Find equation of parabola give aos and y-intercept

Question

I know how to find the parabola quadratic equation given the roots. However, in this problem I'm given the y-intercept of (0,3) {So, now I know C value} and the axis of symmetry of x=-3/8. From this I know that 3a=4b. But, it seems I need one more point, since I have 3 unknowns, I'd need 3 INDEPENDENT equations.

So, I said, ah, I know another point, it's on the other side of the aos. So, I used (-3/4, 3). However, when I plug this in, it appears to not be an independent equation. So, I'm left with just c=3 and 3a=4b. How, do I work this from here?

Answer 1

The symetry axis gives the point where the derivative is zero.

let $y=f(x)=ax^2+bx+c$ be the equation of the parabola.

the derivative is $2ax+b$ and gives

$2a\frac{-3}{8}+b=0$ or

$3a=4b$

the point $(0,3)$ satisfy the condition

$3=c$.

finally, we get the expression

$y=f(x)=ax^2+\frac{3a}{4}x+3$.

you have an infinite number of solutions. try it graphically.

Answer 2

The point you propose to use, (-3/4,3) just follows from symmetry of the parabola about x = -3/8. So, you are not actually fixing a degree of freedom of the parabola. In a sense, the parabola can slide up and down. Let me give 2 examples, to illustrate. Consider, y = $64(x-3/8)^2/3$ which follows all the properties above, and y = $64(x+3/8)^2-6$ which again satisfies the properties. So, the solution to the question will not be a parabola, it will be a family of parabolas, $y_t(x) = t(x+3/8)^2+3-9t/64$.