Question

F(x) is a quadratic function with x-intercepts at (−1, 0) and (−3, 0). If the range of f(x) is [−4, ∞) and g(x) = 2x2 + 8x + 6, compare f(x) and g(x). Select the statement that is not correct?

A) Both functions have the same vertex.

B) Both functions have the same domain.

C) Both functions have the same x-intercepts.

D) Both functions are decreasing on the interval (−∞, −2).

Answer 1

The answer is A

Explanation:

Answer 2

B) Both functions have the same domain.

C) Both functions have the same x-intercepts.

D) Both functions are decreasing on the interval (−∞, −2).

Explanation:

step 1

Find the vertex of f(x)

we know that

The x-coordinate of the vertex is the midpoint of the roots

we have

x=-1 and x=-3

so

the midpoint is

(-1-3)/2=-2

The y-coordinate of the vertex is -4 (because the range is [−4, ∞))

therefore

The vertex of f(x) is the point (-2,-4)

Is a vertical parabola open upward

The vertex is a minimum

The domain is all real numbers

step 2

we have

`g(x)=2x^2+8x+6`

Find the vertex

Factor the leading coefficient

`g(x)=2(x^2+4x)+6`

Complete the square

`g(x)=2(x^2+4x+4)+6-8`

`g(x)=2(x^2+4x+4)-2`

Rewrite as perfect squares

`g(x)=2(x+2)^2-2`

Is a vertical parabola open upward

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

The domain is all real numbers

The range is the interval [−2, ∞)

Find the x-intercepts

For g(x)=0

`0=2(x+2)^2-2`

`2(x+2)^2=2`

`(x+2)^2=1\\x+2=\pm1\\x=-2\pm1`

so

The roots or x-intercepts are

x=-1 and x=-3

Verify each statement

A) Both functions have the same vertex

The statement is false

The vertex of f(x) is (-2,-4) and the vertex of g(x) is (-2,-2)

B) Both functions have the same domain.

The statement is true

The domain is all real numbers

C) Both functions have the same x-intercepts

The statement is true (see the explanation)

D) Both functions are decreasing on the interval (−∞, −2)

The statement is true

Because the x-coordinate of the vertex is the same in both functions, and both functions open upward