Unit 3 · Topic 2.2
Change in Linear and Exponential Functions
Recognize constant differences in linear functions and constant ratios or percent change in exponential functions. Develop the idea through symbolic, numerical, graphical, and contextual representations.
Learning Goals
- Extend arithmetic and geometric patterns to real-number inputs.
- Compare constant additive and constant proportional change.
- Construct linear and exponential functions from an anchor point and a rate.
1. Essential Structure
Recognize constant differences in linear functions and constant ratios or percent change in exponential functions.
Read the formula together with its domain, units, starting input, and the interval length over which change is measured.
2. Core Ideas
- For equal steps \(h\), \(L(x+h)-L(x)=mh\) is constant.
- For an exponential function, \(E(x+h)/E(x)=b^h\) is constant.
- Point forms \(L(x)=y_i+m(x-x_i)\) and \(E(x)=y_ib^{x-x_i}\) preserve a known pair.
- A factor \(b=1+r\) represents growth rate \(r\); \(b=1-r\) represents decay rate \(r\).
3. Graph and Representation
Use graphical evidence together with an algebraic or numerical reason. A graphing window alone does not establish domain, end behavior, or model validity.
4. Original Worked Example
Interpret the change in \(E(t)=100(1.05)^t\).
Check the result in the original representation and explain its meaning, including units when the quantities are contextual.
5. AP Reasoning Workflow
- Verify equal input intervals.
- Compare output differences and ratios.
- Identify an initial value or anchor point.
- Construct the model in a rate-revealing form.
- Interpret the rate with units and interval length.
A strong AP response shows the mathematical evidence first and then states a precise conclusion.
6. Extended Example and Application
Two accounts begin at 500 dollars. \(L(t)=500+60t\) adds 60 dollars yearly, while \(E(t)=500(1.08)^t\) grows 8% yearly. Exponential dollar increases become larger even though the percent stays constant.
7. Technology and Validation
Use tables with equal input steps before selecting a regression. Visual curvature by itself is not enough evidence.
When technology is used, record the model or result, preserve sufficient precision, and explain why the output answers the question.
8. From Sequences to Real-Input Functions
An arithmetic sequence and a linear function share additive structure, while a geometric sequence and an exponential function share multiplicative structure. Their domains are the important difference.
| Discrete pattern | Related real-input function | Domain change |
|---|---|---|
| \(a_n=a_0+dn\) | \(L(x)=a_0+dx\) | Whole-number indices become real inputs. |
| \(g_n=g_0r^n\) | \(E(x)=g_0r^x\), \(r>0\) | Whole-number indices become real inputs. |
A geometric sequence may have a negative common ratio and alternate signs, but a real-valued exponential function \(ab^x\) defined for every real \(x\) requires a positive base \(b\).
9. Change over Any Equal-Length Interval
The defining patterns are not limited to one-unit steps. For any fixed interval length \(h\), a linear function has a constant output difference, while an exponential function has a constant output ratio.
If \(h=3\), a linear model adds \(3m\) and an exponential model multiplies by \(b^3\), regardless of the starting input \(x\). Always verify that compared input intervals have the same length.
10. Construct Each Model from an Anchor Point
Suppose both models pass through \((2,45)\). A linear rate of 7 output units per input unit gives
An exponential growth factor of 1.12 per input unit gives
Substituting \(x=2\) makes the added or exponentiated change zero, so both formulas return the anchor output 45. This point form is often clearer than first converting to an initial-value form.
11. Read Tables When Input Steps Are Not One
The table compares models at two-unit input steps.
| \(x\) | 0 | 2 | 4 | Two-unit evidence | One-unit parameter |
|---|---|---|---|---|---|
| \(L(x)\) | 10 | 18 | 26 | Difference \(+8\) | Slope \(8/2=4\) |
| \(E(x)\) | 10 | 22.5 | 50.625 | Ratio \(2.25\) | Factor \(\sqrt{2.25}=1.5\) |
A constant ratio of 2.25 over two input units does not mean the one-unit factor is 2.25. Convert an \(h\)-unit ratio \(R\) to a one-unit factor with \(R^{1/h}\).
12. Growth Factor, Growth Rate, and Percent Change
In \(E(x)=ab^x\), the base \(b\) is a factor, not a percent. The decimal rate is \(r=b-1\), so the percent rate is \(100(b-1)\%\).
| Factor \(b\) | Type | Rate | Meaning per unit |
|---|---|---|---|
| 1.18 | Growth | 0.18 | Increase 18% |
| 0.82 | Decay | \(-0.18\) | Decrease 18% |
| 1 | Constant | 0 | No change |
For decay, 0.82 is the proportion that remains. The proportion lost is \(1-0.82=0.18\).
13. Compare Short-Term and Long-Term Behavior
Linear growth may initially exceed exponential growth even when the exponential model eventually becomes larger. Consider
Both begin at 100. At \(t=5\), \(L(5)=250\) and \(E(5)\approx248.83\), so the linear output is still slightly larger. At \(t=6\), \(L(6)=280\) while \(E(6)\approx298.60\), so the exponential model has overtaken it.
The exact intersection can be estimated graphically or numerically. A statement such as “exponential always grows faster” is incomplete: the comparison depends on parameters, starting values, and the input interval.
14. Context and AP-Ready Justification
A service plan costs 24 dollars initially and adds 6 dollars each month. This is linear because equal one-month intervals add the same 6-dollar amount. A different account begins at 24 dollars and grows 6% monthly. This is exponential because equal one-month intervals multiply the balance by 1.06.
Linear response pattern: “For every \(h\)-unit increase in the input, the output changes by the constant amount \(mh\), so the model has constant additive change.”
Exponential response pattern: “For every \(h\)-unit increase in the input, the output is multiplied by the constant factor \(b^h\), so the model has constant proportional change.”
Include the input interval, units, direction of change, and contextual domain. Do not justify the model only from the visual shape of its graph.
15. Common Errors
- Do not interpret an exponential growth factor of 1.05 as a 105% increase.
- Calling curved data exponential without checking proportional change.
- Saying a factor of 0.92 means 92% decay instead of an 8% decrease.
- Giving a numerical result without a domain check, units, or interpretation.
Key Takeaways
- Recognize constant differences in linear functions and constant ratios or percent change in exponential functions.
- Core relationship: \(L(x)=a+mx,\qquad E(x)=ab^x\)
- Error check: Do not interpret an exponential growth factor of 1.05 as a 105% increase.
- Outputs 80, 68, 57.8, and 49.13 occur at consecutive integers. Identify and justify a model family.
- Write an exponential function through \((3,250)\) with 6% growth per unit.
- Explain the structural difference between slope and growth factor.