Mastering Exponential Functions: Graphing, Transformations, and Applications

PreAlgebra Grades High School 3:42 Video
Print Lesson Plan Watch Video

Lesson Description

Explore exponential functions, including their graphs, transformations, growth/decay properties, and real-world applications. Learn to identify key features like asymptotes, domain, and range.

Video Resource

Exponential Functions (How to Graph)

Mario's Math Tutoring

Duration: 3:42
Watch on YouTube

Key Concepts

  • Exponential Growth and Decay
  • Horizontal Asymptotes
  • Graph Transformations (Translations)

Learning Objectives

  • Students will be able to graph exponential functions of the form y = a(b)^(x-h) + k.
  • Students will be able to identify whether an exponential function represents growth or decay based on its equation.
  • Students will be able to determine the horizontal asymptote, domain, and range of an exponential function.
  • Students will be able to describe how changes to parameters a, b, h, and k affect the graph of y = a(b)^(x-h) + k.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the definition of exponential functions and their general form. Introduce the video by Mario's Math Tutoring as a resource for learning how to graph these functions.
  • Exponential Growth and Decay (10 mins)
    Play the video from 0:05 to 1:10. Discuss the difference between exponential growth and decay based on the base 'b' of the function. Emphasize that b > 1 represents growth and 0 < b < 1 represents decay. Show examples and ask students to identify growth vs. decay.
  • Horizontal Asymptotes (10 mins)
    Play the video from 1:10 to 2:25. Explain the concept of a horizontal asymptote and how it relates to the x-axis for basic exponential functions. Discuss how the horizontal asymptote shifts with vertical translations. Provide examples and have students identify the horizontal asymptote in different equations.
  • Graph Transformations (15 mins)
    Play the video from 2:25 to 4:24. Explain how the parameters 'h' and 'k' in the equation y = a(b)^(x-h) + k affect the graph (horizontal and vertical translations). Work through examples of graphing transformed exponential functions. Emphasize the shifting of the horizontal asymptote. Have students practice graphing transformations on provided worksheets.
  • Domain and Range (10 mins)
    Play the video from 3:33 to 4:24 and 6:50 to 7:05. Define domain and range in the context of exponential functions. Explain why the domain is typically all real numbers and how the range is affected by the horizontal asymptote and vertical translations. Practice identifying the domain and range of various exponential functions.
  • Exponential Decay Example (10 mins)
    Play the video from 4:24 to 5:24. Walk through the example of graphing an exponential decay function. Reinforce the concepts of transformations and asymptotes in the context of decay. Have students compare and contrast graphing growth vs. decay functions.
  • Wrap-up and Q&A (5 mins)
    Summarize the key concepts covered in the lesson. Allow time for students to ask questions and clarify any misunderstandings.

Interactive Exercises

  • Graphing Challenge
    Provide students with a set of exponential functions with varying transformations. Have them graph the functions and identify the key features (asymptote, intercepts, domain, range).
  • Transformation Match
    Present students with equations and corresponding graph transformations (e.g., shift right 2 units, shift up 1 unit). Have them match the correct transformation to each equation.

Discussion Questions

  • How does the value of 'b' in y = a(b)^x determine whether the function represents growth or decay?
  • What is a horizontal asymptote, and how does it affect the graph of an exponential function?
  • How do the parameters 'h' and 'k' in y = a(b)^(x-h) + k transform the graph of the parent function y = a(b)^x?
  • What are some real-world applications of exponential growth and decay?

Skills Developed

  • Graphing exponential functions
  • Analyzing function transformations
  • Identifying key features of exponential functions (asymptotes, domain, range)

Multiple Choice Questions

Question 1:

Which of the following functions represents exponential decay?

Correct Answer: y = 5(0.7)^x

Question 2:

The horizontal asymptote of the function y = 2(3)^(x-1) + 4 is:

Correct Answer: y = 4

Question 3:

The graph of y = 5^x is shifted 3 units to the left. The new equation is:

Correct Answer: y = 5^(x+3)

Question 4:

What is the domain of the function y = 4(2)^(x+2) - 1?

Correct Answer: All real numbers

Question 5:

What is the range of the function y = 3(0.5)^x + 2?

Correct Answer: y > 2

Question 6:

In the exponential function y = a(b)^(x-h) + k, what does the parameter 'k' represent?

Correct Answer: Vertical shift

Question 7:

If an exponential function has a base 'b' such that 0 < b < 1, the function represents:

Correct Answer: Exponential decay

Question 8:

Which transformation does the equation y = 2^(x) - 3 represent from the parent function y = 2^(x)?

Correct Answer: Shift down 3 units

Question 9:

The y-intercept of the function y = 2(3)^x is:

Correct Answer: (0, 2)

Question 10:

Given the function y = 4(2)^(x+1) - 5, what is the equation of the horizontal asymptote?

Correct Answer: y = -5

Fill in the Blank Questions

Question 1:

If the base 'b' of an exponential function is greater than 1, the function represents exponential _________.

Correct Answer: growth

Question 2:

The line that an exponential function approaches but never crosses is called the _________ _________.

Correct Answer: horizontal asymptote

Question 3:

In the equation y = a(b)^(x-h) + k, the parameter 'h' represents a _________ shift.

Correct Answer: horizontal

Question 4:

The _________ of an exponential function y = a(b)^x is all real numbers.

Correct Answer: domain

Question 5:

A vertical translation of an exponential function shifts the _________ _________.

Correct Answer: horizontal asymptote

Question 6:

The y-intercept of the parent function y = b^x is always (0, _________).

Correct Answer: 1

Question 7:

If the base of an exponential function is between 0 and 1, it represents exponential _________.

Correct Answer: decay

Question 8:

The value of 'k' in the equation y = a(b)^(x-h) + k represents the vertical _________.

Correct Answer: translation

Question 9:

Changing the value of 'h' in y = a(b)^(x-h) + k will shift the graph _________.

Correct Answer: horizontally

Question 10:

For the exponential function y = a(b)^(x-h) + k, the range is y > _________.

Correct Answer: k

Educational Standards

PC.F.1:Analyze functions and relations. PC.F.1.2:Sketch the graph of a function that models a relationship between two quantities, identifying key features. PC.F.1.4:Describe end behavior, asymptotic behavior, and points of discontinuity. PC.F.3.1:Predict solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans