Transforming Parent Functions: Shifts, Stretches, and Reflections

Algebra 2 Grades High School 8:57 Video
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Lesson Description

Explore transformations of parent functions including vertical/horizontal shifts, stretches/compressions, and reflections. Learn to graph functions by understanding how these transformations affect the parent graph.

Video Resource

Parent Function Graphs Transformations (Shift, Stretch, Reflect)

Mario's Math Tutoring

Duration: 8:57
Watch on YouTube

Key Concepts

  • Parent Functions
  • Horizontal and Vertical Shifts
  • Vertical Stretches and Compressions
  • Reflections over the x-axis
  • Asymptotes of Rational Functions

Learning Objectives

  • Identify the parent function from a given transformed equation.
  • Determine the horizontal and vertical shifts from the equation.
  • Determine the vertical stretch or compression factor from the equation.
  • Determine if a reflection over the x-axis has occurred.
  • Graph transformed functions using a shifted origin and a table of values.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing common parent functions (absolute value, square root, reciprocal). Briefly discuss the general forms of these functions.
  • Video Presentation (15 mins)
    Play the Mario's Math Tutoring video 'Parent Function Graphs Transformations (Shift, Stretch, Reflect)'. Encourage students to take notes on the general forms of equations and how each parameter affects the graph.
  • Guided Practice (20 mins)
    Work through the examples from the video as a class, pausing to answer questions and clarify concepts. Focus on identifying the transformations and how they relate to the equation. Emphasize the 'shifted origin' concept. Use the table of values method to plot points.
  • Independent Practice (15 mins)
    Provide students with a worksheet of similar problems (ideally the one linked in the video description). Have them work individually or in pairs to graph the transformed functions.
  • Wrap-up (5 mins)
    Review the key concepts and address any remaining questions. Preview the upcoming topic (e.g., transformations of other types of functions).

Interactive Exercises

  • Desmos Graphing Challenge
    Give students equations of transformed functions and challenge them to graph them accurately on Desmos. They can then compare their graphs to the actual graphs to check their understanding.
  • Transformation Matching Game
    Create cards with equations of transformed functions and cards with descriptions of the transformations. Students must match the equation to the correct description.

Discussion Questions

  • How does changing the value inside the function, like f(x + c), affect the graph compared to changing the value outside the function, like f(x) + c?
  • Why do we consider a 'shifted origin' when graphing transformed functions?
  • How can you quickly identify a reflection over the x-axis from the equation?
  • What are asymptotes and why are they important when graphing rational functions?

Skills Developed

  • Equation Analysis
  • Graphing Techniques
  • Visual Reasoning
  • Problem-Solving

Multiple Choice Questions

Question 1:

Which transformation does the '-2' represent in the equation y = |x| - 2?

Correct Answer: Shift down 2 units

Question 2:

In the equation y = 3√(x), what effect does the '3' have on the graph of the parent function?

Correct Answer: Vertical stretch by a factor of 3

Question 3:

What transformation does a negative sign in front of a function, like y = -f(x), represent?

Correct Answer: Reflection over the x-axis

Question 4:

The graph of y = √(x + 4) is the graph of y = √x shifted:

Correct Answer: 4 units to the left

Question 5:

Which of the following equations represents a vertical compression of the absolute value function by a factor of 1/2?

Correct Answer: y = (1/2)|x|

Question 6:

What is the horizontal asymptote of the function y = 5/(x-2) + 1?

Correct Answer: y = 1

Question 7:

What point should be considered as the 'shifted origin' for the absolute value function y = |x - 3| + 2?

Correct Answer: (3, 2)

Question 8:

Which transformation is represented by the equation y = f(-x)?

Correct Answer: Reflection across the y-axis

Question 9:

The range of the function y = |x| - 5 is:

Correct Answer: y ≥ -5

Question 10:

Which function has a vertical asymptote at x = -1?

Correct Answer: y = 1/(x+1)

Fill in the Blank Questions

Question 1:

The transformation that shifts a graph left or right is called a __________ shift.

Correct Answer: horizontal

Question 2:

The general form for vertical shift is f(x) + k, where a positive k shifts the graph ___________.

Correct Answer: up

Question 3:

If a > 1 in the equation y = a|x|, the absolute value graph experiences a vertical __________.

Correct Answer: stretch

Question 4:

A reflection over the x-axis is represented by placing a __________ sign in front of the function.

Correct Answer: negative

Question 5:

In y = √(x - 2), the graph of y = √x is shifted 2 units to the __________.

Correct Answer: right

Question 6:

The point from which we plot the transformed graph is called the shifted ___________.

Correct Answer: origin

Question 7:

In the function y = a/x, the lines that the graph approaches but never touches are called ___________.

Correct Answer: asymptotes

Question 8:

The ________ of a function is the set of all possible input values (x-values).

Correct Answer: domain

Question 9:

The ________ of a function is the set of all possible output values (y-values).

Correct Answer: range

Question 10:

If 0 < a < 1 in the function y = a*f(x), the graph experiences 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.1.7: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. A2.F.1.6:Graph a rational function and identify the domain (including holes), range, x- and y-intercepts, vertical and horizontal asymptotes, using various methods and tools that may include a graphing calculator or other appropriate technology (excluding slant or oblique asymptotes).

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