Chapter 1.3, Problem 11E

Solution

To match: the given graph with its function and verify with graphing window $\left[-4.7,4.7\right]\times \left[-3.1\times 3.1\right]$ .

The graph matches with option (l) $y=2-\frac{4}{1+e^{-x}}$ .

Given information: The given graph is,

  

Formula used:

The graph of the function $y=f\left(x\right)-k$ is found by shifting the graph of the function $y=f\left(x\right)$ by $k$ units downward.

The graph of the function $y=cf\left(x\right)$ is found by stretching the graph of the function $y=f\left(x\right)$ by a factor $c$ .

The graph of the function $y=-f\left(x\right)$ is found by reflecting the graph of the function $y=f\left(x\right)$ through $x$ -axis.

Calculation:

See the shape of the graph.

The graph resembles to the graph of $y=e^{-x}$ .

Consider the basic function $y=e^{-x}$ .

Shift the graph 1 unit to upward such that the $y$ -intercept is $1$ .

Thus, the graph of the function is $y=1+e^x$

Reflect the graph through $x$ -axis.

New graph will be $y=-\frac{1}{1+e^{-x}}$ .

After stretching and shifting 2 unit downward again, the final graph is $y=2-\frac{4}{1+e^{-x}}$

The given graph matches with option (l).

Graph the function in the graphing window $\left[-4.7,4.7\right]\times \left[-3.1\times 3.1\right]$, by using graphing calculator:

  

Hence, the graph matches with option (l) $y=2-\frac{4}{1+e^{-x}}$ .