Question

1. if the cubic parent function, f(x) = x³, is shifted right 4 units and reflected vertically over the x-axis, 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 and reflection factor (negative for vertical reflection),
• \( h \) is the horizontal shift (positive for right shift, negative for left shift),
• \( k \) is the vertical shift (positive for up, negative for down).

Step2: Determine the value of \( a \)

The function is reflected vertically over the \( x \)-axis. A vertical reflection of a function \( y = f(x) \) is given by \( y = -f(x) \), so \( a = -1 \).

Step3: Determine the value of \( h \)

The parent function is shifted right 4 units. For a horizontal shift, if we shift right by \( c \) units, the transformation is \( f(x - c) \). So, \( h = 4 \) (since the formula is \( (x - h) \), a right shift of 4 means \( h = 4 \)).

Step4: Determine the value of \( k \)

There is no vertical shift mentioned, so \( k = 0 \).

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

Substitute \( a = -1 \), \( h = 4 \), and \( k = 0 \) into the general form:
\( g(x) = -1(x - 4)^3 + 0 \), which simplifies to \( g(x) = -(x - 4)^3 \).

Answer:

\( a = -1 \), \( h = 4 \), \( k = 0 \)

\( g(x) = -(x - 4)^3 \)