Chapter 2.5, Problem 84AYU

a)

To determine

To find: The corresponding location of the given point on the graph of $y=f\left(x+3\right)-5$ .

Answer to Problem 84AYU

The corresponding location of the given point on the graph of $y=f\left(x+3\right)-5$ is $\left(-2,-2\right)$ .

Explanation of Solution

Given:

A point $\left(1,3\right)$ on the graph of $y=f\left(x\right)$

Concept used:

For any point $\left(a,b\right)$ on the graph of $y=f\left(x\right)$ the corresponding location on the graph of $y=cf\left(x+h\right)+k$ can be obtained by substituting

  $\begin{array}{l}x+h=a\\ cf\left(x+h\right)=b\end{array}$

And simplifying $x,y$

Calculation:

Consider the given point and the new graph.

  $\left(1,3\right)$ and $y=f\left(x+3\right)-5$

Now, let us take

  $\begin{array}{c}x+3=1\\ x=-2\end{array}$

Now, let us take

  $\begin{array}{c}y=f\left(1\right)-5\\ =3-5\\ =-2\end{array}$

Hence, the corresponding location of the given point on the graph of $y=f\left(x+3\right)-5$ is $\left(-2,-2\right)$ .

b)

To determine

To find: The corresponding location of the given point on the graph of $y=-2f\left(x-2\right)+1$ .

Answer to Problem 84AYU

The corresponding location of the given point on the graph of $y=-2f\left(x-2\right)+1$ is $\left(3,-5\right)$ .

Explanation of Solution

Given:

A point $\left(1,3\right)$ on the graph of $y=f\left(x\right)$

Concept used:

For any point $\left(a,b\right)$ on the graph of $y=f\left(x\right)$ the corresponding location on the graph of $y=cf\left(x+h\right)+k$ can be obtained by substituting

  $\begin{array}{l}x+h=a\\ cf\left(x+h\right)=b\end{array}$

And simplifying $x,y$

Calculation:

Consider the given point and the new graph.

  $\left(1,3\right)$ and $y=-2f\left(x-2\right)+1$

Now, let us take

  $\begin{array}{c}x-2=1\\ x=3\end{array}$

Now, let us take

  $\begin{array}{c}y=-2f\left(1\right)+1\\ =-2\times 3+1\\ =-6+1\\ =-5\end{array}$

Hence, the corresponding location of the given point on the graph of $y=-2f\left(x-2\right)+1$ is $\left(3,-5\right)$ .

c)

To determine

To find: The corresponding location of the given point on the graph of $y=f\left(2x+3\right)$ .

Answer to Problem 84AYU

The corresponding location of the given point on the graph of $y=f\left(2x+3\right)$ is $\left(-1,3\right)$ .

Explanation of Solution

Given:

A point $\left(1,3\right)$ on the graph of $y=f\left(x\right)$

Concept used:

For any point $\left(a,b\right)$ on the graph of $y=f\left(x\right)$ the corresponding location on the graph of $y=cf\left(x+h\right)+k$ can be obtained by substituting

  $\begin{array}{l}x+h=a\\ cf\left(x+h\right)=b\end{array}$

And simplifying $x,y$

Calculation:

Consider the given point and the new graph.

  $\left(1,3\right)$ and $y=f\left(2x+3\right)$

Now, let us take

  $\begin{array}{c}2x+3=1\\ 2x=-2\\ x=-1\end{array}$

Now, let us take

  $\begin{array}{c}y=f\left(1\right)\\ =3\end{array}$

Hence, the corresponding location of the given point on the graph of $y=f\left(2x+3\right)$ is $\left(-1,3\right)$ .