Q. 93 - Precalculus Enhanced with Graphing Utilities

Question

Suppose $\left(-3,5\right)$ is a point on the graph of $y=g\left(x\right)$.

(a) What point is on the graph of $y=g(x+1)-3$?

(b) What point is on the graph of $y=-3g\left(x-4\right)+3$?

(c) What point is on the graph of$y=g\left(3x+9\right)$?

Step-by-Step Solution

Answer

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

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

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

Part (a) Step 1. Finding the point on y = g ( x + 1 ) - 3

It is given that $\left(-3,5\right)$ is a point on the graph  $y=g\left(x\right)$.

So, substituting $x+1=-3$ in the function$y=g(x+1)-3$:

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

Now, we have:

$$\begin{gathered} y=g(-3)-3 \\ y=5-3 \left[g(-3)=5\right] \\ y=2 \end{gathered}$$

Hence, $\left(-3,5\right)$ is on the graph $y=g(x+1)-3$ 

Part (b). Step 1. Finding the point on the graph y = - 3 g x - 4 + 3

It is given that $\left(-3,5\right)$ is a point on the graph  $y=g\left(x\right)$.

So, substituting $x-4=-3$in the function $y=-3g\left(x-4\right)+3$:

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

Now , we have:

$$\begin{gathered} y=-3g(-3)+3 \\ y-3\times 5+3 \\ y=-12 \end{gathered}$$

Hence, $\left(1,-12\right)$ is on the graph $y=-3g(x-4)+3$ 

Part (c) Step 1. Finding the point on the graph y = g ( 3 x + 9 )

It is given that $\left(-3,5\right)$ is a point on the graph  $y=g\left(3x+9\right)$.

Substituting $3x+9=-3$ in the function $y=g\left(3x+9\right)$:

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

Now, we have:

$$\begin{gathered} y=g(-3) \\ y=5 \end{gathered}$$

Hence, $\left(-4,5\right)$ is on the graph 

$y=g(3x+9)$