Transformations of Parent Functions: Shift, Stretch, Reflect!

Algebra 2 Grades High School 44:33 Video
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Lesson Description

Explore how to manipulate parent functions by shifting, stretching, and reflecting them. Learn to identify transformations through function notation and create accurate graphs.

Video Resource

Shift, Stretch, Reflect Parent Functions by Identifying Transformations and Graph

Mario's Math Tutoring

Duration: 44:33
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Key Concepts

  • Parent Functions
  • Vertical and Horizontal Shifts
  • Vertical and Horizontal Stretches/Shrinks
  • Reflections over the x and y axes
  • Function Notation for Transformations
  • Order of Transformations

Learning Objectives

  • Identify and graph common parent functions (absolute value, square root, quadratic, cubic, reciprocal, greatest integer).
  • Describe transformations of parent functions using proper terminology (shift, stretch, shrink, reflection).
  • Apply function notation to represent transformations.
  • Graph transformed functions accurately by applying transformations in the correct order.
  • Determine the domain and range of transformed functions.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of a function and its graph. Introduce the idea of parent functions as the basic building blocks for more complex functions. Briefly discuss the different types of transformations that can be applied to these parent functions (shifts, stretches, and reflections).
  • Parent Functions and Tables (10 mins)
    Focus on the absolute value, square root, quadratic, cubic, reciprocal, and greatest integer functions. For each function, create a table of values and plot the corresponding points on a graph. Emphasize the characteristic shape of each parent function.
  • Transformations: Shifts (15 mins)
    Explain vertical and horizontal shifts. Use examples like f(x) + c (vertical shift) and f(x + c) (horizontal shift). Highlight the 'opposite effect' of horizontal shifts. Demonstrate how shifts affect the table of values and the graph.
  • Transformations: Stretches and Shrinks (15 mins)
    Introduce vertical and horizontal stretches and shrinks. Explain how these transformations are represented in function notation (cf(x) and f(cx)). Discuss how to determine whether a transformation is a stretch or a shrink based on the value of 'c'. Demonstrate the effect on the table and the graph.
  • Transformations: Reflections (10 mins)
    Explain reflections over the x-axis (-f(x)) and the y-axis (f(-x)). Show how reflections change the signs of the y-values (x-axis reflection) or the x-values (y-axis reflection). Demonstrate the impact on the graph.
  • Order of Transformations and Nested Functions (15 mins)
    Discuss the importance of applying transformations in the correct order. Introduce the concept of 'nested functions' and how they can help determine the correct order. Emphasize working from the 'inside out,' but also considering how the function is built up (like Russian dolls).
  • Graphing Transformed Functions (15 mins)
    Work through several examples of graphing transformed functions by applying multiple transformations in the correct order. Use tables of values to track the changes in x and y coordinates. Emphasize accuracy in plotting points and sketching the transformed graph.
  • Advanced Examples and Function Notation (15 mins)
    Tackle more challenging examples involving multiple transformations and function notation. Show students how to rewrite functions to make the transformations more apparent. Reiterate the thought process for identifying the order of transformations.
  • Conclusion (5 mins)
    Summarize the key concepts of transformations of parent functions. Encourage students to practice graphing transformed functions on their own. Remind them of the resources available for further learning.

Interactive Exercises

  • Graphing Challenge
    Provide students with a list of transformed functions and have them graph the functions by hand or using graphing software. Check their graphs for accuracy.
  • Transformation Identification
    Present students with graphs of transformed functions and ask them to identify the transformations that were applied.

Discussion Questions

  • How does the function notation help us understand transformations?
  • Why is the order of transformations important?
  • Can you give an example of a real-world application of transformations of functions?

Skills Developed

  • Analytical Skills
  • Problem-Solving Skills
  • Visual Representation Skills
  • Algebraic Manipulation Skills

Multiple Choice Questions

Question 1:

What transformation does the function f(x) + c represent?

Correct Answer: Vertical Shift

Question 2:

What transformation does the function f(x + c) represent?

Correct Answer: Horizontal Shift

Question 3:

What transformation does the function -f(x) represent?

Correct Answer: Reflection over the x-axis

Question 4:

What transformation does the function f(-x) represent?

Correct Answer: Reflection over the y-axis

Question 5:

What type of transformation is represented by the function cf(x) when c > 1?

Correct Answer: Vertical Stretch

Question 6:

What type of transformation is represented by the function f(cx) when 0 < c < 1?

Correct Answer: Horizontal Stretch

Question 7:

The function g(x) = (x - 2)² + 3 is a transformation of the parent function f(x) = x². What is the vertex of g(x)?

Correct Answer: (2, 3)

Question 8:

Which of the following transformations affects the x-values of a function?

Correct Answer: Horizontal Shift

Question 9:

Which of the following transformations affects the y-values of a function?

Correct Answer: Vertical Shift

Question 10:

What is the parent function of f(x) = |x + 3| - 2?

Correct Answer: f(x) = |x|

Fill in the Blank Questions

Question 1:

A vertical ______ by a factor of 'a' occurs when you multiply the entire function by 'a'.

Correct Answer: stretch

Question 2:

The parent function of f(x) = x³ is called the ______ function.

Correct Answer: cubic

Question 3:

When a number is grouped with 'x' inside parentheses, it affects the _______ direction.

Correct Answer: horizontal

Question 4:

The function f(x) = √x is called the _______ _______ function.

Correct Answer: square root

Question 5:

A reflection over the ______ axis makes the y-values the opposite.

Correct Answer: x

Question 6:

The function that always rounds down to the nearest integer is known as the ______ ______ function.

Correct Answer: greatest integer

Question 7:

The ______ is a good visual aid to see what a graph generally looks like.

Correct Answer: table

Question 8:

The order of transformation should generally happen from the ______ out.

Correct Answer: inside

Question 9:

Multiply or dividing causes a graph to ______ or shrink.

Correct Answer: stretch

Question 10:

Adding or subtracting causes a graph to ______.

Correct Answer: shift

Educational Standards

A2.F.1.2:Identify the parent forms of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [𝑓(𝑥 + 𝑐), 𝑓(𝑥) + 𝑐, 𝑓(𝑐𝑥), and 𝑐𝑓(𝑥)] algebraically and graphically. A2.F.1.7:Graph a radical function (square root and cube root only). Identify the domain, range, and x- and y-intercepts using various methods and tools that may include a graphing calculator or other appropriate technology.

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