Transforming Absolute Value Functions: A Visual Guide

Algebra 2 Grades High School 6:52 Video
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Lesson Description

Explore graphing absolute value functions of the form y=a|x-h|+k, understanding transformations, and identifying key features like vertex, shifts, stretches, shrinks, and reflections.

Video Resource

Graphing Absolute Value Functions (y=a|x-h|+k)

Mario's Math Tutoring

Duration: 6:52
Watch on YouTube

Key Concepts

  • Parent function of absolute value: y = |x|
  • Vertex form of absolute value function: y = a|x-h|+k
  • Transformations: vertical/horizontal shifts, vertical stretch/shrink, reflection

Learning Objectives

  • Students will be able to graph absolute value functions in vertex form.
  • Students will be able to identify and describe the transformations applied to the parent function.
  • Students will be able to determine the vertex, domain, and range of an absolute value function.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of absolute value and its effect on numbers. Introduce the parent function y = |x| and its characteristic V-shape. Briefly discuss the concept of transformations in general.
  • Video Presentation (15 mins)
    Play the Mario's Math Tutoring video on graphing absolute value functions. Encourage students to take notes on the key concepts and examples presented. Pause the video at strategic points to clarify and answer questions.
  • Vertex and Transformations (15 mins)
    Explain the role of 'h', 'k', and 'a' in the vertex form y = a|x-h|+k. Emphasize that 'h' causes a horizontal shift (opposite sign), 'k' causes a vertical shift (same sign), and 'a' causes a vertical stretch/shrink/reflection. Work through examples, highlighting how each parameter affects the graph.
  • Graphing Examples (15 mins)
    Provide additional examples of absolute value functions in vertex form. Guide students through the process of identifying the vertex, applying the transformations, and sketching the graph. Encourage students to use graph paper or graphing software for accuracy.
  • Domain and Range (5 mins)
    Discuss how to determine the domain and range of absolute value functions based on their graphs. Reinforce that the domain is typically all real numbers, and the range is determined by the vertex and the direction of opening.

Interactive Exercises

  • Graphing Challenge
    Provide students with a set of absolute value functions in vertex form and challenge them to graph them accurately. Have students compare their graphs with each other.
  • Transformation Matching
    Create a matching activity where students match equations of absolute value functions with descriptions of their transformations (e.g., shifted right 3, stretched vertically by a factor of 2).

Discussion Questions

  • How does changing the value of 'a' affect the graph of the absolute value function?
  • Why does the 'h' value in the vertex form cause a horizontal shift in the opposite direction?
  • How can you determine the range of an absolute value function just by looking at its equation?

Skills Developed

  • Graphing functions
  • Identifying transformations
  • Analyzing equations
  • Problem-solving

Multiple Choice Questions

Question 1:

What is the vertex of the absolute value function y = |x - 2| + 3?

Correct Answer: (2, 3)

Question 2:

The function y = -2|x + 1| - 4 is reflected over which axis?

Correct Answer: x-axis

Question 3:

What transformation does 'h' cause in the equation y = a|x - h| + k?

Correct Answer: Horizontal shift

Question 4:

The graph of y = |x| is transformed to y = 3|x|. This is a:

Correct Answer: Vertical stretch

Question 5:

What is the domain of the function y = 5|x - 7| + 2?

Correct Answer: All real numbers

Question 6:

What is the range of y = |x| - 5?

Correct Answer: y ≥ -5

Question 7:

In the equation y = a|x - h| + k, which variable controls vertical translation?

Correct Answer: k

Question 8:

The graph of y = |x + 4| is shifted:

Correct Answer: Left 4 units

Question 9:

Which equation represents an absolute value function that opens downwards?

Correct Answer: y = -|x|

Question 10:

What is the vertex of the function y = -|x|?

Correct Answer: (0, 0)

Fill in the Blank Questions

Question 1:

The parent function of an absolute value function is y = ____.

Correct Answer: |x|

Question 2:

In the equation y = a|x - h| + k, the _____ value determines the vertical shift.

Correct Answer: k

Question 3:

If 'a' is negative in y = a|x - h| + k, the graph is reflected over the _____.

Correct Answer: x-axis

Question 4:

The point where the absolute value graph bends is called the _____.

Correct Answer: vertex

Question 5:

A vertical stretch occurs when 'a' is _____ than 1.

Correct Answer: greater

Question 6:

The horizontal shift is determined by the ______ value.

Correct Answer: h

Question 7:

The _____ of an absolute value function is all real numbers.

Correct Answer: domain

Question 8:

If 'a' is between 0 and 1, the absolute value graph experiences a vertical _____.

Correct Answer: shrink

Question 9:

The absolute value function y = |x - 5| is shifted 5 units to the _____.

Correct Answer: right

Question 10:

In the vertex form of an absolute value equation, y=a|x-h|+k, the coordinates of the vertex are (_____, ______).

Correct Answer: h, k

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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