The question in your image asks to fill in the blanks to correctly complete the following sentence:

"To obtain the graph of $f(x) = (x + 8)^3 - 2$, shift the graph of $y = x^3$ to the left __ units and down __ units."

Explanation:

The function $f(x) = (x + 8)^3 - 2$ is a transformation of the basic cubic function $y = x^3$.

1. Horizontal shift: The term $(x + 8)$ indicates a horizontal shift. Since it's $x + 8$, this means the graph shifts 8 units to the left (opposite of the sign).
2. Vertical shift: The term $- 2$ outside the cubic function indicates a vertical shift downward by 2 units.

Conclusion:

• Left shift: 8 units
• Downward shift: 2 units

So, the answer should be:

• Shift to the left 8 units
• Shift down 2 units

Would you like more details or have any other questions?

5 Related Questions:

1. How do you shift other types of functions horizontally or vertically?
2. What happens if the transformation inside the parentheses was $x - 8$?
3. How does the general transformation of $a(x + h)^n + k$ affect graphs?
4. What is the effect of a reflection on the graph of $y = x^3$?
5. How do horizontal stretches and compressions affect the graph of a cubic function?

Tip:

When analyzing transformations, always pay attention to the signs inside and outside the parentheses to determine the direction of shifts or reflections!