Math Problem Statement

Below is a portion of the graph of a transformation 𝑓(π‘₯) of π‘₯3 . Following the Black, Purple, Blue, Red color scheme that is used in the textbook (as illustrated in Example 6 in Section 2.6), give the Black, Purple, Blue, Red sequence of functions that ends with the function that is graphed below. Enter your answer as a sequence of four functions separated by commas. The first term in your sequence should be π‘₯3 . The last term in your sequence should be the formula that would complete this sentence: The formula for this transfomation is 𝑓(π‘₯)=

Below is a portion of the graph of a transformation 𝑓(π‘₯) of π‘₯^3. Following the Black, Purple, Blue, Red color scheme that is used in the textbook, give the Black, Purple, Blue, Red sequence of functions that ends with the function that is graphed below.

Solution

Based on the graph you uploaded, it appears to be a transformation of the basic cubic function f(x)=x3f(x) = x^3. To determine the sequence of transformations, we will analyze the graph and describe how the cubic function was shifted, stretched, or reflected.

Let’s break down the transformations step by step, following a color scheme:

  1. Black: The base function, f(x)=x3f(x) = x^3. This is the starting point of the sequence.

  2. Purple: We notice that the graph has been reflected along the x-axis, indicating a negative sign in front of x3x^3. The function becomes: f(x)=βˆ’x3f(x) = -x^3

  3. Blue: The next transformation involves a vertical shift. Observing the graph, the cubic curve appears to have been moved downward, indicating a vertical shift. Based on the graph's behavior, we estimate that it has shifted by 6 units downward: f(x)=βˆ’x3βˆ’6f(x) = -x^3 - 6

  4. Red: Finally, there is a horizontal shift to the right, indicating the addition of a constant inside the argument of the function. The shift is by 5 units to the right: f(x)=βˆ’(xβˆ’5)3βˆ’6f(x) = -(x - 5)^3 - 6

Thus, the final transformation is: f(x)=βˆ’(xβˆ’5)3βˆ’6f(x) = -(x - 5)^3 - 6

Final answer (color sequence):

x3,βˆ’x3,βˆ’x3βˆ’6,βˆ’(xβˆ’5)3βˆ’6x^3, -x^3, -x^3 - 6, -(x - 5)^3 - 6

This is the sequence of functions ending with the given transformation.

Do you have any questions or need further clarifications?

Related Questions:

  1. What is the general rule for horizontal and vertical shifts in transformations?
  2. How do you determine whether a function has been reflected?
  3. What are the effects of stretching or compressing a function on its graph?
  4. Can you explain how to detect transformations based on the graph's shape and key points?
  5. How do you apply the sequence of transformations to other base functions, like x2x^2?

Tip:

Always pay attention to changes in symmetry when analyzing reflections, as they can indicate transformations involving a negative coefficient.

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Math Problem Analysis

Mathematical Concepts

Algebra
Cubic Functions
Graph Transformations

Formulas

Base cubic function: f(x) = x^3
Reflection: f(x) = -x^3
Vertical shift: f(x) = -x^3 - 6
Horizontal shift: f(x) = -(x - 5)^3 - 6

Theorems

Transformation of functions
Reflections
Shifts

Suitable Grade Level

Grades 9-12

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