Week 9, Session 1: Comparing Linear, Quadratic, and Exponential Models

Session Focus

Today you will compare model families by first differences, second differences, ratios, and long-run behavior. This session is written for independent study, but it should feel like a teacher is sitting beside you and asking the right questions at the right time. Read slowly, keep paper beside you, and do each example before you look at the explanation.

Key Language

linear model, quadratic model, exponential model, difference, ratio

Teacher Explanation

Different model families can look similar over a short interval. The disciplined approach is to examine how outputs change, not only how large they are at one point. This is a high-difficulty course, so the goal is not only to get an answer. The goal is to understand why the method works, what restrictions are present, and how the same idea appears in symbolic, graphical, numerical, and verbal forms.

When you study this session, use a three-pass routine. On the first pass, read the explanation and copy the main rule in your own words. On the second pass, redo the worked examples from a blank page. On the third pass, complete the independent practice without checking notes until you have a full attempt.

Visual Study Cue

Use the image above as a quick map of the session. Before solving, name the main object in the visual, identify what is changing, and connect the picture to the rule below. This habit helps you move between graphical, symbolic, numerical, and verbal reasoning.

Core Rule

Linear models have constant first differences, quadratics have constant second differences, and exponentials have constant ratios for equal input steps.

Worked Example 1: Classifying from a table

Suppose the y-values for equally spaced x-values are \(2, 5, 10, 17, 26\). The first differences are \(3, 5, 7, 9\). The second differences are \(2, 2, 2\), so a quadratic model fits the pattern. If instead the y-values are \(2, 6, 18, 54, 162\), the ratios are all 3, so an exponential model fits.

Pause after the example and ask yourself what made the solution move forward. Was it a definition, a theorem, a graph feature, an algebraic manipulation, or a domain restriction? Naming the reason matters because it helps you transfer the method to harder problems.

Worked Example 2: A second angle on the same idea

Long-run behavior also matters. A positive exponential function with base greater than 1 eventually outgrows any linear or quadratic model, even if it starts smaller. But on a small interval, a quadratic may temporarily be larger.

Notice how the second example uses the same core idea but changes the surface details. That is deliberate. Advanced algebra and precalculus become difficult when a familiar idea is hidden inside a new representation. Your task is to recognize the structure underneath the wording.

Independent Practice

• Classify \(10, 20, 40, 80, 160\).
• Classify \(3, 8, 15, 24, 35\).
• Compare \(100+20x\) and \(100(1.2)^x\) at x=0,1,2,10.

Error-Log Reflection

After you finish, write one sentence beginning with "The step I must watch most carefully is..." Then write one sentence beginning with "The clue that tells me to use this method is..." These two sentences convert practice into self-learning because they make your decision process visible.

References

• Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e
• College Board. (2023). AP Precalculus course and exam description. https://apcentral.collegeboard.org/media/pdf/ap-precalculus-course-and-exam-description.pdf
• National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. https://corestandards.org/wp-content/uploads/2023/09/Math_Standards1.pdf