Answer:
B) Both functions have the same domain.
C) Both functions have the same x-intercepts.
D) Both functions are decreasing on the interval (−∞, −2).
Step-by-step 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](https://tex.z-dn.net/?f=g%28x%29%3D2x%5E2%2B8x%2B6)
Find the vertex
Factor the leading coefficient
![g(x)=2(x^2+4x)+6](https://tex.z-dn.net/?f=g%28x%29%3D2%28x%5E2%2B4x%29%2B6)
Complete the square
![g(x)=2(x^2+4x+4)+6-8](https://tex.z-dn.net/?f=g%28x%29%3D2%28x%5E2%2B4x%2B4%29%2B6-8)
![g(x)=2(x^2+4x+4)-2](https://tex.z-dn.net/?f=g%28x%29%3D2%28x%5E2%2B4x%2B4%29-2)
Rewrite as perfect squares
![g(x)=2(x+2)^2-2](https://tex.z-dn.net/?f=g%28x%29%3D2%28x%2B2%29%5E2-2)
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](https://tex.z-dn.net/?f=0%3D2%28x%2B2%29%5E2-2)
![2(x+2)^2=2](https://tex.z-dn.net/?f=2%28x%2B2%29%5E2%3D2)
![(x+2)^2=1\\x+2=\pm1\\x=-2\pm1](https://tex.z-dn.net/?f=%28x%2B2%29%5E2%3D1%5C%5Cx%2B2%3D%5Cpm1%5C%5Cx%3D-2%5Cpm1)
so
The roots or x-intercepts are
x=-1 and x=-3
<u><em>Verify each statement</em></u>
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