Chapter 5.3, Problem 79SR

(a)

To determine

To Find: The comparison of the graph of the function to the parent function.

Answer to Problem 79SR

The graph opens downward and it is narrower than the parent function and the vertex is $\left(1,8\right)$ .

Explanation of Solution

Given:

The water lands 3 feet away from the jet.

The given graph is shown in Figure 1

  

Figure 1

Calculation:

The graph shows that the vertex of the parabola is (1,8) and then use the parabola form to solve the required result $h\left(d\right)=a{\left(d-1\right)}^2+8$ .

Consider the graph passes through the point (3,0). Thus, substitute the values in the parabola form as,

  $\begin{array}{l}0=a{\left(3-1\right)}^2+8\\ a=-2\end{array}$

Then, the required function is,

  $\begin{array}{c}h\left(d\right)=-2{\left(d-1\right)}^2+8\\ =-2d^2+4d+6\end{array}$

The graph for the above function is shown in Figure 2

  

Figure 2

Since, the graph opens downward and it is narrower than the parent function and the vertex is $\left(1,8\right)$ .

(b)

To determine

To Find: The new quadratic function for $h\left(d\right)$ and the way in which it changes in $h$ and $k$ affect the shape of the graph.

Answer to Problem 79SR

The graph shifts upward 4.5 ft. and 3 in to the right.

Explanation of Solution

Given:

The stream height is increased by 12.5 feet at the distance of 15 inches from the jet.

The water lands 3.75 feet from the jet.

Calculation:

Consider the part of the vertex is at $\left(15\text{in,}12.5\text{in}\right)$ and convert it to the feet and then the coordinate becomes $\left(1.25\text{ft,}12.5\text{ft}\right)$ . Thus, the equation is,

  $\begin{array}{c}h\left(d\right)=-2{\left(d-1.25\right)}^2+1.25\\ =-2d^2+5d-1.875\end{array}$

The graph for the above function is shown in Figure 3

  

Figure 3

Thus, the graph shifts upward 4.5 ft. and 3 in to the right.