Math Problem Statement

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.

Solution

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)32f(x) = (x + 8)^3 - 2, shift the graph of y=x3y = x^3 to the left __ units and down __ units."

Explanation:

The function f(x)=(x+8)32f(x) = (x + 8)^3 - 2 is a transformation of the basic cubic function y=x3y = x^3.

  1. Horizontal shift: The term (x+8)(x + 8) indicates a horizontal shift. Since it's x+8x + 8, this means the graph shifts 8 units to the left (opposite of the sign).
  2. Vertical shift: The term 2- 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 x8x - 8?
  3. How does the general transformation of a(x+h)n+ka(x + h)^n + k affect graphs?
  4. What is the effect of a reflection on the graph of y=x3y = 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!

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Math Problem Analysis

Mathematical Concepts

Algebra
Function Transformations
Cubic Functions

Formulas

f(x) = (x + h)^3 + k

Theorems

Transformation Theorem
Horizontal and Vertical Shifts

Suitable Grade Level

Grades 9-12