Question

1. if the parent function, f(x) = x³, is shifted left 3 units, down 6 units, and it is vertically compressed by a factor is 1/5, write the new function g(x):

a=____ h=__ k=____
g(x)=____________________

Explanation:

Step1: Recall the transformation formula

The general form of a transformed cubic function is \( g(x) = a(x - h)^3 + k \), where:

• \( a \) is the vertical stretch/compression factor (if \( |a|>1 \) it's a stretch, if \( 0<|a|<1 \) it's a compression)
• \( h \) is the horizontal shift (right if \( h>0 \), left if \( h<0 \))
• \( k \) is the vertical shift (up if \( k>0 \), down if \( k<0 \))

Step2: Determine the value of \( a \)

The function is vertically compressed by a factor of \( \frac{1}{5} \), so \( a=\frac{1}{5} \).

Step3: Determine the value of \( h \)

The parent function is shifted left 3 units. For horizontal shifts, shifting left by \( c \) units means \( h=-c \) (since the formula is \( x - h \), so to shift left 3, we have \( x - (-3)=x + 3 \)). So \( h=-3 \).

Step4: Determine the value of \( k \)

The function is shifted down 6 units. Shifting down by \( c \) units means \( k=-c \), so \( k = - 6 \).

Step5: Write the function \( g(x) \)

Substitute \( a=\frac{1}{5} \), \( h = - 3 \), and \( k=-6 \) into the formula \( g(x)=a(x - h)^3 + k \):
\[
g(x)=\frac{1}{5}(x - (-3))^3+(-6)=\frac{1}{5}(x + 3)^3-6
\]

Answer:

\( a=\frac{1}{5} \), \( h = - 3 \), \( k=-6 \)

\( g(x)=\frac{1}{5}(x + 3)^3-6 \)