Learning Objectives
Distinguish linear, quadratic, and exponential patterns in equally spaced data.
Interpret the parameters of a simple exponential model.
Find restrictions and interpret a simple rational or absolute-value function.
Prerequisites: function notation, exponent rules, and quadratic forms. This lesson compares common nonlinear structures. A small table can suggest a model, but it does not prove that the pattern continues beyond the given data.
1. Compare Changes over Equal Input Steps
A linear function has constant first differences when input steps are equal. A quadratic has constant second differences. An exponential function Abˣ has a constant output ratio over equal unit steps, provided the relevant outputs are nonzero.
x | Linear: 2x + 1 | Quadratic: x² | Exponential: 2ˣ |
|---|---|---|---|
0 | 1 | 0 | 1 |
1 | 3 | 1 | 2 |
2 | 5 | 4 | 4 |
3 | 7 | 9 | 8 |
The linear column has differences 2, 2, 2. The quadratic column has differences 1, 3, 5, whose differences are 2, 2. The exponential column multiplies by 2 each time. Check equal spacing in x before using these tests.
2. Exponential Models Multiply Repeatedly
Worked Example: Growth from an Initial Amount
A model starts at 300 and grows by 10% per stage. Let t = 0 denote the initial stage. The model is Q(t) = 300(1.1)ᵗ.
After two stages, Q(2) = 300(1.21) = 363. The increase is 63, not 60, because the second 10% is taken from 330 rather than 300.
The initial value is 300 and the multiplier is 1.1. A 10% decay model would instead use 0.9.
For A > 0, a base b > 1 gives growth, while 0 < b < 1 gives decay. The model assumes the same multiplier applies to every equal time interval. Keep the time unit and initial stage explicit.
3. Rational Functions Have Denominator Restrictions
Worked Example: A Reciprocal Model
Suppose y = 12/x. Its algebraic domain excludes x = 0. If x represents a positive group size, the context further restricts x to positive values, possibly integers.
When x doubles from 2 to 4, y halves from 6 to 3. The product xy stays 12, not the sum x + y.
This is an inverse-proportion relationship. It is not a linear function: the changes in y are not constant over equal changes in x.
A cancelled factor does not restore an excluded input. For example, (x² − 1)/(x − 1) equals x + 1 only for x ≠ 1. The original rational function still has a hole at x = 1.
4. Absolute Value Measures Distance
The expression |x − h| is the distance between x and h on the number line. Therefore |x − 3| = 5 asks for points 5 units from 3. They are x = 8 and x = −2. The graph y = |x − 3| has a V-shaped minimum at (3, 0).
Common Mistakes
Constant differences suggest additive change; constant ratios suggest multiplicative change.
A growth rate of 10% uses multiplier 1.1, not 0.1.
Track the original domain when simplifying a rational function.
An absolute value is nonnegative, so |x − 3| = −5 has no real solution.
5. Practice
1. Outputs at x = 0, 1, 2, 3 are 5, 15, 45, 135. Give a simple exponential model.
2. A quantity starts at 80 and decreases by 25% per stage. Find its value after two stages.
3. State the domain restriction before simplifying (x² − 9)/(x − 3).
4. Solve |x + 2| = 6.
Worked Solutions
1. Each output is three times the previous one, and the value at zero is 5. The model y = 5·3ˣ fits the table.
2. The retained fraction is 0.75. The value is 80(0.75)² = 45.
3. Exclude x = 3. Factoring gives (x − 3)(x + 3)/(x − 3) = x + 3 for x ≠ 3.
4. x + 2 = 6 or x + 2 = −6, so x = 4 or x = −8. Both are 6 units from −2.
6. Summary
- Compare differences and ratios only with appropriate input spacing.
- Read an exponential model's starting value and per-step multiplier.
- Rational functions require nonzero denominators.
- Absolute value connects equations with distance on a number line.