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Jan 29, 2021

#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.

Jan 29, 2021
#2
+120023
+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

Jan 29, 2021
edited by CPhill  Jan 29, 2021
edited by CPhill  Jan 29, 2021
edited by CPhill  Jan 29, 2021
#3
0

I like your use of symmetry to figure out the other x-intercept. I wish I had thought of this ingenuity...

Guest Jan 29, 2021