Question

a. write the coordinates of the vertex
b. write the equation of the graph in vertex form
c. write the equation of the graph in factored form
d. write the equation of the graph in standard form
e. verify the roots are at ( x = 1 ) and ( x = -1 ) algebraically
f. find ( f(3) )

Explanation:

Step1: Identify vertex coordinates

The vertex is the highest point of the parabola, visible at $(0,3)$.

Step2: Write vertex form

Vertex form is $y=a(x-h)^2+k$, where $(h,k)=(0,3)$. Substitute a root $(1,0)$ to solve for $a$:
$0=a(1-0)^2+3$
$a=-3$
Equation: $y=-3(x-0)^2+3 = -3x^2+3$

Step3: Write factored form

Roots are $x=-1$ and $x=1$, so factored form is $y=a(x+1)(x-1)$. Use vertex $(0,3)$ to find $a$:
$3=a(0+1)(0-1)$
$3=-a \implies a=-3$
Equation: $y=-3(x+1)(x-1)$

Step4: Write standard form

Expand vertex/factored form:
$y=-3(x^2-1) = -3x^2+3$

Step5: Verify roots algebraically

Set $y=0$ in standard form:
$0=-3x^2+3$
$3x^2=3$
$x^2=1$
$x=\pm1$

Step6: Calculate $f(3)$

Substitute $x=3$ into $f(x)=-3x^2+3$:
$f(3)=-3(3)^2+3 = -27+3$

Answer:

a. $(0, 3)$
b. $y=-3x^2+3$
c. $y=-3(x+1)(x-1)$
d. $y=-3x^2+3$
e. By solving $-3x^2+3=0$, we get $x^2=1$, so $x=1$ and $x=-1$, which matches the given roots.
f. $-24$