Math Problem Statement

Use transformations of the standard cubic function, f(x) = x^3, to graph the function h(x) = 1/5 * (x - 7)^3 + 6. What transformations are needed?

Solution

It appears that you uploaded the same image again. Based on my previous explanation and after re-examining the question in the image, the transformations for h(x)=15(x7)3+6h(x) = \frac{1}{5}(x - 7)^3 + 6 are still the same:

  • Horizontal Shift: Right by 7 units (due to x7x - 7).
  • Vertical Shrinking: Shrunk by a factor of 15\frac{1}{5} (because of the 15\frac{1}{5} coefficient).
  • Vertical Shift: Upward by 6 units (because of the +6).

Thus, the correct answer is B: "The graph of f(x)=x3f(x) = x^3 should be horizontally shifted to the right by 7 units, shrunk vertically by a factor of 15\frac{1}{5}, and shifted vertically up by 6 units."

Would you like more details or have further questions?

Follow-up Questions:

  1. What is the difference between horizontal and vertical shifts in function transformations?
  2. Why does multiplying a function by a number less than 1 result in a vertical compression?
  3. How would the transformation change if the function had +7+7 instead of 7-7 inside the parentheses?
  4. Can you explain how these transformations would affect the shape of the graph visually?
  5. How would the graph change if the function were 15(x7)3+6-\frac{1}{5}(x - 7)^3 + 6?

Tip:

Always remember that horizontal transformations (inside the parentheses) affect the x-direction in a counterintuitive way: xax - a shifts right, while x+ax + a shifts left.

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Math Problem Analysis

Mathematical Concepts

Transformations of functions
Cubic functions
Horizontal and vertical shifts
Vertical compression

Formulas

h(x) = a * (x - h)^3 + k
Transformation formula for cubic functions

Theorems

Transformation Theorem for Graphs of Functions

Suitable Grade Level

Grades 10-12