Graphing Exponential Functions with Natural Base 'e'

Algebra 2 Grades High School 7:39 Video
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Lesson Description

Learn how to graph exponential functions involving the natural base 'e' using transformations and tables. This lesson covers identifying key features such as asymptotes, domain, and range.

Video Resource

How to Graph Exponential Functions with Natural Base e

Mario's Math Tutoring

Duration: 7:39
Watch on YouTube

Key Concepts

  • The natural base 'e' as an irrational number approximately equal to 2.7.
  • The parent function y = e^x and its graph.
  • Transformations of exponential functions: vertical and horizontal shifts, stretches, and reflections.
  • Horizontal asymptotes and their role in exponential functions.
  • Domain and range of exponential functions.

Learning Objectives

  • Students will be able to graph the parent exponential function y = e^x.
  • Students will be able to apply transformations (shifts, stretches, and reflections) to graph exponential functions with base 'e'.
  • Students will be able to identify the domain, range, and horizontal asymptote of an exponential function with base 'e'.
  • Students will be able to analyze an equation and determine the correct transformations to perform.

Educator Instructions

  • Introduction (5 mins)
    Begin by discussing the number 'e' and its approximate value (2.7). Explain that 'e' is similar to Pi in that it's an irrational number. Introduce the concept of the natural exponential function y = e^x as a parent function.
  • Graphing the Parent Function y = e^x (10 mins)
    Create a table of values for y = e^x using values like x = -1, 0, 1, and 2. Approximate e^-1 as 1/3, e^0 as 1, e^1 as 2.7, and e^2 as 7.4. Plot these points and sketch the graph. Emphasize the horizontal asymptote at y = 0. Define the domain as all real numbers and the range as y > 0.
  • Transformations of Exponential Functions (20 mins)
    Explain how transformations affect the graph of y = e^x. Cover vertical shifts (y = e^x + k), horizontal shifts (y = e^(x - h)), vertical stretches/compressions (y = a*e^x), and reflections across the x-axis (y = -e^x) and y-axis (y = e^-x). Work through Example 2 from the video: y = 2e^(x-1) - 1. Identify each transformation step-by-step. Work through Example 3 from the video: y = -e^(-x) + 2, identifying each transformation, and the corresponding changes to the graph.
  • Identifying Domain, Range, and Asymptotes (10 mins)
    For each transformed function, practice identifying the domain, range, and horizontal asymptote. Stress that vertical shifts affect the horizontal asymptote and the range. Discuss how reflections affect the range. For Example 2, the domain is all real numbers, the range is y > -1, and the horizontal asymptote is y = -1. For Example 3, the domain is all real numbers, the range is y < 2, and the horizontal asymptote is y = 2.
  • Wrap-up and Q&A (5 mins)
    Summarize the key concepts and answer any remaining questions. Reiterate the importance of understanding transformations for graphing exponential functions. Assign practice problems for homework.

Interactive Exercises

  • Transformation Matching
    Provide students with a set of exponential functions with base 'e' and a list of corresponding transformations. Students must match each function with the correct set of transformations.
  • Graphing Challenge
    Give students an exponential function with base 'e' and have them graph it on graph paper. They should also identify the domain, range, and horizontal asymptote.

Discussion Questions

  • How does the value of 'e' influence the graph of y = e^x?
  • How do different transformations (shifts, stretches, reflections) affect the domain, range, and asymptote of an exponential function with base 'e'?
  • Why is understanding the order of transformations important when graphing exponential functions?
  • Can you describe a real-world situation that can be modeled by an exponential function with base 'e'?

Skills Developed

  • Graphing exponential functions
  • Applying transformations to functions
  • Identifying domain, range, and asymptotes
  • Analytical thinking

Multiple Choice Questions

Question 1:

What is the approximate value of the natural base 'e'?

Correct Answer: 2.7

Question 2:

Which of the following transformations shifts the graph of y = e^x upwards?

Correct Answer: y = e^x + 2

Question 3:

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

Correct Answer: y = 0

Question 4:

What is the domain of y = e^x - 3?

Correct Answer: All real numbers

Question 5:

The graph of y = e^(-x) is a reflection of y = e^x over which axis?

Correct Answer: y-axis

Question 6:

Which transformation would cause the graph of y = e^x to vertically stretch by a factor of 3?

Correct Answer: y = 3e^x

Question 7:

What is the range of y = e^x + 5?

Correct Answer: y > 5

Question 8:

What transformation occurs in the equation y = e^(x+4)?

Correct Answer: Horizontal shift left 4 units

Question 9:

If y = -e^x, which transformation occurred?

Correct Answer: Reflection over the x-axis

Question 10:

What is the horizontal asymptote of y = e^x - 2?

Correct Answer: y = -2

Fill in the Blank Questions

Question 1:

The natural base 'e' is approximately equal to ____.

Correct Answer: 2.7

Question 2:

The horizontal asymptote of the parent function y = e^x is ____.

Correct Answer: y=0

Question 3:

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

Correct Answer: real

Question 4:

The transformation y = e^(x - 3) shifts the graph 3 units to the ____.

Correct Answer: right

Question 5:

The transformation y = -e^x reflects the graph over the ____-axis.

Correct Answer: x

Question 6:

In the equation y = 2e^x, the 2 represents a vertical ____ by a factor of 2.

Correct Answer: stretch

Question 7:

The range of the function y = e^x + 4 is y > ____.

Correct Answer: 4

Question 8:

The graph of y=e^(-x) is reflected across the ____ axis.

Correct Answer: y

Question 9:

Adding a constant outside the function, such as in y = e^x + c, results in a ____ shift.

Correct Answer: vertical

Question 10:

The horizontal asymptote of y = e^x + 3 is y = ____.

Correct Answer: 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.1.4: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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