Chapter 4, Problem 1RE

(a)

To determine

The way to obtain the graph of the function $y=-\frac{1}{x}+6$ from the graph of the function $y=\frac{1}{x}$.

Answer to Problem 1RE

The graph of $y=-\frac{1}{x}+6$ is obtained by reflecting the curve $y=\frac{1}{x}$ across the $x$ axis and translation of the curve $y=-\frac{1}{x}$$6$ units upwards towards the positive direction of $y$ axis.

Explanation of Solution

Given:

The function is $y=-\frac{1}{x}+6$.

Concept used:

Shifting of graphs:

“Shifting is a rigid translation because it does not change the size and shape of the curve. Shifting is used to move the curve vertically or horizontally without any change in shape and size of the curve.”

The function $y=f\left(x+a\right)\text{and }y=f\left(x-a\right)$ is a shift of the curve $y=f\left(x\right)$ horizontally towards negative and positive direction of $x$ axis respectively.

The function $y=f\left(x\right)+a\text{ and }y=f\left(x\right)-a$ is a shift of the curve $y=f\left(x\right)$ vertically towards positive and negative direction of $y$ axis respectively.

Scaling of graph:

Scaling is used to change the shape and size of the curve. A vertical scaling multiplies or divides ever $y$ coordinate by a constant while leaving the $x$ coordinate unchanged.

$y=a\cdot f\left(x\right)$

If $a$ is negative in $y=a\cdot f\left(x\right)$ then the graph is reflected across the $x$ axis.

A horizontal scaling multiplies or divides ever $x$ coordinate by a constant while leaving the $y$ coordinate unchanged.

$y=f\left(b\cdot x\right)$

Calculation:

The given function is $y=-\frac{1}{x}+6$.

The given function is of the form $y=-f\left(x\right)+c$.

The graph of $y=-\frac{1}{x}+6$ is obtained from the graph of $y=\frac{1}{x}$ in two steps.

First obtain the graph of $y=-\frac{1}{x}$ from the graph of $y=\frac{1}{x}$.

The graph of $y=-\frac{1}{x}$ is obtained from the graph of $y=\frac{1}{x}$ when each point on the curve of $y=\frac{1}{x}$ is reflected across the $x$ axis.

Now obtain the graph of $y=-\frac{1}{x}+6$ from the graph of $y=-\frac{1}{x}$.

The graph of $y=-\frac{1}{x}+6$ is obtained from the graph of $y=-\frac{1}{x}$ when each point on the curve of $y=-\frac{1}{x}$ is shifted $6$ units upwards.

Thus, the graph of $y=-\frac{1}{x}+6$ is obtained by the reflection of curve $y=\frac{1}{x}$ across the $x$ axis and translation of the curve $y=-\frac{1}{x}$$6$ units upwards.

(b)

To determine

The graph of the function $y=-\frac{1}{x}+6$.

Explanation of Solution

Given:

The function is $y=-\frac{1}{x}+6$.

Calculation:

Draw the graph of the function $y=-\frac{1}{x}+6$.

Figure $1$ represents the graph of the function $y=-\frac{1}{x}+6$.

(c)

To determine

The graph of the function $y=-\frac{1}{x}+6$ by graphing calculator.

Explanation of Solution

Given:

The function is $y=-\frac{1}{x}+6$.

Calculation:

To graph the function $y=-\frac{1}{x}+6$, first turn on the calculator and set the window as shown below.

Press the button $\text{Y}=$ and insert the function as shown below.

Now press the $\text{GRAPH}$ button and the graph of the function is obtained as follows:

In the above screen the graph of the function $y=-\frac{1}{x}+6$ is sketched with no $x$-intercept and no $y$-intercept.