Chapter ISG, Problem 7.6.1P

(a)

To determine

To find: The equation for the given graph of a function.

Answer to Problem 7.6.1P

The equation that describes the given graph is $y=-2^x$ .

Explanation of Solution

Given information: The below graph shows the transformation of a function from its parent function $y=2^x$ as:

  

Figure (1)

Calculation:

The given function $y=2^x$ is of the form:

  $y=a\cdot b^{x-h}+k$

Here, $a=1$ , $b=2$ , $h=0$ and $k=0$ .

The graph of the function $y=2^x$ can be drawn as shown below:

  

Figure (2)

Compare the graph of the function $y=2^x$ with the given graph. It can be observed that the given graph shows reflection of the graph of function $y=2^x$ across the $x-$ axis.

The reflection across the $x-$ axis is described by $-a$ . The value of a is 1. So, the equation that describes the given graph is:

  $y=-2^x$

Therefore, the equation that describes the given graph is $y=-2^x$ .

(b)

To determine

To find: The equation for the given graph of a function.

Answer to Problem 7.6.1P

The equation that describes the given graph is $y=\frac{1}{8}{\left(2\right)}^x$ .

Explanation of Solution

Given information: The below graph shows the transformation of a function from its parent function $y=2^x$ as:

  

Figure (1)

Calculation:

The given function $y=2^x$ is of the form:

  $y=a\cdot b^x+d$ …… (1)

From the given graph it can be observed that the asymptote is $y=0$ . Substitute 0 for $d$ and 2 for $b$ in equation (1).

  $y=a{\left(2\right)}^x$ …… (2)

A point of the graph of the function is $\left(3,1\right)$ . Substitute 3 for $x$ and 1 for $y$ in the above equation.

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

Substitute $\frac{1}{8}$ for a in equation (2). So, the equation that describes the given graph is:

  $y=\frac{1}{8}{\left(2\right)}^x$

Therefore, the equation that describes the given graph is $y=\frac{1}{8}{\left(2\right)}^x$ .

(c)

To determine

To find: The equation for the given graph of a function.

Answer to Problem 7.6.1P

The equation that describes the given graph is $y={\left(2\right)}^{-x}+5$ .

Explanation of Solution

Given information: The below graph shows the transformation of a function from its parent function $y=2^x$ as:

  

Figure (1)

Calculation:

The given function $y=2^x$ is of the form:

  $y=a\cdot b^x+d$ …… (1)

From the given graph it can be observed that the asymptote is $y=5$ . Substitute 5 for $d$ and 2 for $b$ in equation (1).

  $y=a{\left(2\right)}^x+5$ …… (2)

The intersection point on $y-$ axis is $\left(0,6\right)$ . Substitute 0 for $x$ and 6 for $y$ in the above equation.

  $\begin{array}{c}6=a\cdot {\left(2\right)}^0+5\\ 6=a+5\\ a=6-5\\ a=1\end{array}$

Substitute 1 for a in equation (2). So,

  $y={\left(2\right)}^x+5$

It can be observed that the given graph is mirror image of the function $y={\left(2\right)}^x+5$ . Thus, the equation that describes the given graph is:

  $y={\left(2\right)}^{-x}+5$

Therefore, the equation that describes the given graph is $y={\left(2\right)}^{-x}+5$ .

(d)

To determine

To find: The equation for the given graph of a function.

Answer to Problem 7.6.1P

The equation that describes the given graph is $y=8{\left(2\right)}^x+4$ .

Explanation of Solution

Given information: The below graph shows the transformation of a function from its parent function $y=2^x$ as:

  

Figure (1)

Calculation:

The given function $y=2^x$ is of the form:

  $y=a\cdot b^x+d$ …… (1)

From the given graph it can be observed that the asymptote is $y=4$ . Substitute 4 for $d$ and 2 for $b$ in equation (1).

  $y=a{\left(2\right)}^x+4$ …… (2)

The intersection point on $y-$ axis is $\left(0,12\right)$ . Substitute 0 for $x$ and 12 for $y$ in the above equation.

  $\begin{array}{c}12=a\cdot {\left(2\right)}^0+4\\ 12=a+4\\ a=12-4\\ a=8\end{array}$

Substitute 8 for a in equation (2). So,

  $y=8{\left(2\right)}^x+4$

Therefore, the equation that describes the given graph is $y=8{\left(2\right)}^x+4$ .