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Introduction
Comparing Linear and Exponential Models helps you connect a mathematical rule to a result you can explain and check. Compare a linear and exponential prediction at specific inputs. Avoid assuming the same model is larger for every real input. 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 comparing linear and exponential models.
- 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: Real-World Exponential Models
Key mathematical vocabulary
- constant
- A value fixed within a problem.
- domain
- The set of permitted inputs.
- range
- The set of outputs produced.
- exponent
- A number specifying a power.
- factor
- An expression multiplied by another to form a product.
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
Compare a linear and exponential prediction at specific inputs. Avoid assuming the same model is larger for every real input.
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. The linear model adds 340 per step; the exponential model multiplies by 2.
- 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. The exponential factor is repeatedly multiplied; it exceeds t+1 at this specified t.
- 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
Compare L(t)=340+340t and E(t)=340·2t. Which is larger at t=5?
- The linear model adds 340 per step; the exponential model multiplies by 2.
- At t=5, L=2040, while E=10880.
- Since 10880>2040, the exponential model is larger at this input. Do not assume it is larger at every real input.
Answer: E(t)
Verify: The exponential factor is repeatedly multiplied; it exceeds t+1 at this specified t.
Avoid this mistake: Using the percent rate itself as the growth factor.
Example 2 · Basic
Compare L(t)=350+350t and E(t)=350·2t. Which is larger at t=5?
- The linear model adds 350 per step; the exponential model multiplies by 2.
- At t=5, L=2100, while E=11200.
- Since 11200>2100, the exponential model is larger at this input. Do not assume it is larger at every real input.
Answer: E(t)
Verify: The exponential factor is repeatedly multiplied; it exceeds t+1 at this specified t.
Avoid this mistake: Adding the same amount each period in an exponential model.
Example 3 · Intermediate
Compare L(t)=360+360t and E(t)=360·2t. Which is larger at t=5?
- The linear model adds 360 per step; the exponential model multiplies by 2.
- At t=5, L=2160, while E=11520.
- Since 11520>2160, the exponential model is larger at this input. Do not assume it is larger at every real input.
Answer: E(t)
Verify: The exponential factor is repeatedly multiplied; it exceeds t+1 at this specified t.
Avoid this mistake: Including zero in the range of an unshifted nonzero exponential.
Example 4 · Advanced
Compare L(t)=250+250t and E(t)=250·2t. Which is larger at t=6?
- The linear model adds 250 per step; the exponential model multiplies by 2.
- At t=6, L=1750, while E=16000.
- Since 16000>1750, the exponential model is larger at this input. Do not assume it is larger at every real input.
Answer: E(t)
Verify: The exponential factor is repeatedly multiplied; it exceeds t+1 at this specified t.
Avoid this mistake: Using the percent rate itself as the growth factor.
Example 5 · Challenge
Compare L(t)=260+260t and E(t)=260·2t. Which is larger at t=6?
- The linear model adds 260 per step; the exponential model multiplies by 2.
- At t=6, L=1820, while E=16640.
- Since 16640>1820, the exponential model is larger at this input. Do not assume it is larger at every real input.
Answer: E(t)
Verify: The exponential factor is repeatedly multiplied; it exceeds t+1 at this specified t.
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
Compare a linear and exponential prediction at specific inputs. Avoid assuming the same model is larger for every real input.
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 comparing linear and exponential models?
Evaluate powers and convert percentages to decimals.
How can I check my work?
The exponential factor is repeatedly multiplied; it exceeds t+1 at this specified t. 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
- Compare a linear and exponential prediction at specific inputs. Avoid assuming the same model is larger for every real input.
- 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