Math Problem Statement

The graph to the right was created by horizontally shifting the graph of the basic function

yequals=x cubedx3.

Write a function that describes the graph to the right.

-10-5510-10-5510xy

left parenthesis 3 comma 0 right parenthesis(3, 0)

f left parenthesis x right parenthesisf(x)

A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 10 to 10 in increments of 1. From left to right, a curve labeled f(x) rises in the fourth quadrant, decreases in slope as it approaches the x-axis and passes through the point (3,0), then increases in slope as it rises upward in the first quadrant. A point labeled (3,0) is plotted on the curve at (3,0).

Solution

The graph in question describes a transformation of the basic cubic function f(x)=x3f(x) = x^3, and it has been shifted horizontally. Based on the description you provided, it passes through the point (3,0)(3, 0), indicating a horizontal shift of 3 units to the right.

The general form of a horizontal shift of the function f(x)=x3f(x) = x^3 would be:

f(x)=(x−h)3f(x) = (x - h)^3

Where hh represents the horizontal shift. If the graph is shifted 3 units to the right, then h=3h = 3. Therefore, the function describing the transformed graph is:

f(x)=(x−3)3f(x) = (x - 3)^3

This function shifts the basic cubic graph 3 units to the right, and the point (3,0)(3, 0) is a result of this shift, as f(3)=(3−3)3=0f(3) = (3 - 3)^3 = 0.

Summary:

The function that describes the graph is:

f(x)=(x−3)3f(x) = (x - 3)^3

Would you like to see more details or have any questions about horizontal shifts and transformations?

Related Questions:

  1. How do other transformations (vertical shifts, reflections, stretches) affect the graph of a cubic function?
  2. How do we determine the new graph when a function is shifted horizontally to the left or right?
  3. What would happen if we reflected the graph of f(x)=x3f(x) = x^3 across the x-axis?
  4. How can we solve for the intersection points between the transformed cubic function and the x-axis?
  5. How does the behavior of the function f(x)=x3f(x) = x^3 change after a horizontal shift?

Tip: A horizontal shift (x−h)3(x - h)^3 shifts the graph to the right by hh units, while (x+h)3(x + h)^3 shifts it to the left by hh units.

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Math Problem Analysis

Mathematical Concepts

Algebra
Cubic Functions
Transformations

Formulas

f(x) = (x - h)^3

Theorems

Transformation of functions
Horizontal shift of functions

Suitable Grade Level

Grades 8-10

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