9.2: Model Selection | Precalculus | Code Scholars Tutoring

How it works

Choose a model using both the numerical pattern and the context. Equal first differences suggest a linear model, equal second differences at equally spaced inputs suggest a quadratic, and a constant positive output ratio suggests an exponential. A fitted pattern does not justify unlimited extrapolation.

\text{constant difference: linear; constant ratio: exponential}

Worked examples

Example 1

Data at $\displaystyle x=0,1,2$ are $\displaystyle 5,8,11$ . Choose a simple model.

The differences are $\displaystyle 3,3$ , so a linear model is $\displaystyle y=3x+5$ .

Example 2

Data at $\displaystyle x=0,1,2$ are $\displaystyle 2,6,18$ . Choose a simple model.

The ratios are $\displaystyle 3,3$ , so an exponential model is $\displaystyle y=2\cdot3^x$ .

Problems

Problem 9.2.1

Data at $\displaystyle x=0,1,2,3$ are $\displaystyle 1,4,9,16$ . Identify a simple model.

Show worked solution

First differences are $\displaystyle 3,5,7$ and second differences are $\displaystyle 2,2$ . A fitting quadratic is $\displaystyle y=(x+1)^2$ .

Problem 9.2.2

Data at $\displaystyle t=0,1,2$ are $\displaystyle 100,80,64$ . Identify the factor and a model.

Show worked solution

The ratio is $\displaystyle 0.8$ , giving $\displaystyle y=100(0.8)^t$ .

Problem 9.2.3

The graph compares $\displaystyle L(t)=10+5t$ and $\displaystyle E(t)=10(1.5)^t$ . Which has constant additive change and which has constant multiplicative change?

The lines and curves show L(t) = 10 + 5t and E(t) = 10 times 1.5 to the t. — Compare equal additive and equal multiplicative change.

Show worked solution

$\displaystyle L$ increases by $\displaystyle 5$ per time unit. $\displaystyle E$ multiplies by $\displaystyle 1.5$ per time unit, a $\displaystyle 50\%$ increase. The models agree initially but later separate.

The lines and curves show L(t) = 10 + 5t and E(t) = 10 times 1.5 to the t. L adds 5 per step; E multiplies by 1.5 per step. — L adds 5 per step; E multiplies by 1.5 per step.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 9.2.1

Problem 9.2.2

Problem 9.2.3

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