Question

(2.2 - 4.5)
9 of 19
this test: 90 point(s) possible
this question: 6 point(s) possible
graph the following function using the techniques of shifting,
compressing, stretching, and/or reflecting. start with the graph of the
basic function. be sure to identify at least three key points. find the
domain and the range of the function.
f(x) = -((x - 1)^3) - 1
complete the table of coordinates that lie on the graph of y = x^3 and the
corresponding points that lie on the graph of f(x) = -((x - 1)^3) - 1.
(type ordered pairs. simplify your answers.)
use the graphing tool to graph the equation.

Explanation:

Step1: Find \( y \) - values for \( y = x^{3} \)

For \( x=-1 \), \( y=(-1)^{3}=-1 \).
For \( x = 0 \), \( y=(0)^{3}=0 \).
For \( x = 1 \), \( y=(1)^{3}=1 \).

Step2: Use transformation rules

The transformation from \( y = x^{3} \) to \( f(x)=-(x - 1)^{3}-1 \) involves three steps:

1. Horizontal shift: \( y=x^{3}\to y=(x - 1)^{3} \) (shift 1 unit to the right).
2. Reflection: \( y=(x - 1)^{3}\to y=-(x - 1)^{3} \) (reflect about the \( x \) - axis).
3. Vertical shift: \( y=-(x - 1)^{3}\to y=-(x - 1)^{3}-1 \) (shift 1 unit down).

If \( (x_1,y_1) \) is on \( y = x^{3} \), then for \( f(x)=-(x - 1)^{3}-1 \), the new \( x \) - coordinate \( x_2=x_1 + 1 \) and the new \( y \) - coordinate \( y_2=-y_1-1 \).

For \( (-1,-1) \) on \( y = x^{3} \):
\( x=-1+1 = 0 \), \( y=-(-1)-1=0 \). The point is \( (0,0) \).

For \( (0,0) \) on \( y = x^{3} \):
\( x=0 + 1=1 \), \( y=-0-1=-1 \). The point is \( (1,-1) \).

For \( (1,1) \) on \( y = x^{3} \):
\( x=1 + 1=2 \), \( y=-1-1=-2 \). The point is \( (2,-2) \).

Answer:

Points that lie on the graph of \( y = x^{3} \)	Corresponding points that lie on the graph of \( f(x)=-(x - 1)^{3}-1 \)
\( (0,0) \)	\( (1,-1) \)
\( (1,1) \)	\( (2,-2) \)