Math Homework NEED HELP!?!?!?

Question

Find the formula for a parabola with its vertex at (2,7) and with a second derivative of -4.

Find the formula for a function of the form y = A\sin(Bx) + C with a maximum at (3,6), a minimum at (9,-4), and no critical points between these two points.

Find the formula for a function of the form y = be^{-(x-a)^2/2} with its maximum at the point (0,8).

Find the formula for a quartic polynomial whose graph is symmetric about the y-axis and has local maxima at (-3,5) and (3,5) and a y-intercept of 2.

I am so confused. My math professor is horrible. I have no idea how to do these problems. I have looked in my book and I have asked other people and we cant figure out how to do these. PLEASE EXPLAIN!!! Thank you so mcuh!

Answer 1

1.) The equation would have this form:
y - 7 = p(x - 2)^2
y = p(x - 2)^2 + 7
y ' = 2p(x - 2)
y ' = 2px - 4p
y '' = 2p = -4
p = -2
y - 7 = -2(x - 2)^2

2.) 3 ===> π/2
9 ===> 3π/2
sin(πx/6)
(6 + (-4))/2 = 1
sin(πx/6) + 1
(6 - (-4))/2 = 5
The formula is:
y =5(sin(πx/6)) + 1

3.)

4.) Symmetry with respect to the y axis ===> even function. That is, no odd powers of x.
y ' has zeros at x = 3, 0, -3 :
y ' = x( x + 3)(x - 3)
y ' = x(x^2 - 9)
y ' = x^3 - 9x
Integrate and note: the first 2 terms may have a multiplier, say A. This multiplier must be the same for both terms otherwise when you take y ' again it won't have factors of (x + 3) and (x - 3).
y = A[(x^4)/4 - (9x^2)/2] + C
(0, 2) ===>2 = C
(3, 5) ===> 5 = A(81/4 - 81/2) + C
5 = (-81/4)A + 2
3 = (-81/4)A
4 = -27A
A = -4/27
The formula is:
y = -4/27[(x^4)/4 - (9x^2)/2] + 2
y = (-1/27)x^4 + (2/3)x^2 + 2