Question

The negatives are confusing me in this problem. Write the equation of the quadratic function with roots -1 and -7 and a vertex at (-4, 7)

Answer 1

roots -1 and -7

that means x= -1 and x= -7

you can rewrite x= -1 by adding +1 to both sides . you get
x+1 = -1+1 or
x+1= 0

and x= -7 means (x+7)=0

if we multiply (x+1)(x+7) we know this is still equal to 0
(x+1)(x+7)=0
if we multiply by some number, (call it a), we get
a(x+1)(x+7)= a*0 or
0= a(x+1)(x+7)
to make a quadratic equation, let this be equal to y
y= a(x+1)(x+7)

the last step is find a. use the vertex info (-4, 7), which means when x=-4, y = 7

can you do that ?

Answer 2

Um, 7= a(-4 + 1)(-4+7)?

Answer 3

as a first step. simplify

Answer 4

simplify 7= a(-4 + 1)(-4+7)?

Answer 5

yes.

Answer 6

7=a(-3)(3)
7=a(-9)?

Answer 7

one more step. divide both sides by -9

Answer 8

a=-7/9

Answer 9

so one form of the equation is
y= a(x+1)(x+7) with a= -7/9

y = (-7/9) (x+1)(x+7)

if this is multiple choice it may not be easy to match this to your choices, if the choices are in a different form.

Answer 10

The other way to get the equation is use the vertex form :

y= a(x-h)^2 + k
where (h,k) is the vertex
you replace h= -4, and k= 7
y = a(x- (-4))^2 + 7
or
y= a(x+4)^2 + 7

a will be the same as before, but if we did not know it. try the point (-1,0) (one of the roots)

Answer 11

for example,
\[ y= a(x+4)^2 + 7
\]
use (-1,0) to get
\[ 0 = a(-1+4)^2 + 7\\ 0= 9a + 7\\ 9a= -7\\ a= -\frac{7}{9} \]

Answer 12

It is a written question buy that is not the form we are supposed to write the final equation in..

Answer 13

what form do they want ?

Answer 14

well an example is roots being 6 and 8 and vertex (7, -5)
The final equation is
y=5x^2-70x+240

Answer 15

The hardest part for me is writing the final equation correctly, I have done a ton of these problems.

Answer 16

using vertex form for the problem
well an example is roots being 6 and 8 and vertex (7, -5)

y = a( x - 7)^2 -5
use (6,0) to find a:
0= a(6-7)^2 -5
0 = a* (-1)^2 -5
5= 1a
a=5

y= 5(x-7)^2 -5

to change from vertex form to standard form
expand the (x-7)^2= x^2 -14x +49

y= 5(x^2 -14x +49)-5
y = 5x^2 -70 x +(5*49-5)
y = 5x^2 -70 x + 240

Answer 17

How would I come up with the final equation for the other problem in this form?

Answer 18

\[ y= - \frac{7}{9} \left(x+4\right)^2 + 7 \]

expand (x+4)(x+4)= x^2 +8x +16

\[ y= - \frac{7}{9} \left(x^2 +8x +16\right) + 7 \]
I would make the final 7 = 63/9 so we can write this as
\[ y= - \frac{7}{9} \left(x^2 +8x +16\right) + \frac{63}{9}\\ y= \frac{1}{9}\left(-7x^2 -56x-112 +63\right)\]
\[y= \frac{-7x^2 -56x-49}{9}\]

Answer 19

Thank You so much!