Quadratic Functions

Question

I'm .1 points away from getting full credit on my online homework, but this last problem is killing me.

 

Define the formula for a parabola (a quadratic function) that has horizontal intercepts (roots) at x=−3 and x=3 and passes through the point (0,−6) .

 

So if I know my roots, we could go back and rewrite it like this ( or I think so)

 

f(x)=a(x-r)(x-s)

 

r and s represent our x-intercepts or roots so

 

f(x)=a(x+3)(x-3)

 

I know our Y-intercept is going to be -6, which should be  C in our quadratic function ax^2+bx+c, but when I FOIL out (x+3)(x-3) I get x^2-9 which isn't right.

 

I could be doing this completely wrong, but I figured I'd let you know my thought process so you can correct me. Any help is much appreciated.

 

 

Answer 1

Hi Justin, start with y = a(x - m)(x - n). The x-intercepts are (-3, 0) and (3, 0). The y-intercept is (0, -6).

 

First find the value of a using the x and y-components of the x and y-intercept points.

 

-6 = a * (0 - (-3)) * (0 -3)

 

-6 = a * (3) * (-3)

 

-6 = a (-9)

 

Divide both sides by -9 gives:

 

6/9 = a, note the two minus signs cancel out.

 

a = 2/3 (reduce 6/9)

 

Then:

 

y = 2/3 * (x - (-3)) * (x - 3)

 

y = 2/3 * (x + 3) * (x - 3)

 

y = 2/3 * (x² - 9)

 

If you distribute the 2/3 you get:

 

y = 2/3(x²) - 6

 

Questions?

Answer 2

f(x) = a(x+3)(x-3)

f(x) = a(x²-9)

 

The y-intercept is (0,-6)

 

So when x=0, y=-6

Plug these numbers in and solve for "a"

 

-6 = a[(0)²-9]

-6 = a(-9)

-6 = -9a

-6/-9 = a

6/9 = a

2/3 = a

 

Now that you know what "a" is, plug it back into f(x) = a(x²-9)

 

f(x) = 2/3(x²-9)

 

Distribute/FOIL

 

f(x) = 2⁄3 x²-6  <----Final Answer

 

Its a bit difficult to write but the x² is either in the numerator or on the side of the fraction, NOT in the denominator. You probably knew this I just wanted to clarify just in case. Only .1 points away. Great job, Justin!

 

Answer 3

The y intercept is the value at f(0).

 

You've done most of the heavy lifting so far, and just need to plug in 0 to your function f(x)=a(x+3)(x-3) and set f(0)=-6 in order to solve for a.

 

To check your work, I will provide my work below:

 

f(0)=a(3)(-3)=-9a=-6

a=6/9=2/3

 

Therefore after expanding

f(x) = 2⁄3x²-6 

 

I hope to answer more of your questions soon.

-cldp