Math Problem Statement

The graphs below have the same shape. What is the equation of the blue graph? Given F(x) = x^3.

Solution

The problem asks for the equation of the blue graph G(x)G(x), which is stated to have the same shape as the red graph F(x)=x3F(x) = x^3. This implies that G(x)G(x) is a transformation of F(x)F(x), specifically a horizontal shift, since the two graphs appear identical except for their position along the x-axis.

Analysis:

  1. The red graph F(x)=x3F(x) = x^3 passes through the origin, whereas the blue graph G(x)G(x) seems to have the same shape but is shifted left.
  2. A horizontal shift of a cubic function x3x^3 can be described by G(x)=(xh)3G(x) = (x - h)^3, where hh is the shift. From the graph, it appears the blue graph is shifted 3 units to the left, which gives the equation: G(x)=(x+3)3G(x) = (x + 3)^3

Conclusion:

The equation of the blue graph is G(x)=(x+3)3G(x) = (x + 3)^3, which corresponds to option C.

Let me know if you want more details or have any questions! Here are five follow-up questions related to this topic:

  1. How does a horizontal shift affect the graph of a function?
  2. What is the difference between horizontal and vertical shifts in transformations?
  3. How can you identify a vertical shift from a graph?
  4. What happens to the graph of f(x)=xnf(x) = x^n when you apply a reflection over the y-axis?
  5. Can you describe how other types of transformations (like stretching or compressing) would modify the graph of x3x^3?

Tip: When analyzing graph transformations, always look at the vertex (or the key point) and check if the movement is horizontal, vertical, or both.

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Math Problem Analysis

Mathematical Concepts

Algebra
Graph Transformations
Cubic Functions

Formulas

Horizontal shift: G(x) = (x + h)^3

Theorems

Function Transformation Theorem

Suitable Grade Level

Grades 9-11