downward quadratic function is when its curve resembles a frown face. This also means that the a-value in f(x) = ax^2 is negative (or <0 as mathematically noted). So:
$$f(x) = x ^{2} ~~~is ~~positive ~(+) ~~or ~~upward \left( \smile \right)$$
To make this quadratic function negative, we just have to multiply to a negative value which in our case is k = -1.
$$f(x) \times k = x ^{2} \times \left( -1 \right) = -x ^{2}$$
$$f(x) = -x ^{2}$$
$$f(x) ~~~is ~~now ~~negative \left( - \right) ~~or ~~downward ~~(\frown)$$

iwanttogotostanford · Answer

@Cardinal_Carlo I'm still confused, would it be f(kx) then?

anonymous · Answer

Well if we did f(kx), we would get:
$$f(x) = x ^{2}$$
$$f(kx) = \left( kx \right)^{2} = \left( -x \right)^{2} = x ^{2}$$
As  you can see,  -1 will always become a positive 1 after it's squared. This is not our answer.

iwanttogotostanford · Answer

oh ok

anonymous · Answer

Your actual answer is kf(x) which is basically the product of k and f(x). From this, we get:
$$kf(x) = \left( -1 \right)x ^{2} = -x ^{2}$$
which is what we need to make f(x) negative. Is this making any sense so far?