Exponential Functions and Their Graphs

Active learning helps students move from abstract symbols to concrete understanding of exponential and logarithmic relationships. By manipulating equations and testing rules with real numbers, students build intuition that textbooks alone cannot provide.

• Grade 12
• Mathematics
• 3 activities
• 20 min–40 min

Learning Objectives

1. Analyze the effect of the base 'b' on the growth or decay rate of an exponential function y = ab^x.
2. Compare the graphical features, including domain, range, and asymptotes, of exponential growth and decay functions.
3. Explain the relationship between the base 'b' and the y-intercept of an exponential function.
4. Predict the long-term behavior (as x approaches infinity or negative infinity) of an exponential function based on its equation.
5. Identify the horizontal asymptote of an exponential function from its equation and graph.

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: The Inverse Connection

Students are given an exponential table of values and must work in pairs to create the corresponding logarithmic table. They discuss the swapping of x and y values and what this means for the graph's asymptotes.

Prepare & details

Analyze the impact of the base 'b' on the growth or decay rate of an exponential function.

Facilitation Tip: During Think-Pair-Share: The Inverse Connection, circulate to listen for pairs who confuse the inverse relationship and redirect them by asking, 'If 2^3 equals 8, what does log base 2 of 8 equal?'

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Inquiry Circle

⏱ 40 min·Small Groups

Inquiry Circle: Law Discovery

Provide groups with calculators and a list of log expressions (e.g., log 2 + log 5). Students calculate the values and look for patterns to 'discover' the product, quotient, and power laws before they are formally taught.

Prepare & details

Compare the graphical features of exponential growth functions with those of exponential decay functions.

Facilitation Tip: In Collaborative Investigation: Law Discovery, provide calculators and pre-selected equations so students can test each law with specific values and see the equality hold.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Peer Teaching

⏱ 25 min·Pairs

Peer Teaching: Change of Base

After a brief introduction, students work in pairs where one student explains how to solve a base-10 log and the other explains how to use the change of base formula for a base-5 log. They then swap roles with new problems.

Prepare & details

Predict the long-term behavior of an exponential function based on its equation.

Facilitation Tip: For Peer Teaching: Change of Base, assign groups different bases so they can compare results and notice patterns in the change of base formula.

Setup: Presentation area at front, or multiple teaching stations

Materials: Topic assignment cards, Lesson planning template, Peer feedback form, Visual aid supplies

• Understand
• Apply
• Analyze
• Create
• Self-Management
• Relationship Skills

Teaching This Topic

Teachers approach this topic by grounding logarithms in exponential equations first. Avoid introducing logarithmic laws before students understand why they exist. Research shows that students grasp these concepts better when they derive the rules themselves through guided investigation rather than memorizing formulas. Emphasize the connection to real-world applications, like pH levels or sound intensity, to make the abstract tangible.

What to Expect

Successful learning looks like students confidently converting between exponential and logarithmic forms, applying logarithmic laws correctly, and explaining why these rules hold true. They should also recognize the domain restrictions and the practical implications of exponential growth and decay.

Watch Out for These Misconceptions

Common Misconception

During Collaborative Investigation: Law Discovery, watch for students who incorrectly distribute a logarithm over addition, thinking log(A + B) equals log A + log B.

What to Teach Instead

Have them test this with assigned values, like log 10 + log 10 and log 20, using calculators to show the inequality. Prompt them to recall the product law and why it applies to multiplication, not addition.

Common Misconception

During Think-Pair-Share: The Inverse Connection, watch for students who believe logarithms can have negative arguments.

What to Teach Instead

Ask them to recall the exponential form (b^y = x) and test with a positive base raised to various powers. Use peer discussion to reinforce that exponential functions with positive bases never yield negative outputs, so logarithms cannot have negative arguments.

Assessment Ideas

Exit Ticket

After Think-Pair-Share: The Inverse Connection, collect students' explanations of how exponentials and logarithms serve as inverses. Look for accuracy in describing the domain and range of each function type.

Quick Check

During Collaborative Investigation: Law Discovery, ask groups to present one law they tested and how their calculations proved the law. Listen for correct application of the laws and explanations of why the law works.

Discussion Prompt

After Peer Teaching: Change of Base, facilitate a class discussion where students share their observations about how the base affects the steepness of the graph. Assess their ability to connect the algebraic change of base formula to the visual representation of the function.

Extensions & Scaffolding

• Challenge students to find a real-world context for a logarithmic function with a fractional base.
• For students who struggle, provide a partially completed table that relates exponential and logarithmic forms to scaffold the conversion process.
• Deeper exploration: Ask students to research how logarithms are used in computer science, such as in binary search algorithms, and present their findings.

Key Vocabulary

Exponential Growth Function	A function of the form y = ab^x where a > 0 and b > 1. The function's value increases as x increases.
Exponential Decay Function	A function of the form y = ab^x where a > 0 and 0 < b < 1. The function's value decreases as x increases.
Base (b)	In an exponential function y = ab^x, the base 'b' determines the rate of growth or decay. If b > 1, it's growth; if 0 < b < 1, it's decay.
Horizontal Asymptote	A horizontal line that the graph of a function approaches but never touches. For exponential functions of the form y = ab^x, the asymptote is typically the x-axis (y=0).
Domain	The set of all possible input values (x-values) for a function. For most exponential functions, the domain is all real numbers.
Range	The set of all possible output values (y-values) for a function. For exponential functions of the form y = ab^x with a > 0, the range is y > 0.