Unlocking Exponential Functions: Graphing with Base 'e'

Algebra 2 Grades High School 6:23 Video
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Lesson Description

This lesson explores graphing exponential functions with the natural base 'e', covering its properties, graphing techniques using tables, identifying asymptotes, and understanding transformations.

Video Resource

Graphing Exponential Functions with Base e

Mario's Math Tutoring

Duration: 6:23
Watch on YouTube

Key Concepts

  • The number 'e' as an irrational number.
  • Graphing exponential functions using a table of values.
  • Identifying and understanding horizontal asymptotes.
  • Transformations of exponential functions (horizontal and vertical shifts).
  • Domain and range of exponential functions.

Learning Objectives

  • Students will be able to graph exponential functions with base 'e' using a table of values.
  • Students will be able to identify and explain the significance of the horizontal asymptote in exponential functions.
  • Students will be able to graph exponential functions with base 'e' that have been transformed (shifted horizontally and vertically).
  • Students will be able to determine the domain and range of exponential functions with base 'e'.

Educator Instructions

  • Introduction (5 mins)
    Begin by introducing the number 'e' as an irrational number, similar to pi. Discuss its approximate value (2.71) and its significance in exponential functions. Briefly recap exponential functions.
  • Graphing y = e^x (15 mins)
    Demonstrate how to graph y = e^x by creating a table of values (e.g., x = -1, 0, 1, 2). Plot the points and connect them to form the exponential curve. Discuss the exponential growth pattern.
  • Asymptotes (5 mins)
    Explain the concept of a horizontal asymptote and how it applies to the function y = e^x. Emphasize that the graph approaches the x-axis (y = 0) but never touches or crosses it. Define domain and range of y=e^x
  • Graphing y = e^-x (10 mins)
    Explain how the negative exponent affects the graph, resulting in exponential decay. Create a table of values (e.g., x = -1, 0, 1) and plot the points. Discuss how this is a decay function.
  • Graphing y = 2e^(x-1) + 3 (15 mins)
    Introduce transformations. Explain how (x-1) shifts the graph right by 1 unit and how +3 shifts the graph up by 3 units. Sketch the new asymptote. Create a small table around the new origin and plot points to sketch the final transformed graph. Define domain and range of the function.
  • Wrap-up and Key Takeaways (5 mins)
    Summarize the key concepts learned in the video. Reinforce the importance of 'e', graphing techniques, asymptotes, and transformations. Address any remaining questions.

Interactive Exercises

  • Graphing Practice
    Provide students with several exponential functions with base 'e' (with and without transformations) to graph on their own. Encourage them to use a table of values and identify the asymptote.
  • Transformation Challenge
    Give students a base function y=e^x and ask them to apply a set of transformations (e.g., shift right 2 units, shift up 1 unit) and write the equation of the transformed function.

Discussion Questions

  • How does the value of 'e' affect the growth of the exponential function?
  • What real-world phenomena can be modeled using exponential functions with base 'e'?
  • How do transformations change the asymptote of an exponential function?

Skills Developed

  • Graphing exponential functions
  • Identifying asymptotes
  • Understanding transformations of functions
  • Analyzing domain and range

Multiple Choice Questions

Question 1:

What is the approximate value of 'e'?

Correct Answer: 2.71

Question 2:

What is the horizontal asymptote of the function y = e^x?

Correct Answer: y = 0

Question 3:

Which of the following transformations shifts the graph of y = e^x to the right by 3 units?

Correct Answer: y = e^(x-3)

Question 4:

What is the domain of the function y = e^x?

Correct Answer: All real numbers

Question 5:

What is the range of the function y = e^x + 2?

Correct Answer: y > 2

Question 6:

Which of the following functions represents exponential decay?

Correct Answer: y = e^-x

Question 7:

The graph of y = e^(x-1) + 3 has been shifted. What is the new horizontal asymptote?

Correct Answer: y = 3

Question 8:

What is the y-intercept of the function y = e^x?

Correct Answer: (0, 1)

Question 9:

What does the '2' do to the graph in the function y = 2e^x?

Correct Answer: Stretches it vertically by a factor of 2

Question 10:

Which of the following is an irrational number similar to 'e'?

Correct Answer: Pi

Fill in the Blank Questions

Question 1:

The line that an exponential function approaches but never touches or crosses is called an ________.

Correct Answer: asymptote

Question 2:

The function y = e^-x represents exponential ________.

Correct Answer: decay

Question 3:

The domain of y=e^x is all ________ numbers.

Correct Answer: real

Question 4:

The range of y=e^x is y greater than ________.

Correct Answer: 0

Question 5:

The base of the natural exponential function is ________.

Correct Answer: e

Question 6:

A shift to the left is a ________ transformation.

Correct Answer: horizontal

Question 7:

The approximate value of 'e' is ________.

Correct Answer: 2.71

Question 8:

If a function is shifted up 2 units, the range changes by ________.

Correct Answer: 2

Question 9:

In the equation y = e^(x-a), 'a' is the ________ shift.

Correct Answer: horizontal

Question 10:

When graphing, creating a ________ is a helpful way to plot points.

Correct Answer: table

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.1.4:Graph exponential and logarithmic functions. Identify the domain, range, asymptotes, and x- and y-intercepts using various methods and tools that may include calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically.

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