Let's solve the given problem step-by-step:

Given Function:

$f(x) = x^3$

Transformed Function:

$g(x) = a \left[ k(x - d) \right]^3 + c$

We'll choose the following values for the parameters:

• $a = 2$ (vertical stretch by a factor of 2)
• $k = 1$ (no horizontal stretch or compression)
• $d = 3$ (horizontal shift to the right by 3 units)
• $c = -1$ (vertical shift downward by 1 unit)

Transformed Function Equation:

$g(x) = 2[1(x - 3)]^3 - 1 = 2(x - 3)^3 - 1$

Describe the Transformations:

1. Vertical Stretch: The graph is stretched vertically by a factor of 2.
2. Horizontal Shift: The graph is shifted 3 units to the right.
3. Vertical Shift: The graph is shifted 1 unit downward.

Key Points Transformation:

Let's start with some key points from $f(x) = x^3$:

• Original points: $(-1, -1)$, $(0, 0)$, $(1, 1)$, $(2, 8)$

Apply the transformations to these points:

1. For $(-1, -1)$:

   • Horizontal shift: $(-1 + 3, -1) = (2, -1)$
   • Vertical stretch: $(-1) \times 2 = -2$
   • Vertical shift: $-2 - 1 = -3$
   • New point: $(2, -3)$

2. For $(0, 0)$:

   • Horizontal shift: $(0 + 3, 0) = (3, 0)$
   • Vertical stretch: $0 \times 2 = 0$
   • Vertical shift: $0 - 1 = -1$
   • New point: $(3, -1)$

3. For $(1, 1)$:

   • Horizontal shift: $(1 + 3, 1) = (4, 1)$
   • Vertical stretch: $1 \times 2 = 2$
   • Vertical shift: $2 - 1 = 1$
   • New point: $(4, 1)$

4. For $(2, 8)$:

   • Horizontal shift: $(2 + 3, 8) = (5, 8)$
   • Vertical stretch: $8 \times 2 = 16$
   • Vertical shift: $16 - 1 = 15$
   • New point: $(5, 15)$

Graphing the Transformed Function:

1. Plot the new points: $(2, -3)$, $(3, -1)$, $(4, 1)$, $(5, 15)$
2. Draw the curve passing through these points to represent the function $g(x) = 2(x - 3)^3 - 1$

Here is a summary to fill in the table:

• $a = 2$
• $k = 1$
• $d = 3$
• $c = -1$
• Transformed function equation: $g(x) = 2(x - 3)^3 - 1$

Graph:

(You would need to plot this on the provided graph paper.)

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. What would the graph look like if $a = -2$ instead of $2$?
2. How does changing $k$ from $1$ to $0.5$ affect the graph?
3. What if $d$ were a negative value?
4. How does the value of $c$ influence the graph's position?
5. Can you describe the transformations if $a = 1$, $k = 2$, $d = 0$, and $c = 3$?
6. What are the key points if the original function is $f(x) = x^2$ instead?
7. How do you determine the new coordinates of a point after transformation?
8. What are the effects of combining multiple transformations in one function?

Tip:

When graphing transformed functions, always start with the base function and apply transformations step-by-step for accuracy.