Math Problem Statement

graph

Graph the equation shown below by transforming the given graph of the parent function y = x^3 - 5.

Solution

The equation given is y=x35y = x^3 - 5, which is a transformation of the parent function y=x3y = x^3.

Steps to understand the transformation:

  • Parent function: The graph of y=x3y = x^3 is a cubic curve that passes through the origin (0,0) and has the characteristic "S" shape.
  • Transformation: The equation y=x35y = x^3 - 5 represents a vertical shift of the parent function. The entire graph is shifted down by 5 units.

How this affects the graph:

  • The key feature is that each point of the graph of y=x3y = x^3 is moved 5 units downward.
  • The inflection point, originally at (0,0)(0, 0), moves to (0,5)(0, -5).
  • Other key points follow this pattern of shifting downwards by 5 units.

Confirming with the graph:

  • The graph shows that the new curve passes through points that match the transformation. For example:
    • The inflection point is at (0,5)(0, -5).
    • Other points, such as (2,13)(-2, -13) and (2,3)(2, 3), also follow this transformation of the cubic function being shifted down.

Would you like further details or explanations?

Here are five related questions to expand your understanding:

  1. What are the general transformations of a cubic function?
  2. How does shifting a function vertically affect its graph?
  3. Can a cubic function be transformed horizontally?
  4. What other types of transformations can be applied to cubic functions?
  5. How would you graph y=(x2)3y = (x - 2)^3 or other variations?

Tip: Always focus on how transformations like shifts, stretches, and reflections affect the critical points of a function.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Graph Transformations

Formulas

y = x^3
y = x^3 - 5

Theorems

Transformation of functions
Vertical shifts in graphs

Suitable Grade Level

Grades 9-12