Mastering Transformations: Parent Functions Unleashed!

Algebra 2 Grades High School 13:57 Video
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Lesson Description

Explore advanced transformations of parent functions, focusing on the order of operations for horizontal shifts, stretches, and compressions. Learn to apply the formulas g(x) = af(b(x-h)) +k and g(x)=af(bx-h) + k and use tables to graph transformations accurately.

Video Resource

Transformations of Parent Graphs (Advanced)

Mario's Math Tutoring

Duration: 13:57
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Key Concepts

  • Parent functions (absolute value, square root, quadratic, exponential)
  • Horizontal and vertical shifts, stretches, and compressions
  • Order of transformations based on function form (factored vs. distributed)
  • Reflection over the x and y axis
  • Using tables to track coordinate transformations

Learning Objectives

  • Students will be able to identify the correct order of transformations based on the function's form (g(x) = af(b(x-h)) + k vs. g(x)=af(bx-h) + k).
  • Students will be able to accurately graph transformed parent functions using tables and applying transformations step-by-step.
  • Students will be able to describe the transformation that occurs to a parent function given an equation.
  • Students will be able to distinguish between horizontal stretches/compressions and horizontal shifts, understanding their reciprocal relationship.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing parent functions and basic transformations (shifting, stretching, reflecting). Briefly introduce the complexity of combined horizontal transformations and the importance of order of operations. Show the video to the students.
  • Video Analysis and Discussion (15 mins)
    After watching the video, discuss the two main forms of transformed functions (factored and distributed B value). Emphasize the rules for determining the order of transformations in each case. Highlight the 'opposite effect' of horizontal stretches/compressions and shifts.
  • Worked Examples (20 mins)
    Work through the examples from the video (absolute value, square root, quadratic, exponential) on the board, emphasizing the step-by-step application of transformations. Use different colors to highlight each transformation on the graph. Explain clearly why the order matters.
  • Table Method Practice (15 mins)
    Guide students through creating tables to track the transformation of key points on parent functions. Provide additional examples and encourage students to work independently or in pairs.
  • Wrap-up and Q&A (5 mins)
    Summarize the key takeaways regarding the order of transformations. Address any remaining questions and preview the upcoming practice problems/homework.

Interactive Exercises

  • Card Sort Activity
    Create cards with equations of transformed functions and corresponding descriptions of the transformations (e.g., 'horizontal shift right 2', 'vertical stretch by 3'). Students match the equations to the correct transformation descriptions, paying attention to the order.
  • Graphing Challenge
    Provide students with a set of parent functions and a list of transformations. Students work individually or in groups to graph the transformed functions accurately using the table method.

Discussion Questions

  • Why is it important to understand the order of transformations?
  • How does the factored vs. distributed form of the function affect the transformation process?
  • Explain the reciprocal effect of the 'b' value in horizontal stretches/compressions.
  • Can you think of real-world applications of transformations?

Skills Developed

  • Analytical thinking
  • Problem-solving
  • Visual representation
  • Attention to detail

Multiple Choice Questions

Question 1:

Which transformation should be performed FIRST on the function g(x) = 2f(3(x-1)) + 4?

Correct Answer: Horizontal compression by 1/3

Question 2:

The function g(x) = f(2x + 6) can be rewritten as which of the following to clearly show the horizontal transformations?

Correct Answer: g(x) = f(2(x + 3))

Question 3:

What is the effect of a negative 'a' value in the transformation g(x) = af(x)?

Correct Answer: Reflection over the x-axis

Question 4:

For the function g(x) = f(bx), if 'b' is greater than 1, what type of transformation occurs?

Correct Answer: Horizontal compression

Question 5:

What transformation does the '+ k' represent in the function g(x) = f(x) + k?

Correct Answer: Vertical shift up

Question 6:

In the function g(x)=af(bx-h)+k, which transformation should be applied first?

Correct Answer: Horizontal Shift

Question 7:

Which of the following represents a vertical compression?

Correct Answer: 1/2 * f(x)

Question 8:

Which of the following represents a reflection over the y axis?

Correct Answer: f(-x)

Question 9:

What does the 'h' in f(x-h) represent?

Correct Answer: Horizontal Shift

Question 10:

Which of the following equations represents a horizontal stretch by a factor of 3?

Correct Answer: f(1/3 * x)

Fill in the Blank Questions

Question 1:

In the function g(x) = af(b(x-h)) + k, the 'a' value represents a ______ stretch or compression.

Correct Answer: vertical

Question 2:

A negative sign in front of the function, such as -f(x), results in a reflection over the ______ axis.

Correct Answer: x

Question 3:

The 'h' value in f(x - h) represents a horizontal _______.

Correct Answer: shift

Question 4:

If b > 1 in f(bx), the graph experiences a horizontal _______.

Correct Answer: compression

Question 5:

The formula g(x) = af(b(x-h)) + k indicates that the horizontal shift occurs _______ the horizontal stretch/compression.

Correct Answer: after

Question 6:

In g(x)=af(bx-h) + k, the horizontal shift should be applied _______ the horizontal stretch/compression.

Correct Answer: before

Question 7:

A function f(x)+k will have a _______ shift if k is positive.

Correct Answer: vertical

Question 8:

The parent function f(x)=x^2 is a _______ function.

Correct Answer: quadratic

Question 9:

The reciprocal effect impacts _______ transformations.

Correct Answer: horizontal

Question 10:

Multiplying f(x) by a constant between 0 and 1 results in a vertical _______.

Correct Answer: compression

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.1.1:Use algebraic, interval, and set notations to specify the domain and range of various types of functions, and evaluate a function at a given point in its domain.

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