Question

1. (3) what is the equation of the parabola with vertex (1, -11) and that passes through (-4, 64)? express the final answer in a standard form.

Explanation:

Step1: Use vertex form of parabola

The vertex form of a parabola is $f(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex. Given vertex $(1,-11)$, substitute $h=1$, $k=-11$:
$f(x)=a(x-1)^2-11$

Step2: Solve for $a$ using given point

Substitute the point $(-4,64)$ (i.e., $x=-4$, $f(x)=64$) into the equation:
$64=a(-4-1)^2-11$
Simplify the term inside the square:
$64=a(-5)^2-11$
$64=25a-11$
Add 11 to both sides:
$64+11=25a$
$75=25a$
Solve for $a$:
$a=\frac{75}{25}=3$

Step3: Expand to standard form

Substitute $a=3$ back into the vertex form:
$f(x)=3(x-1)^2-11$
Expand $(x-1)^2$ using $(x-h)^2=x^2-2hx+h^2$:
$f(x)=3(x^2-2x+1)-11$
Distribute the 3:
$f(x)=3x^2-6x+3-11$
Combine constant terms:
$f(x)=3x^2-6x-8$

Answer:

$f(x)=3x^2-6x-8$