Q. 71 - Precalculus Enhanced with Graphing Utilities

Question

Graph of a function is illustrated. Use the graph of f as the first step toward graphing each of the following functions:

Part (a): $F\left(x\right)=f\left(x\right)+3$

Part (b): $G\left(x\right)=f\left(x+2\right)$

Part (c): $P\left(x\right)=-f\left(x\right)$

Part (d): $H\left(x\right)=f\left(x+1\right)-2$

Part (e): $Q\left(x\right)=\frac{1}{2}f\left(x\right)$

Part (f): $g\left(x\right)=f\left(-x\right)$

Part (g): $h\left(x\right)=f\left(2x\right)$

Step-by-Step Solution

Answer

Part (a): The required graph is given below,

Part (b): The required graph is given below,

Part (c): The required graph is given below,

Part (d): The required graph is given below,

Part (e): The required graph is given below,

Part (f): The required graph is given below,

Part (g): The required graph is given below,

Part (a) Step 1. Plot the graph.

Consider the given graph,

If the function is shifted vertically upwards by $3units$, then the function is $F\left(x\right)=f\left(x\right)+3$.

On plotting the graph,

Part (b) Step 1. Plot the graph.

Consider the given graph,

If the function is shifted to the horizontally left by $2units$, then the function is $G\left(x\right)=f\left(x+2\right)$.

On plotting the graph,

Part (c) Step 1. Plot the graph.

Consider the given graph,

On reflecting the graph about the x-axis as $f\left(x\right)$ is multiplied by $-1$, the function transforms into $P\left(x\right)=-f\left(x\right)$.

On plotting the graph,

Part (d) Step 1. Plot the graph.

Consider the given graph,

First, if the function is shifted horizontally to the left by $1units$and then shifted downwards by $2units$is $H\left(x\right)=f\left(x+1\right)-2$.

On plotting the graph,

Part (e) Step 1. Plot the graph.

Consider the given graph,

On compressing the graph by dividing the function by $2$, then the function is $Q\left(x\right)=\frac{1}{2}f\left(x\right)$.

On plotting the graph,

Part (f) Step 1. Plot the graph.

Consider the given graph,

On reflecting the graph about y-axis, the function transforms into $g\left(x\right)=f\left(-x\right)$.

On plotting the graph,

Part (g) Step 1. Plot the graph.

Consider the given graph,

On compressing the graph horizontally by multiplying the function by $2$, then the function is $h\left(x\right)=f\left(2x\right)$.

On plotting the graph,