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Introduction
Graphing Exponential Functions helps you connect a mathematical rule to a result you can explain and check. Build a table at negative, zero, and positive inputs and graph the corresponding smooth curve. Identify the y-intercept and horizontal asymptote. This lesson belongs to exponential functions and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in graphing exponential functions.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Evaluate powers and convert percentages to decimals.
Previous lesson: Exponential Function Notation
Key mathematical vocabulary
- constant
- A value fixed within a problem.
- intercept
- A point where a graph meets an axis.
- domain
- The set of permitted inputs.
- range
- The set of outputs produced.
- exponent
- A number specifying a power.
Formulas and rules
- f(x)=abx, b>0, b≠1
- Growth factor=1+r; decay factor=1−r
- A=P(1+r/n)nt
Exponential growth and decay
Build a table at negative, zero, and positive inputs and graph the corresponding smooth curve. Identify the y-intercept and horizontal asymptote.
An exponential model has the form f(t)=abt, where b>0 and b≠1 for a nonconstant exponential. The initial value is a=f(0). Each increase of 1 in t multiplies the output by b rather than adding a constant.
For a positive initial amount, b>1 models growth and 0<b<1 models decay. A rate r written as a decimal gives growth factor 1+r or decay factor 1−r. Compound interest is A=P(1+r/n)nt, with compatible units. For a≠0, the mathematical domain is all real numbers; the sign of a determines whether the range is y>0 or y<0. Context may restrict time to t≥0.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. At t=−1, f=100; at t=0, f=200; at t=1, f=400. Plot these points.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. Divide by the accumulated factor to recover the initial amount.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Graph f(t)=200·2t using points at t=−1, 0, 1, then evaluate f(3). State its y-intercept and asymptote.
- At t=−1, f=100; at t=0, f=200; at t=1, f=400. Plot these points.
- Draw a smooth increasing curve that stays above y=0. The y-intercept is (0,200) and the horizontal asymptote is y=0.
- The initial amount is 200; the multiplicative factor is 2.
- After 3 periods, multiply by 2 a total of 3 times: 200·23.
- 23=8; the output is 1600. A factor greater than 1 means growth.
Answer: 1600; growth
Verify: Divide by the accumulated factor to recover the initial amount.
Avoid this mistake: Using the percent rate itself as the growth factor.
Example 2 · Basic
Graph f(t)=210·2t using points at t=−1, 0, 1, then evaluate f(3). State its y-intercept and asymptote.
- At t=−1, f=105; at t=0, f=210; at t=1, f=420. Plot these points.
- Draw a smooth increasing curve that stays above y=0. The y-intercept is (0,210) and the horizontal asymptote is y=0.
- The initial amount is 210; the multiplicative factor is 2.
- After 3 periods, multiply by 2 a total of 3 times: 210·23.
- 23=8; the output is 1680. A factor greater than 1 means growth.
Answer: 1680; growth
Verify: Divide by the accumulated factor to recover the initial amount.
Avoid this mistake: Adding the same amount each period in an exponential model.
Example 3 · Intermediate
Graph f(t)=220·2t using points at t=−1, 0, 1, then evaluate f(3). State its y-intercept and asymptote.
- At t=−1, f=110; at t=0, f=220; at t=1, f=440. Plot these points.
- Draw a smooth increasing curve that stays above y=0. The y-intercept is (0,220) and the horizontal asymptote is y=0.
- The initial amount is 220; the multiplicative factor is 2.
- After 3 periods, multiply by 2 a total of 3 times: 220·23.
- 23=8; the output is 1760. A factor greater than 1 means growth.
Answer: 1760; growth
Verify: Divide by the accumulated factor to recover the initial amount.
Avoid this mistake: Including zero in the range of an unshifted nonzero exponential.
Example 4 · Advanced
Graph f(t)=230·2t using points at t=−1, 0, 1, then evaluate f(3). State its y-intercept and asymptote.
- At t=−1, f=115; at t=0, f=230; at t=1, f=460. Plot these points.
- Draw a smooth increasing curve that stays above y=0. The y-intercept is (0,230) and the horizontal asymptote is y=0.
- The initial amount is 230; the multiplicative factor is 2.
- After 3 periods, multiply by 2 a total of 3 times: 230·23.
- 23=8; the output is 1840. A factor greater than 1 means growth.
Answer: 1840; growth
Verify: Divide by the accumulated factor to recover the initial amount.
Avoid this mistake: Using the percent rate itself as the growth factor.
Example 5 · Challenge
Graph f(t)=240·2t using points at t=−1, 0, 1, then evaluate f(3). State its y-intercept and asymptote.
- At t=−1, f=120; at t=0, f=240; at t=1, f=480. Plot these points.
- Draw a smooth increasing curve that stays above y=0. The y-intercept is (0,240) and the horizontal asymptote is y=0.
- The initial amount is 240; the multiplicative factor is 2.
- After 3 periods, multiply by 2 a total of 3 times: 240·23.
- 23=8; the output is 1920. A factor greater than 1 means growth.
Answer: 1920; growth
Verify: Divide by the accumulated factor to recover the initial amount.
Avoid this mistake: Adding the same amount each period in an exponential model.
Common mistakes and corrections
- Using the percent rate itself as the growth factor.
- Adding the same amount each period in an exponential model.
- Including zero in the range of an unshifted nonzero exponential.
Real-world applications
- Model a population with a constant percent change. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Predict a repeated depreciation or compound-interest balance. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Build a table at negative, zero, and positive inputs and graph the corresponding smooth curve. Identify the y-intercept and horizontal asymptote.
An exponential model has the form f(t)=abt, where b>0 and b≠1 for a nonconstant exponential. The initial value is a=f(0). Each increase of 1 in t multiplies the output by b rather than adding a constant.
Frequently asked questions
What should I know before graphing exponential functions?
Evaluate powers and convert percentages to decimals.
How can I check my work?
Divide by the accumulated factor to recover the initial amount. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Using the percent rate itself as the growth factor. Adding the same amount each period in an exponential model. Including zero in the range of an unshifted nonzero exponential.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Model a population with a constant percent change.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Build a table at negative, zero, and positive inputs and graph the corresponding smooth curve. Identify the y-intercept and horizontal asymptote.
- Using the percent rate itself as the growth factor.
- Adding the same amount each period in an exponential model.
- Including zero in the range of an unshifted nonzero exponential.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
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Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials