Transforming Absolute Value Functions: From Parent to Complex Graphs

Algebra 2 Grades High School 3:44 Video
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Lesson Description

Explore the absolute value function, its parent graph, and how transformations like shifts, stretches, compressions, and reflections affect its graph. Learn to graph absolute value functions from their equations.

Video Resource

Absolute Value Function (How to Graph)

Mario's Math Tutoring

Duration: 3:44
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Key Concepts

  • Absolute Value Function
  • Parent Function
  • Transformations (Shifting, Stretching, Compressing, Reflecting)
  • Vertex of an Absolute Value Function

Learning Objectives

  • Graph the parent absolute value function.
  • Identify and apply transformations to absolute value functions given an equation.
  • Determine the vertex of a transformed absolute value function.
  • Sketch the graph of transformed absolute value functions.
  • Determine the domain and range of graphed absolute value functions.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of absolute value: the distance from zero. Introduce the absolute value notation (|x|). Briefly discuss why absolute values are always non-negative.
  • Graphing the Parent Function (10 mins)
    Demonstrate how to graph the parent function, y = |x|, by creating a table of values with both negative and positive inputs. Plot the points and connect them to form the V-shape. Emphasize the vertex at (0,0) and the symmetry about the y-axis.
  • Understanding Transformations (15 mins)
    Introduce the general form of an absolute value function: y = a|x - h| + k. Explain the role of each parameter: 'a' (vertical stretch/compression and reflection), 'h' (horizontal shift), and 'k' (vertical shift). Explain that h has the opposite effect of the sign, and k has the same effect of the sign. Provide examples of how each transformation affects the graph, relating it back to the video.
  • Graphing Transformed Functions (20 mins)
    Work through examples of graphing transformed absolute value functions. Start by identifying the vertex (h, k). Use 'a' as the "slope" to find other points on the graph. Emphasize the symmetry of the graph. Work through examples similar to those in the video, such as y = -2|x - 1| + 4 and y = (1/3)|x + 2| - 1.
  • Domain and Range (5 mins)
    Review how to determine the domain and range of an absolute value function from its graph. Discuss that the domain of all absolute value functions is all real numbers. Explain that the range depends on the vertex and whether the function opens upward or downward.
  • Practice and Review (10 mins)
    Provide students with practice problems to graph transformed absolute value functions. Circulate to provide assistance and answer questions. Review the key concepts and address any remaining uncertainties.

Interactive Exercises

  • Graphing Challenge
    Present students with equations of transformed absolute value functions and have them graph the functions on graphing paper or using graphing software. Then, ask them to state the vertex, domain, and range of each function.
  • Transformation Game
    Create a game where students must identify the transformations applied to an absolute value function based on its equation or graph. For example, "What transformations are applied to y = |x| to get y = 2|x + 3| - 5?"

Discussion Questions

  • How does changing the 'a' value affect the graph of the absolute value function?
  • Why is the absolute value of a number always non-negative?
  • Explain how to find the vertex of an absolute value function given its equation.
  • How can you determine the range of an absolute value function from its graph?

Skills Developed

  • Graphing absolute value functions
  • Identifying and applying transformations
  • Determining domain and range
  • Analytical thinking

Multiple Choice Questions

Question 1:

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

Correct Answer: (2, 3)

Question 2:

Which transformation does the 'a' value in y = a|x - h| + k control?

Correct Answer: Vertical stretch/compression and reflection

Question 3:

The graph of y = -|x| is a reflection of y = |x| over which axis?

Correct Answer: x-axis

Question 4:

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

Correct Answer: All real numbers

Question 5:

What is the range of the function y = |x| + 3?

Correct Answer: y ≥ 3

Question 6:

Which equation represents a vertical stretch of the parent absolute value function?

Correct Answer: y = 2|x|

Question 7:

Which transformation shifts the graph of y = |x| to the left?

Correct Answer: y = |x + c|

Question 8:

If the 'a' value is negative, the absolute value graph opens:

Correct Answer: Downward

Question 9:

The vertex of the absolute value function is also:

Correct Answer: Minimum or Maximum Point

Question 10:

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

Correct Answer: Vertical Shift

Fill in the Blank Questions

Question 1:

The parent function of the absolute value function is y = _______.

Correct Answer: |x|

Question 2:

The _______ of an absolute value function is always all real numbers.

Correct Answer: domain

Question 3:

The vertex of the graph y = |x + 3| - 2 is at the point (_______, _______).

Correct Answer: -3,-2

Question 4:

A negative 'a' value in y = a|x - h| + k causes a _______ across the x-axis.

Correct Answer: reflection

Question 5:

The 'h' value in the equation y = a|x - h| + k shifts the graph _______.

Correct Answer: horizontally

Question 6:

If |a| > 1, then the graph of y = a|x| is a vertical _______.

Correct Answer: stretch

Question 7:

If 0 < |a| < 1, then the graph of y = a|x| is a vertical _______.

Correct Answer: compression

Question 8:

The 'k' value in the equation y = a|x - h| + k shifts the graph _______.

Correct Answer: vertically

Question 9:

The sharp corner of an absolute value function is called the _______.

Correct Answer: vertex

Question 10:

The absolute value of any number is its _______ from zero.

Correct Answer: distance

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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