Question

The function ƒ is defined by ƒ(x) = (x + 3)(x + 1). The graph of ƒ in the xy-plane is a parabola. Which interval contains the x-coordinate of the vertex of the graph of ƒ?

Answer 1

Answer:

All the intervals that contain the number $-2$ are solution of the problem

Step-by-step explanation:

we know that

The equation of a vertical parabola into vertex form is equal to

$f(x)=a(x-h)^{2}+k$

where

(h,k) is the vertex of the parabola

In this problem we have

$f(x)=(x+3)(x+1)$

Convert to vertex form

$f(x)=x^{2}+x+3x+3$

$f(x)=x^{2}+4x+3$

Group terms that contain the same variable, and move the constant to the opposite side of the equation

$f(x)-3=x^{2}+4x$

Complete the square. Remember to balance the equation by adding the same constants to each side.

$f(x)-3+4=x^{2}+4x+4$

$f(x)+1=x^{2}+4x+4$

Rewrite as perfect squares

$f(x)+1=(x+2)^2$

$f(x)=(x+2)^2-1$ -------> equation in vertex form

The vertex is the point $(-2,-1)$

The x-coordinate of the vertex is $-2$