Math Problem Statement

Given the image below, which function best represents y2?

Solution

The graph provided shows several cubic functions, with a focus on identifying which function best represents y2y_2. Based on the graph, we can analyze the different functions:

  1. The curve y1y_1 is shifted to the left compared to the standard cubic function y=x3y = x^3, indicating a horizontal transformation.
  2. The curve y2y_2 seems to have a similar shape but is further shifted horizontally to the left compared to y1y_1. This is a key clue that it involves a transformation of the form y=(x+c)3y = (x + c)^3, where cc represents a horizontal shift.

By observing the positions of the curves and knowing that cubic functions have the form y=(x+a)3y = (x + a)^3, we can eliminate options that don't involve such shifts.

The function y=(x+3)3y = (x + 3)^3 (which is highlighted in the image) best matches the curve for y2y_2, as it represents a cubic function shifted 3 units to the left. This aligns with the position of y2y_2 on the graph.

Thus, the correct function for y2y_2 is:

y2=(x+3)3y_2 = (x + 3)^3

Would you like further clarification on the transformations involved?


Here are five related questions to expand on this concept:

  1. How does the graph of y=x3y = x^3 change when you add a constant inside the parentheses, such as y=(x+2)3y = (x + 2)^3?
  2. What is the effect of adding or subtracting a constant outside the cubic function, like in y=x3+2y = x^3 + 2?
  3. How would the graph of y=x3y = -x^3 differ from the graph of y=x3y = x^3?
  4. Can you describe how vertical stretching or compression affects the graph of a cubic function?
  5. How does the graph of a cubic function behave as xx \to \infty and xx \to -\infty?

Tip: When working with transformations, always identify whether the shift occurs horizontally (inside the function) or vertically (outside the function).

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Math Problem Analysis

Mathematical Concepts

Cubic Functions
Function Transformations
Horizontal Shifts

Formulas

y = x^3
y = (x + c)^3

Theorems

Transformation Theorem: Horizontal shifts in cubic functions

Suitable Grade Level

Grades 9-12