Transformations of Parent Functions: Shift, Stretch, Reflect!

Algebra 2 Grades High School 11:09 Video
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Lesson Description

Explore how to shift, stretch, and reflect parent functions to create new graphs. Understand the impact of constants within equations on graphical transformations.

Video Resource

Shifting, Stretching and Reflecting Parent Function Graphs

Mario's Math Tutoring

Duration: 11:09
Watch on YouTube

Key Concepts

  • Parent functions
  • Horizontal and vertical shifts
  • Vertical and horizontal stretches/compressions
  • Reflections over the x-axis and y-axis

Learning Objectives

  • Identify and graph six basic parent functions: square root, absolute value, reciprocal, cubic, step, and quadratic.
  • Determine the transformations applied to a parent function based on its equation.
  • Graph transformed functions by applying shifts, stretches/compressions, and reflections.
  • Express transformations in general form: y = a * f(x - h) + k.

Educator Instructions

  • Introduction (5 mins)
    Begin by introducing the concept of parent functions and their importance as a foundation for understanding more complex graphs. Briefly review the six basic parent functions: square root, absolute value, reciprocal, cubic, step, and quadratic. Show the diagrams of each function.
  • Horizontal Shifts (10 mins)
    Explain how adding or subtracting a constant within the function (i.e., inside the parentheses or absolute value) affects the graph horizontally. Emphasize that the shift is in the *opposite* direction of the sign (e.g., x - 2 shifts the graph right 2 units). Use examples of square root and absolute value functions to illustrate.
  • Vertical Stretches/Compressions (10 mins)
    Explain how multiplying a function by a constant affects the graph vertically. If the constant is greater than 1, it's a vertical stretch. If the constant is between 0 and 1, it's a vertical compression. Use the reciprocal function as an example.
  • General Form and Examples (15 mins)
    Introduce the general form of a transformed function: y = a * f(x - h) + k. Explain that 'a' controls vertical stretch/compression and reflection over the x-axis, 'h' controls horizontal shift, and 'k' controls vertical shift. Work through several examples, including transforming a cubic function and a square root function. Demonstrate the shortcut of shifting the origin first.
  • Reflections (10 mins)
    Explain reflections over the x-axis (multiplying the entire function by -1) and the y-axis (replacing x with -x). Use examples like the step function and quadratic function to illustrate. Highlight that reflection over the y-axis affects the x-values, while reflection over the x-axis affects the y-values.
  • Summary and Conclusion (5 mins)
    Summarize the different types of transformations and their effects on parent functions. Reiterate the importance of recognizing the parent function and understanding how the constants in the equation alter its graph.

Interactive Exercises

  • Graphing Challenge
    Provide students with equations of transformed functions (e.g., y = 2(x + 1)^2 - 3). Have them sketch the graphs, identifying the parent function and the transformations applied. They can use graphing calculators or online graphing tools to check their answers.
  • Transformation Match
    Create cards with equations of transformed functions and matching cards with descriptions of the transformations (e.g., 'Shifted left 3 units, vertically stretched by a factor of 2'). Have students match the equations to the correct transformations.

Discussion Questions

  • How does changing the value of 'h' in y = f(x - h) affect the graph of the function?
  • What is the difference between a vertical stretch and a horizontal compression?
  • Can you give an example of a real-world scenario where transformations of functions might be used to model data?
  • How does the order of transformations affect the final graph? Is there an order of operations for transformations?

Skills Developed

  • Function analysis
  • Graphical representation
  • Algebraic manipulation
  • Problem-solving

Multiple Choice Questions

Question 1:

What transformation does the '+3' represent in the function y = |x + 3|?

Correct Answer: Shift left 3 units

Question 2:

What type of transformation occurs when a function is multiplied by a constant greater than 1 (e.g., y = 2f(x))?

Correct Answer: Vertical stretch

Question 3:

Which of the following functions represents a reflection of y = x^2 over the x-axis?

Correct Answer: y = -x^2

Question 4:

In the general form y = a * f(x - h) + k, what does the 'h' represent?

Correct Answer: Horizontal shift

Question 5:

What is the parent function of y = √(x - 5)?

Correct Answer: y = √x

Question 6:

What transformation does the '2' represent in the function y = (2x)^2?

Correct Answer: Horizontal compression by a factor of 1/2

Question 7:

Which transformation results from replacing x with -x in a function?

Correct Answer: Reflection over the y-axis

Question 8:

What transformation moves every point on a graph the same distance in the same direction?

Correct Answer: Translation

Question 9:

If the parent function y = x^3 is transformed to y = (x - 2)^3 + 1, which transformations occur?

Correct Answer: Shift right 2, shift up 1

Question 10:

Which equation represents a vertical compression of the function y = |x| by a factor of 1/3?

Correct Answer: y = (1/3)|x|

Fill in the Blank Questions

Question 1:

The function y = f(x) + k represents a vertical _______ if k is positive.

Correct Answer: shift

Question 2:

A vertical _______ occurs when a function is multiplied by a constant between 0 and 1.

Correct Answer: compression

Question 3:

Replacing x with -x in a function results in a _______ over the y-axis.

Correct Answer: reflection

Question 4:

In the general form y = a * f(x - h) + k, the 'k' value determines the _______ shift.

Correct Answer: vertical

Question 5:

The graph of y = √(x + 4) is the graph of y = √x shifted 4 units to the _______.

Correct Answer: left

Question 6:

If a function is multiplied by -1, it is reflected over the _______-axis.

Correct Answer: x

Question 7:

A horizontal stretch occurs when the x-values are _______ by a number greater than one.

Correct Answer: multiplied

Question 8:

A rigid transformation that shifts the graph without changing its shape is called a ______.

Correct Answer: translation

Question 9:

In the equation y = |x - 3|, the vertex of the absolute value function is shifted _______ units to the right.

Correct Answer: 3

Question 10:

The parent function of y = (x + 2)^3 - 5 is y = _______.

Correct Answer: x^3

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.

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