We must compare two quadratic functions.
1) First, we find the equation of the quadratic function g(x).
The general quadratic equation has the form:
[tex]g(x)=a\cdot x^2+b\cdot x+c.[/tex]
We must find the values of a, b and c. To do that, we take three points from the table:
[tex]\begin{gathered} g(-1)=-3=a\cdot(-1)^2+b\cdot(-1)+c\Rightarrow-3=a-b+c, \\ g(0)=-8=a\cdot0^2+b\cdot0+c\Rightarrow c=-8, \\ g(1)=-15=a\cdot1^2+b\cdot1+c\Rightarrow-15=a+b+c\text{.} \end{gathered}[/tex]
Replacing the value c = -8 in the first and second equation, we have:
[tex]\begin{gathered} -3=a-b-8\Rightarrow a-b=5, \\ -15=a+b-8\Rightarrow a+b=-7. \end{gathered}[/tex]
Summing the equations, we have:
[tex]2a=5-7=-2\Rightarrow a=-1.[/tex]
Replacing the value a = -1 in one of the equations above, we get:
[tex]b=-7-a=-7+1=-6.[/tex]
So we have the following equation for g(x):
[tex]g(x)=-x^2-6x-8.[/tex]
2) We plot the graph of f(x) and g(x):
A. From the graphs, we see that:
• f(x) is symmetric respect to x = 0,
,
• g(x) is symmetric respect to x = -3.
So statement A is false.
B. From the graphs, we see that:
• the minimum value of f(x) is f(0) = 3, the maximum value of g(x) is g(-3) = 1.
So statement B is true.
C. From the graphs, we see that:
• f(-2) = 7 ≠ -1,,
,
• g(-2) = 0.
We see that not all the data of statement C is true.
D. From the graphs, we see that:
• f(x) is symmetric respect to x = 0,
,
• g(x) is symmetric respect to x = -3.
So the axis of symmetry of g(x) is to the left, and statement D is false.
Answer
B. The minimum value of f(x) is greater than the maximum value of g(x) because f(0) > g(-3).
