Math Problem Statement

Determine the parent function from which the graph of the function shown below can be obtained. Next, identify each transformation that can be applied to the parent function in order to obtain the graph of the function shown below. f(x) = ∛x - 3.

Solution

Let's go through this step-by-step to solve the given problem.

Given Function:

f(x)=x33f(x) = \sqrt[3]{x} - 3

Part a: Parent Function

The function involves a cube root, so the parent function is: y=x3y = \sqrt[3]{x}

Thus, the correct parent function is:

  • y=x3y = \sqrt[3]{x}

Part b: Reflections

There is no negative sign in front of the cube root, so there is no reflection.

Part c: Stretches/Compressions

The function does not have a coefficient multiplying x3\sqrt[3]{x}, so there is no vertical stretch or compression.

Part d: Vertical Shifts

The function has 3-3 added outside the cube root, which represents a vertical shift down by 3 units.

Part e: Horizontal Shifts

There is no transformation inside the cube root (i.e., no term added or subtracted from xx), so there is no horizontal shift.

Summary:

  1. Parent function: y=x3y = \sqrt[3]{x}
  2. Reflections: None.
  3. Stretches/Compressions: None.
  4. Vertical Shifts: Down 3 units.
  5. Horizontal Shifts: None.

Let me know if you'd like further details or have any questions.

Here are some related questions for practice:

  1. What would the transformation be if the function was f(x)=x33f(x) = -\sqrt[3]{x} - 3?
  2. How would a horizontal shift affect the function f(x)=x33f(x) = \sqrt[3]{x} - 3?
  3. What is the difference between vertical and horizontal shifts?
  4. How would a coefficient in front of x3\sqrt[3]{x} affect the graph?
  5. Can a cubic root function have a reflection and a vertical shift simultaneously?

Tip: Always check for transformations both inside and outside the root or any function to understand its graph!

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Math Problem Analysis

Mathematical Concepts

Transformations
Parent Functions
Shifts

Formulas

f(x) = ∛x - 3

Theorems

Transformation of Functions

Suitable Grade Level

Grades 9-12