Math Problem Statement

Use transformations of the cube root function, f(x) = ³√x, to graph the function g(x) = ³√x - 5. Choose the correct graph below.

Solution

The question shown in the image asks to graph the function g(x)=x35g(x) = \sqrt[3]{x} - 5 using transformations of the cube root function f(x)=x3f(x) = \sqrt[3]{x}. Let's go through the steps needed to analyze and select the correct graph.

Function Breakdown:

  • The base function is f(x)=x3f(x) = \sqrt[3]{x}, which is the cube root function.
  • The transformation involved is a vertical translation downward by 5 units. This is due to the "-5" outside the cube root, meaning the entire graph of f(x)f(x) is shifted down.

Steps to graph:

  1. Base Function: Start with the graph of f(x)=x3f(x) = \sqrt[3]{x}. This function passes through the origin (0,0), and its general shape is an increasing curve.
  2. Vertical Shift: Subtracting 5 from f(x)f(x) shifts every point on the graph of x3\sqrt[3]{x} downward by 5 units. So, instead of passing through (0,0), the graph now passes through (0, -5).

Now, check the graphs in the provided options:

  • We are looking for the cube root graph, which has been shifted down by 5 units.

From the options:

  • Option A: Shows a curve with the right behavior, starting at (0, -5).
  • The other options (B, C, D) either do not shift downwards correctly or have a different behavior.

Thus, Option A is the correct graph.


Would you like any further details about graphing transformations?

5 Related Questions:

  1. How do horizontal translations affect the graph of a function?
  2. What is the difference between vertical and horizontal shifts in graphing?
  3. How does scaling (stretching or compressing) affect the graph of a cube root function?
  4. How do you find the domain and range of cube root functions?
  5. How would adding a coefficient to x3\sqrt[3]{x} affect the graph?

Tip:

When graphing transformations, always start by analyzing the base function's key points and behavior before applying shifts or stretches!

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Math Problem Analysis

Mathematical Concepts

Algebra
Transformations of Functions
Cube Root Functions

Formulas

g(x) = ³√x - 5
f(x) = ³√x

Theorems

Vertical Translation

Suitable Grade Level

Grades 9-11