Q. 10 - Precalculus Enhanced with Graphing Utilities

Question

Graph each function using the techniques of shifting,

compressing or stretching, and reflections. Start with the

graph of the basic function and show all stages.

(a) $h\left(x\right)=-2{(x+1)}^3+3$

(b) $g\left(x\right)=|x+4|+2$

Step-by-Step Solution

Answer

(a) graph of $h\left(x\right)=-2{(x+1)}^3+3$ is

(b) Graph of $g\left(x\right)=|x+4|+2$ is

Step 1. Given data

GIven functions are

(a) $h\left(x\right)=-2{(x+1)}^3+3$

(b) $g\left(x\right)=|x+4|+2$

Step 2. Part (a)

Parent function of $h\left(x\right)=-2{(x+1)}^3+3$ is $f\left(x\right)=x^3$

function $h\left(x\right)$ is obtained from $f\left(x\right)$ by transformations-

1. horizontal shifting of $f\left(x\right)$ towards left by one unit
2. its vertical stretch by a factor $2$
3. reflection of it about the y axis
4. Shift it vertically upward by $3$ units.

Step 3. Graph of function

Plot the function $f\left(x\right)=x^3$ and transformed it into $h\left(x\right)=-2{(x+1)}^3+3$

Step 4. Pard (b)

Parent function of $g\left(x\right)=|x+4|+2$ is $f\left(x\right)=\left|x\right|$ 

the function $g\left(x\right)$ is obtained from $f\left(x\right)$ by transformations-

1. horizontal Shifting of $f\left(x\right)$ towards left by four units
2. Shift it vertically downwards by two units

Step 5. Graph of the function

Plot the function $f\left(x\right)=\left|x\right|$ and transformed it into  $g\left(x\right)=|x+4|+2$