Question

water and laser show
at an outdoor shopping mall, there is a water display and a laser display that can be represented by a quadratic function.

• the function for the path of the water display is $f(x)=-\frac{1}{2}x^2 + 4x$ where $f(x)$ represents the height of the water in feet after $x$ seconds traveling in the display.
• the function for the path of the laser display is $g(x)=-\frac{1}{32}(x - 16)^2 + 18$ where $g(x)$ represents the height of the laser in feet after $x$ seconds traveling in the display.

question: 1
which statement represents the height of the water after 3 seconds?
a. $f(3)=7\frac{1}{2}$
b. $f\left(7\frac{1}{2}\
ight)=3$
c. $f(3)=9\frac{3}{4}$
d. $f\left(9\frac{3}{4}\
ight)=3$

question: 2
which characteristics are true of the water display graph, $f(x)$?
a. the water decreases, then increases, resulting in a minimum value at $x = 4$.
b. the water decreases, then increases, resulting in a minimum value at $x = 8$.
c. the water increases, then decreases, resulting in a maximum value at $x = 4$.
d. the water increases, then decreases, resulting in a maximum value at $x = 8$.

Explanation:

Question 1

Step1: Substitute \( x = 3 \) into \( f(x) \)

The function for the water display is \( f(x)=-\frac{1}{2}x^{2}+4x \). We substitute \( x = 3 \) into this function.

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Step1: Analyze the quadratic function form

The function \( f(x)=-\frac{1}{2}x^{2}+4x \) is a quadratic function in the form \( y = ax^{2}+bx + c \), where \( a=-\frac{1}{2}\), \( b = 4 \), and \( c = 0 \). Since \( a=-\frac{1}{2}<0 \), the parabola opens downwards, which means the function has a maximum value.

Step2: Find the vertex (time of maximum height)

For a quadratic function \( y=ax^{2}+bx + c \), the x - coordinate of the vertex (which gives the time at which the maximum or minimum occurs) is given by \( x=-\frac{b}{2a} \).
Substitute \( a = -\frac{1}{2} \) and \( b = 4 \) into the formula:

$$ x=-\frac{4}{2\times(-\frac{1}{2})}=-\frac{4}{-1}=4 $$

Since the parabola opens downwards, the function increases until \( x = 4 \) (reaches the maximum) and then decreases after \( x = 4 \). So the water increases, then decreases, resulting in a maximum value at \( x = 4 \).

Answer:

A. \( f(3)=7\frac{1}{2} \)

Question 2