Chapter 8.4, Problem 3CYU

a.

To determine

To graph: The given function.

Explanation of Solution

Given information:

The following function:

  $P\left(x\right)=\frac{7+x}{11+x}$ .

Graph:

Step 1:

Find the zeroes.

  $\begin{array}{l}7+x=0[\text{Set }a(x)=0]\\ x=-7[\text{Subtract 7 from both sides]}\end{array}$

There is a zero at $x=-7$ .

Step 2:

Find the asymptotes.

  $\begin{array}{l}11+x=0[\text{Set }b(x)=0]\\ x=-11[\text{Subtract 11 from both sides]}\end{array}$

There is a vertical asymptote at $x=-11$ .

Step 3:

Draw the graph

Use a table to find ordered pairs on the graph. Then connect the points

  

  

b.

To determine

To explain: Part of the graph that is meaningful in context of the problem.

Answer to Problem 3CYU

Only values of x greater than or equal to $0$ are meaningful.

Explanation of Solution

Given information:

The following function:

  $P\left(x\right)=\frac{7+x}{11+x}$ .

Graph of the function is obtained as follows:

  

In the problem context, field goals are non-negative values. Only values of x greater than or equal to $0$ are meaningful.

c.

To determine

To explain: Meaning of intercept of vertical axis.

Answer to Problem 3CYU

Intercept of vertical axis is the point where the graph of the line crosses the y-axis. It is equal to $0.63$ units on the y-axis.

Explanation of Solution

Given information:

The following function:

  $P\left(x\right)=\frac{7+x}{11+x}$ .

Intercept of vertical axis is the point where the graph of the line crosses the y-axis. The x-axis coordinate at that point is $0$ . The intercept of the vertical axis is calculated as follows:

  $\begin{array}{l}P\left(x\right)=\frac{7+x}{11+x}[\text{Given]}\\ P(x)=\frac{7+0}{11+0}[\text{Replace }x\text{ with 0]}\\ P(x)=\frac{7}{11}[\text{Add]}\\ P(x)\approx 0.63[\text{Simplify]}\end{array}$

d.

To determine

To explain: The equation of horizontal asymptote with respect to Eduardo’s field goal percentage.

Answer to Problem 3CYU

Equation for the horizontal asymptote is $P(x)=1$ . With increasing consecutive field goals (x), the field goal percent approaches $\text{1 (100\%)}$ .

Explanation of Solution

Given information:

The following function:

  $P\left(x\right)=\frac{7+x}{11+x}$ .

Graph of the given function is obtained as follows:

  

Notice what happens to the y-axis values as x grows larger and as x gets smaller. The y-axis values approach $1$ . So, the equation for the horizontal asymptote is $P(x)=1$ .

In context of the problem, with increasing consecutive field goals (x), the field goal percent approaches $\text{1 (100\%)}$ .