Chapter 1.2, Problem 56PS

Solution

To find: The quadratic function $y=a{(x-h)}^2+k$ for the flight path of the fish, and then explain effects of changes in values of $a$ , $h$ and $k$ .

The model for the flight path is $y=-\frac{5}{1089}{\left(x-33\right)}^2+5$ .

Given information:

The flying fish reaches a maximum height of 5 feet after flying for a horizontal distance of 33 feet.

Concept used:

Vertex form a quadratic function is given as $y=a{\left(x-h\right)}^2+k$ , where $\left(h,k\right)$ is the vertex of the quadratic function.

The vertex of the quadratic function represents the maximum/minimum point of the function.

If $a$ positive, the vertex is the minimum point and if $a$ is negative, the vertex is the maximum point of the function.

Calculation:

Since flying fish reaches a maximum height of 5 feet after flying for a horizontal distance of 33 feet, the vertex of the quadratic function followed by the flying will be $\left(33,5\right)$ .

The quadratic function with vertex at $\left(33,5\right)$ can be given as $y=a{\left(x-33\right)}^2+5$ .

Since the flying fish leaves the water level at the point $\left(0,0\right)$ , the quadratic function must pass through $\left(0,0\right)$ .

Substitute $x=0$ and $y=0$ in the equation $y=a{\left(x-33\right)}^2+5$ , and solve for $a$ .

  $\begin{array}{c}0=a{\left(0-33\right)}^2+5\\ a{\left(0-33\right)}^2+5=0\\ 1089a+5=0\\ 1089a=-5\\ a=-\frac{5}{1089}\end{array}$

Now, substitute $a=-\frac{5}{1089}$ in the equation $y=a{\left(x-33\right)}^2+5$ to get the quadratic model for the path of the flying fish as $y=-\frac{5}{1089}{\left(x-33\right)}^2+5$ .

Effect of changing the value of $a$:

Draw the flight path for different values of $a$ keeping the values of $h$ and $k$ constants.

  

If the value of $a$ is decreased, then it compresses the path of along y-direction. That means, the flying fish will leave the water level at the right of the point $\left(0,0\right)$ .

If the value of $a$ is increased, then it stretches the path of along y-direction. That means, the flying fish will leave the water level to the left of the point $\left(0,0\right)$ .

Effect of changing the value of $h$:

Draw the flight path for different values of $h$ keeping the values of $a$ and $k$ constants.

  

If the value of $h$ is increased, the vertex of the path shifts to the right of the original vertex $\left(33,5\right)$ showing that the fish leaves the water level to the right of the point $\left(0,0\right)$ .

If the value of $h$ is decreased, the vertex of the path shifts to the left of the original vertex $\left(33,5\right)$ showing that the fish leaves the water level to the left of the point $\left(0,0\right)$ .

Effect of changing the value of $k$:

Draw the flight path for different values of $k$ keeping the values of $a$ and $h$ constants.

  

If the value of $k$ is increased, the vertex of the path shifts above the original vertex $\left(33,5\right)$ showing that the fish leaves the water level to the left of the point $\left(0,0\right)$ .

If the value of $k$ is decreased, the vertex of the path shifts below the original vertex $\left(33,5\right)$ showing that the fish leaves the water level to the right of the point $\left(0,0\right)$ .

Conclusion:

The model for the flight path is $y=-\frac{5}{1089}{\left(x-33\right)}^2+5$ .