#1**+1 **

There may be quicker ways to do this question, but this is how I approached it.

The vertex form of a parabola is written \(f(x)=a(x-h)^2+k\) where \((h,k)\) are the coordinates of the vertex of the parabola. The coordinates are given as (2, -54), so h = 2 and k = -54.

\(f(x) = a(x-2)^2-54\\ f(x)=a(x^2-4x+4)-54\\ f(x)=ax^2-4ax+4a-54\)

From the form \(f(x)=\text{__}(x-\text{__})(x-8)\), it is clear that (x-8) is a factor of \(f(x)\), which also implies that \(f(8)=0\).

\(f(8)=64a-32a+4a-54\\ 0=36a-54\\ a=\frac{3}{2} \)

Now we know the value for a, so we should be able to factor \(f(x)\) into a form that looks like \(f(x)=\text{__}(x-\text{__})(x-8)\) now.

\(f(x)=\frac{3}{2}x^2-4*\frac{3}{2}x+4*\frac{3}{2}-54\\ f(x)=\frac{3}{2}x^2-6x-48\\\)

To ease the factoring process, I will factor out 3/2 from all the terms.

\(f(x)=\frac{3}{2}(x^2-4x-32)\\\)

Now, factoring is significantly easier than before.

\(f(x)=\frac{3}{2}(x+4)(x-8)\)

From the factored form of the f(x), the x-intercepts are at (-4, 0) and (8, 0). Also, assuming that the value of a corresponds to the leading coefficient of this quadratic, then a = 3/2.

Good luck on your solving adventures.

Guest Jan 29, 2021

#2**+1 **

One x intercept is x = 8

Because of symmetry, the other x intercept will be 2 - (8 - 2) = 2 -6 = -4

And the point (2, -54) is on the graph so we can solve for "a" thusly

-54 = a ( 2 - - 4) ( 2 - 8)

-54 = a ( 6)(-6)

-54 = a ( -36)

a = 54/36 = 3/2

Here's a graph : https://www.desmos.com/calculator/48496ve0xe

CPhill Jan 29, 2021