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Please complete as soon as possible!               

 Jan 29, 2021
 #1
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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.

 Jan 29, 2021
 #2
avatar+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

 

 

cool cool cool

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

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