Question

Families of Functions

(a) Draw the graphs of the family of functions

$f\left(x\right)=\frac{a}{2}\left(e^{\frac{x}{a}}+e^{-\frac{x}{a}}\right)$

for $a=0.5,1,1.5$, and 2.

(b) How does a larger value of a affect the graph?

Short answer

a. The graphs of the family of functions are:

b.As a increases then the graph goes upwards and it gets more horizontally stretched.

Step-by-step solution

Part a. Step 1. Given.

The given function is: $f\left(x\right)=\frac{a}{2}\left(e^{\frac{x}{a}}+e^{-\frac{x}{a}}\right)$.

Part a. Step 2. To determine.

We have to graph the given function for $a=0.5,1,1.5,2$.

Part a. Step 3. Calculation.

We’ll use a graphing calculator to graph $f\left(x\right)=\frac{a}{2}\left(e^{\frac{x}{a}}+e^{-\frac{x}{a}}\right)$for $a=0.5,1,1.5,2$.

Part b. Step 1. Given.

The given function is: $f\left(x\right)=\frac{a}{2}\left(e^{\frac{x}{a}}+e^{-\frac{x}{a}}\right)$.

Part b. Step 2. To determine.

We have to determine howdoes a larger value of a affect the graph.

Part b. Step 3. Calculation.

From part (a), we see that, as a increases then the graph goes upwards and it gets more horizontally stretched.