Question

compare ( g(x) = -1.3x^2 - 9.1x - 15 ) to ( f ), shown in the graph. which function has a greater maximum value?
choose the correct answer below
a. ( f(x) ); the ( y )-coordinate of the vertex of ( f(x) ) is greater than the ( y )-coordinate of the vertex of ( g(x) )
b. ( g(x) ); the ( x )-coordinate of the vertex of ( g(x) ) is greater than the ( x )-coordinate of the vertex of ( f(x) )
c. ( g(x) ); the ( y )-coordinate of the vertex of ( g(x) ) is greater than the ( y )-coordinate of the vertex of ( f(x) )
d. ( f(x) ); the ( x )-coordinate of the vertex of ( f(x) ) is greater than the ( x )-coordinate of the vertex of ( g(x) )

Explanation:

Step1: Find vertex x of g(x)

For quadratic $ax^2+bx+c$, $x=-\frac{b}{2a}$.
Here $a=-1.3$, $b=-9.1$, so:
$$x=-\frac{-9.1}{2(-1.3)} = -\frac{9.1}{2.6} = -3.5$$

Step2: Find vertex y of g(x)

Substitute $x=-3.5$ into $g(x)$:
$$g(-3.5)=-1.3(-3.5)^2 -9.1(-3.5)-15$$
$$=-1.3(12.25)+31.85-15$$
$$=-15.925+31.85-15=0.925$$

Step3: Analyze f(x) vertex

The graph of $f(x)$ (a downward-opening parabola) has a vertex with a y-coordinate greater than 0.925 (visually, its peak is above the y-value of 0.925 on the axis).

Step4: Compare maximum values

The maximum of a downward parabola is its vertex y-coordinate. Since f(x)'s vertex y > 0.925 = g(x)'s vertex y, f(x) has a greater maximum.

Answer:

A. f(x); The y-coordinate of the vertex of f(x) is greater than the y-coordinate of the vertex of g(x).