2.1: Constant Ratios and Model Choice | AP Precalculus

How it works

At equal input increments, constant output differences suggest a linear model and constant positive output ratios suggest an exponential model. Check spacing before comparing ratios. A finite data set supports a candidate model rather than proving a unique rule.

f(x+h)=b^h f(x)\quad\text{for }f(x)=ab^x

Worked examples

Example 1

At inputs $\displaystyle 0,1,2,3$ , outputs are $\displaystyle 3,6,12,24$ . Give an exponential model.

Each output is twice the previous one, so $\displaystyle f(x)=3\cdot2^x$ fits.

Example 2

At inputs $\displaystyle 0,1,2,3$ , outputs are $\displaystyle 5,9,13,17$ . Choose a model.

Differences are all $\displaystyle 4$ , so $\displaystyle f(x)=5+4x$ is linear.

Problems

Problem 2.1.1

At inputs $\displaystyle 0,1,2,3$ , outputs are $\displaystyle 2,6,18,54$ . Give a model and predict the next output.

Show worked solution

The ratio is $\displaystyle 3$ , so $\displaystyle f(x)=2\cdot3^x$ and $\displaystyle f(4)=162$ .

Problem 2.1.2

At inputs $\displaystyle 0,2,4$ , outputs are $\displaystyle 4,16,64$ . What is the per-unit growth factor?

Show worked solution

A two-unit step multiplies by $\displaystyle 4$ , so a one-unit step multiplies by $\displaystyle \sqrt{4}=2$ . A model is $\displaystyle f(x)=4\cdot2^x$ .

Problem 2.1.3

Why do outputs $\displaystyle 1,2,4$ at inputs $\displaystyle 0,1,3$ not establish a constant per-unit factor?

Show worked solution

The input gaps are $\displaystyle 1$ and $\displaystyle 2$ . Equal output ratios over unequal input changes represent different per-unit factors.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 2.1.1

Problem 2.1.2

Problem 2.1.3

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