Answer:
[tex]y=\frac{1}{4}(x+1)^2-9[/tex]
Step-by-step explanation:
Method 1
we know that
The equation of a vertical parabola in vertex form is equal to
[tex]y=a(x-h)^2+k[/tex]
where
a is the leading coefficient
(h,k) is the vertex
we have
(h,k)=(-1,-9)
substitute
[tex]y=a(x+1)^2-9[/tex]
Remember that
one root is (-7,0)
substitute and solve for a
[tex]0=a(-7+1)^2-9[/tex]
[tex]0=a(-6)^2-9[/tex]
[tex]0=36a-9[/tex]
[tex]a=\frac{1}{4}[/tex]
therefore
[tex]y=\frac{1}{4}(x+1)^2-9[/tex]
Method 2
I use the fact that the roots are the same distance from the vertex
the distance from the given root to the vertex is equal to
6 units
so
If one root is x=-7
then the other root is
x=-1+6=5
The general equation of the quadratic equation is equal to
[tex]y=a(x+7)(x-5)[/tex]
we have the vertex (-1,-9)
substitute the value of x and the value of y and solve for a
[tex]-9=a(-1+7)(-1-5)[/tex]
[tex]-9=a(6)(-6)[/tex]
[tex]-9=-36a[/tex]
[tex]a=\frac{1}{4}[/tex]
[tex]y=\frac{1}{4}(x+7)(x-5)[/tex]
so
Expanded the equation, complete the square and rewrite as vertex form
