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
[tex]g(x)=2x^2+8x+6[/tex]
Find the vertex
Factor the leading coefficient
[tex]g(x)=2(x^2+4x)+6[/tex]
Complete the square
[tex]g(x)=2(x^2+4x+4)+6-8[/tex]
[tex]g(x)=2(x^2+4x+4)-2[/tex]
Rewrite as perfect squares
[tex]g(x)=2(x+2)^2-2[/tex]
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
[tex]0=2(x+2)^2-2[/tex]
[tex]2(x+2)^2=2[/tex]
[tex](x+2)^2=1\\x+2=\pm1\\x=-2\pm1[/tex]
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
