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

Short 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-by-step solution

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 role="math" localid="1645904280709" $h\left(x\right)$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 $g\left(x\right)=|x+4|+2$