Mastering Parent Functions and Transformations

PreAlgebra Grades High School 5:52 Video
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Lesson Description

Explore parent functions and their transformations, including translations, reflections, stretches, and compressions, using examples involving reciprocal, square root, and cubic functions.

Video Resource

Parent Functions Transformations

Mario's Math Tutoring

Duration: 5:52
Watch on YouTube

Key Concepts

  • Parent Functions (reciprocal, square root, cubic)
  • Vertical and Horizontal Translations
  • Vertical Stretches and Compressions
  • Reflections across the x-axis

Learning Objectives

  • Identify and describe common parent functions.
  • Graph transformations of parent functions by applying translations, reflections, stretches, and compressions.
  • Write equations for transformed parent functions given a description of the transformations.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the concept of functions and their graphical representation. Introduce the idea of parent functions as the basic building blocks of more complex functions. Show the video to the students.
  • Parent Function Review (10 mins)
    Discuss the equations and graphs of the reciprocal function (y = 1/x), square root function (y = √x), and cubic function (y = x³). Emphasize their key characteristics and shapes.
  • Transformations: Translations (15 mins)
    Explain how horizontal translations (y = f(x - h)) and vertical translations (y = f(x) + k) affect the graph of a parent function. Use the video's examples to illustrate how to determine the direction and magnitude of the shifts. Provide additional examples and ask students to identify the translations.
  • Transformations: Stretches, Compressions, and Reflections (15 mins)
    Explain how vertical stretches/compressions (y = a * f(x)) and reflections across the x-axis (y = -f(x)) affect the graph of a parent function. Again, use the video's examples, paying close attention to how the 'a' value impacts the graph. Provide examples where students must identify these transformations.
  • Graphing Transformed Functions (20 mins)
    Guide students through the process of graphing transformed functions step-by-step. Emphasize the order of transformations (horizontal shift, stretch/compression/reflection, vertical shift). Utilize the table method demonstrated in the video to plot key points. Provide practice problems for students to work on individually or in pairs.
  • Summary and Q&A (5 mins)
    Summarize the key concepts of parent functions and transformations. Answer any remaining student questions.

Interactive Exercises

  • Transformation Identification
    Present students with equations of transformed parent functions and ask them to identify the transformations that have been applied.
  • Graphing Challenge
    Provide students with equations of transformed parent functions and have them sketch the graphs. Check their work and provide feedback.

Discussion Questions

  • What are the key characteristics of a parent function?
  • How does changing the value of 'h' in y = f(x - h) affect the graph of the function?
  • What is the difference between a vertical stretch and a vertical compression?
  • How does the order of transformations affect the final graph?

Skills Developed

  • Function Analysis
  • Graphical Interpretation
  • Algebraic Manipulation
  • Problem Solving

Multiple Choice Questions

Question 1:

Which of the following is the parent function for y = 2√(x - 3) + 1?

Correct Answer: y = √x

Question 2:

What transformation does the '+3' represent in the function y = (x + 3)²?

Correct Answer: Horizontal shift left 3 units

Question 3:

Which transformation does the coefficient '-2' represent in the function y = -2x³?

Correct Answer: Vertical stretch by a factor of 2 and reflection over x-axis

Question 4:

The graph of y = 1/x is shifted right 2 units and up 3 units. What is the new equation?

Correct Answer: y = 1/(x - 2) + 3

Question 5:

What is the effect of the transformation in y = (1/2)x²?

Correct Answer: Horizontal stretch

Question 6:

Which equation represents a reflection of y = x³ across the x-axis?

Correct Answer: y = -x³

Question 7:

What transformations are applied to y = √x to obtain y = √(x+1) - 2?

Correct Answer: Shift left 1, shift down 2

Question 8:

The graph of y = f(x) is compressed vertically by a factor of 3. The resulting equation is:

Correct Answer: y = (1/3)f(x)

Question 9:

What does the '-4' represent in y = (x - 2)³ - 4?

Correct Answer: Vertical shift down 4

Question 10:

Which transformation is NOT present in the equation y = -2(x + 1)² + 3?

Correct Answer: Horizontal stretch

Fill in the Blank Questions

Question 1:

The general form of the reciprocal function is y = a/(x - h) + k, where 'h' represents the ___________ shift.

Correct Answer: horizontal

Question 2:

In the function y = f(x) + k, a positive value of 'k' shifts the graph ___________.

Correct Answer: upward

Question 3:

A vertical ___________ occurs when a parent function is multiplied by a constant between 0 and 1.

Correct Answer: compression

Question 4:

Reflecting a function across the x-axis is achieved by multiplying the function by ___________.

Correct Answer: -1

Question 5:

The point (h, k) in y = a(x - h)² + k represents the ___________ of the quadratic function.

Correct Answer: vertex

Question 6:

The function y = (x - 5)³ represents a ___________ shift of the parent function y = x³.

Correct Answer: horizontal

Question 7:

In the function y = 3√x, the 3 is a vertical ___________.

Correct Answer: stretch

Question 8:

If a function is reflected over the x-axis, the sign of the ___________ values changes.

Correct Answer: y

Question 9:

In the equation y = (x+7)^2, the function shifts ______ 7 units

Correct Answer: left

Question 10:

The vertical shift of a graph is determined by the _____ value in the general equations

Correct Answer: k

Educational Standards

PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation. PC.F.1.2:Sketch the graph of a function that models a relationship between two quantities, identifying key features. PC.F.1.3:Interpret characteristics of graphs and tables for a function that models a relationship between two quantities in terms of the quantities.

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