Mastering Transformations of Parent Functions

PreAlgebra Grades High School 14:23 Video
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Lesson Description

Explore transformations of parent functions, including vertical stretches/shrinks, reflections, and horizontal/vertical shifts. Learn to graph transformed functions using key points and understand the impact of parameters on the parent function.

Video Resource

Transformations of Parent Functions Strategy Explained

Mario's Math Tutoring

Duration: 14:23
Watch on YouTube

Key Concepts

  • Parent Functions: Understanding the basic shapes and key points of common functions (quadratic, cubic, square root, cube root, reciprocal, absolute value).
  • Transformations: Identifying and applying vertical stretches/shrinks (affecting the y-values), reflections over the x-axis, horizontal shifts (affecting the x-values), and vertical shifts (affecting the y-values).
  • Order of Transformations: Applying transformations in the correct order to achieve the desired graph.

Learning Objectives

  • Identify and graph six common parent functions: quadratic, cubic, square root, cube root, reciprocal, and absolute value.
  • Apply transformations (vertical stretches/shrinks, reflections, horizontal/vertical shifts) to parent functions using tables of values.
  • Determine the equation of a transformed function given its graph, and vice versa.

Educator Instructions

  • Introduction to Parent Functions (10 mins)
    Briefly review the definition of a parent function. Display and discuss the six common parent functions: quadratic (y = x^2), cubic (y = x^3), square root (y = √x), cube root (y = ∛x), reciprocal (y = 1/x), and absolute value (y = |x|). Emphasize the importance of knowing key points for each function (as presented in the video).
  • Understanding Transformations (15 mins)
    Explain the general form of a transformed function: y = a * f(x - h) + k. Discuss the effects of 'a' (vertical stretch/shrink and reflection), 'h' (horizontal shift), and 'k' (vertical shift). Stress that the transformations must be done according to order of operations and the video's recommended order. Use examples to illustrate each transformation individually.
  • Applying Transformations with Tables (20 mins)
    Guide students through the process of transforming parent functions using tables of values. Start with the key points of the parent function. Demonstrate how to apply transformations step-by-step, modifying the x and y values in the table accordingly. Work through the examples provided in the video, emphasizing the order of transformations (vertical stretch/shrink/reflection, horizontal shift, vertical shift).
  • Practice and Examples (20 mins)
    Provide additional examples of transformed functions for students to graph. Encourage students to work independently or in small groups. Circulate to provide assistance and answer questions. Review student work and address any common errors.
  • Wrap up (5 mins)
    Review the key concepts covered in the lesson. Remind students of the importance of understanding parent functions and transformations for future topics in precalculus. Answer any remaining questions.

Interactive Exercises

  • Graphing Challenge
    Provide students with equations of transformed functions and have them graph them using online graphing tools or graph paper. Students should then compare their graphs with the actual graphs generated by the tools.
  • Transformation Matching
    Create a set of cards with equations of transformed functions and another set of cards with corresponding graphs. Have students match the equations with their graphs.

Discussion Questions

  • How does changing the value of 'a' in y = a * f(x) affect the graph of the function?
  • Why does the horizontal shift (x - h) have the 'opposite' effect compared to the vertical shift (+ k)?
  • In what order should transformations be applied to ensure the correct graph?

Skills Developed

  • Graphing Functions
  • Analytical Thinking
  • Problem-Solving

Multiple Choice Questions

Question 1:

Which transformation does the parameter 'a' in the equation y = a*f(x) represent?

Correct Answer: Vertical Stretch/Shrink/Reflection

Question 2:

What is the parent function of y = (x - 2)^2 + 3?

Correct Answer: y = x^2

Question 3:

Which transformation shifts the graph left or right?

Correct Answer: Horizontal Shift

Question 4:

What transformation does a negative 'a' value represent in y = a*f(x)?

Correct Answer: Reflection over the x-axis

Question 5:

If f(x) = x^3, what is the effect of the transformation f(x - 1)?

Correct Answer: Shift right 1 unit

Question 6:

The function y = 2|x| is a vertical ____ of the parent function y = |x|.

Correct Answer: stretch

Question 7:

Which of the following represents a vertical shift up 3 units?

Correct Answer: f(x) + 3

Question 8:

The graph of y = 1/x has vertical and horizontal ____.

Correct Answer: asymptotes

Question 9:

In the transformation y = -f(x), the negative sign causes a reflection over the ____.

Correct Answer: x-axis

Question 10:

What is the key feature that distinguishes the graph of y = |x| from y = x^2?

Correct Answer: sharp V shape

Fill in the Blank Questions

Question 1:

The basic form of a function before any transformations are applied is called the _____ function.

Correct Answer: parent

Question 2:

A vertical ______ occurs when the absolute value of 'a' is greater than 1 in the equation y = a*f(x).

Correct Answer: stretch

Question 3:

The transformation y = f(x) - 2 shifts the graph _____ 2 units.

Correct Answer: down

Question 4:

If a function is reflected over the x-axis, all the y-values are multiplied by ______.

Correct Answer: -1

Question 5:

For the function y = f(x + 4), the graph is shifted to the ______ by 4 units.

Correct Answer: left

Question 6:

The parent function of y = √(x) is the ______ function.

Correct Answer: square root

Question 7:

A vertical shrink is also known as a vertical ______.

Correct Answer: compression

Question 8:

The points used in order to sketch the transformed function are determined by modifying the key points of the ______ function.

Correct Answer: parent

Question 9:

The vertical asymptote for y = 1/x is at x = ______.

Correct Answer: 0

Question 10:

Applying transformations in a different order than indicated may result in a(n) ______ graph.

Correct Answer: incorrect

Educational Standards

PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation. PC.F.1.2:Sketch the graph of a function that models a relationship between two quantities, identifying key features. PC.F.3.2:Graphically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic.

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