Answer:
See below ~
Step-by-step explanation:
Finding the x-intercepts :
Finding the vertex :
Answer:
x-intercepts: x = 1 and x = -3
vertex: (-1, -36)
Step-by-step explanation:
x-intercepts are when y = 0
⇒ 9(x + 3)(x - 1) = 0
Divide both sides by 9:
⇒ (x + 3)(x - 1) = 0
Therefore:
⇒ (x + 3) = 0 ⇒ x = -3
⇒ (x - 1) = 0 ⇒ x = 1
Therefore the x-intercepts are x = 1 and x = -3
Vertex form: [tex]y=a(x-h)^2+k[/tex] where (h, k) is the vertex
Completing the square
[tex]\begin{aligned}y & =ax^2+bx+c\\& =a\left(x^2+\dfrac{b}{a}x\right)+c\\\\& =a\left(x^2+\dfrac{b}{a}x+\left(\dfrac{b}{2a}\right)^2\right)+c-a\left(\dfrac{b}{2a}\right)^2\\\\& =a\left(x-\left(-\dfrac{b}{2a}\right)\right)^2+c-\dfrac{b^2}{4a}\end{aligned}[/tex]
Therefore:
[tex]y=9(x+3)(x-1)[/tex]
[tex]\implies y=9x^2+18x-27[/tex]
Completing the square to rewrite the equation in vertex form:
[tex]\begin{aligned}y & =9x^2+18x-27\\& =9\left(x^2+\dfrac{18}{9}x\right)-27\\\\& =a\left(x^2+\dfrac{18}{9}x+\left(\dfrac{18}{2(9)}\right)^2\right)-27-9\left(\dfrac{18}{2(9)}\right)^2\\\\& =9\left(x-\left(-\dfrac{18}{2(9)}\right)\right)^2-27-\dfrac{18^2}{4(9)}\\\\& =9(x+1)^2-36\end{aligned}[/tex]
Therefore, the vertex is (-1, -36)
