Math Problem Statement

Solve

Describe how the graph of g(x) = (3√x) + 5 can be obtained from the graph of f(x) = 3√x. Then graph the function g(x).

Solution

To solve this problem, let’s break down the questions in the image:

  1. Transformation Question:

    • The function f(x)=x3f(x) = \sqrt[3]{x} represents a basic cubic root function.
    • The function g(x)=x3+5g(x) = \sqrt[3]{x} + 5 involves a vertical shift of the graph of f(x)f(x).
    • Since the transformation adds 5 to the function, it shifts the graph of f(x)f(x) 5 units up.
    • This confirms that the correct answer to the transformation question is D: Shift the graph 5 units up.
  2. Graph Identification:

    • We need to select the correct graph for g(x)=x3+5g(x) = \sqrt[3]{x} + 5.
    • The transformation +5+5 affects only the vertical position, so we are looking for a graph of the cubic root function that is shifted 5 units higher than the usual position of f(x)=x3f(x) = \sqrt[3]{x}.

Upon examining the choices:

  • The correct graph should resemble the cubic root shape but be visibly shifted upward.
  • Based on the graphs provided, the answer should be A as it appears to have the cubic root shape shifted up by 5 units.

Summary:

  • Transformation Answer: D
  • Graph Identification Answer: A

Would you like a detailed explanation on why the graph shifts vertically or how to identify transformations on graphs?

Related Questions

  1. How does a horizontal shift affect the function f(x)=x3f(x) = \sqrt[3]{x}?
  2. What would happen if we added a negative constant to f(x)=x3f(x) = \sqrt[3]{x}?
  3. How can we determine transformations visually without calculations?
  4. What are some common transformations for cubic and square root functions?
  5. How would a reflection over the x-axis affect f(x)=x3f(x) = \sqrt[3]{x}?

Tip: When looking at transformations, remember that changes inside the function (like f(x+a)f(x + a)) affect horizontal positioning, while changes outside (like f(x)+af(x) + a) affect vertical positioning.

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Math Problem Analysis

Mathematical Concepts

Algebra
Graph Transformations
Cube Root Functions

Formulas

g(x) = 3√x + 5
Vertical shift transformation

Theorems

Vertical transformation rule

Suitable Grade Level

Grades 9-11