Math Problem Statement

The function f(x) = ∛x is translated 3 units in the negative y-direction and 8 units in the negative x-direction. Select the correct equation for the resulting function.

Solution

To determine the new function after translating f(x)=x3f(x) = \sqrt[3]{x}:

  1. Translation 3 units in the negative yy-direction:

    • Moving downward by 3 units translates the function vertically. This affects the function by subtracting 3 from f(x)f(x).
    • So, f(x)=x33f(x) = \sqrt[3]{x} - 3.
  2. Translation 8 units in the negative xx-direction:

    • Moving left by 8 units translates the function horizontally. This affects the xx-term inside the function, changing xx to x+8x + 8.
    • So, f(x)=x+833f(x) = \sqrt[3]{x + 8} - 3.

Therefore, the resulting function is: f(x)=x+833f(x) = \sqrt[3]{x + 8} - 3

Answer:

The correct choice is:

  • C) f(x)=x+833f(x) = \sqrt[3]{x + 8} - 3

Would you like a more detailed explanation of each translation?


Here are 5 related questions to further explore this topic:

  1. How does translating a function affect its graph?
  2. What happens if we translate a function in both positive xx- and yy-directions?
  3. How would a reflection over the xx-axis change the function f(x)=x3f(x) = \sqrt[3]{x}?
  4. How do we determine the transformations for different types of functions, such as quadratic or exponential functions?
  5. What are the differences between vertical and horizontal transformations?

Tip: When translating functions, remember that changes inside the function (like x+hx + h) affect the xx-direction, while changes outside the function (like f(x)+kf(x) + k) affect the yy-direction.

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Math Problem Analysis

Mathematical Concepts

Function Transformation
Translations
Cube Root Functions

Formulas

f(x) = ∛(x + h) + k, where h is the horizontal translation and k is the vertical translation

Theorems

Translation of functions theorem

Suitable Grade Level

Grades 9-11