Question

compare the algebraically expressed function ( f(x) = -\frac{3}{2}x^2 + 8x ) to the function shown in the graph to determine which statement is true.
a the algebraic function has a greater maximum value.
b the algebraic function has a lower minimum value.
c the graphed function has a greater maximum value.
d the graphed function has a lower minimum value.

Explanation:

Step1: Analyze the algebraic function

The function is \( f(x) = -\frac{3}{2}x^2 + 8x \). For a quadratic function \( ax^2 + bx + c \), the x - coordinate of the vertex (where maximum/minimum occurs) is \( x = -\frac{b}{2a} \). Here, \( a = -\frac{3}{2} \), \( b = 8 \). So \( x = -\frac{8}{2\times(-\frac{3}{2})}=\frac{8}{3}\approx2.67 \).
To find the maximum value, substitute \( x=\frac{8}{3} \) into \( f(x) \):
\( f(\frac{8}{3})=-\frac{3}{2}(\frac{8}{3})^2 + 8\times\frac{8}{3}=-\frac{3}{2}\times\frac{64}{9}+\frac{64}{3}=-\frac{32}{3}+\frac{64}{3}=\frac{32}{3}\approx10.67 \).

Step2: Analyze the graphed function

From the graph, the parabola opens downward (so it has a maximum). The vertex seems to be around \( y = 4 \) (by visual inspection of the graph's peak).

Step3: Compare maximum values

The algebraic function's maximum is \( \frac{32}{3}\approx10.67 \), and the graphed function's maximum is around 4. So the algebraic function has a greater maximum value.

Answer:

A. The algebraic function has a greater maximum value.