Chapter 2.5, Problem 85AYU

a)

To determine

To find: The corresponding location of the given point on the graph of $y=g\left(x+1\right)-3$ .

Answer to Problem 85AYU

The corresponding location of the given point on the graph of $y=g\left(x+1\right)-3$ is $\left(-4,2\right)$ .

Explanation of Solution

Given:

A point $\left(-3,5\right)$ on the graph of $y=g\left(x\right)$

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.

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

Now, let us take

  $\begin{array}{c}x+1=-3\\ x=-4\end{array}$

Now, let us take

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

Hence, the corresponding location of the given point on the graph of $y=g\left(x+1\right)-3$ is $\left(-4,2\right)$ .

b)

To determine

To find: The corresponding location of the given point on the graph of $y=-3g\left(x-4\right)+3$ .

Answer to Problem 85AYU

The corresponding location of the given point on the graph of $y=-3g\left(x-4\right)+3$ is $\left(1,-12\right)$ .

Explanation of Solution

Given:

A point $\left(-3,5\right)$ on the graph of $y=g\left(x\right)$

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.

  $\left(-3,5\right)$ and $y=-3g\left(x-4\right)+3$

Now, let us take

  $\begin{array}{c}x-4=-3\\ x=-3+4\\ x=1\end{array}$

Now, let us take

  $\begin{array}{c}y=-3g\left(-3\right)+3\\ =-3\times 5+3\\ =-15+3\\ =-12\end{array}$

Hence, the corresponding location of the given point on the graph of $y=-3g\left(x-4\right)+3$ is $\left(1,-12\right)$ .

c)

To determine

To find: The corresponding location of the given point on the graph of $y=g\left(3x+9\right)$ .

Answer to Problem 85AYU

The corresponding location of the given point on the graph of $y=g\left(3x+9\right)$ is $\left(-4,5\right)$ .

Explanation of Solution

Given:

A point $\left(-3,5\right)$ on the graph of $y=g\left(x\right)$

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.

  $\left(-3,5\right)$ and $y=g\left(3x+9\right)$

Now, let us take

  $\begin{array}{c}3x+9=-3\\ 3x=-12\\ x=-4\end{array}$

Now, let us take

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

Hence, the corresponding location of the given point on the graph of $y=g\left(3x+9\right)$ is $\left(-4,5\right)$ .