Linear Function Notation and Modeling

3.6: Linear Function Notation and Modeling. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 6 of 63 parts Practice Flashcards

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Evaluate a linear function

Objective: Use function notation as input substitution.

Prerequisite skills: linear equations and reading input-output pairs. The notation f(t) means the output of function f when the input is t; it is not multiplication of f by t. Substitute the entire input expression everywhere the variable appears. Common error: treating f(a) as f times a. Restrictions and interpretation: A contextual domain may restrict a linear function to nonnegative or integer inputs.

Formulas

$f(a) = ma + b when f(x) = mx + b$

Worked examples

Try itIf f(x) = 3x + 22, what is f(23)?Show answer

Replace x by 23: f(23) = 3(23) + 22 = 69 + 22 = 91. Therefore the requested answer is 91.

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Evaluate a linear function

Objective: Use function notation as input substitution.

Formulas

$f(a) = ma + b when f(x) = mx + b$

Worked examples

Try itIf f(x) = 3x + 22, what is f(23)?Show answer

Replace x by 23: f(23) = 3(23) + 22 = 69 + 22 = 91. Therefore the requested answer is 91.

Interpret a coefficient

Objective: Connect a linear rate to output changes over an interval.

For f(x) = mx + b, the change f(x + h) − f(x) equals mh. The starting value cancels. This lets you compare outputs without computing each complete output. Common error: adding the intercept to an output difference. Restrictions and interpretation: A contextual domain may restrict a linear function to nonnegative or integer inputs.

Formulas

$f(x + h) − f(x) = mh$

Worked examples

Try itA linear model is P(t) = 24t + 40. By how much does P increase as t goes from 5 to 30?Show answer

The input increase is 25. Multiply by the slope: ΔP = 24(25) = 600. The intercept cancels when outputs are subtracted. Therefore the requested answer is 600.

Build a linear model

Objective: Determine a model from a rate and one observed value.

A known rate gives m. Substitute the observed input and output to find the intercept b, then use the model only within a domain justified by the situation. Common error: treating an observation at nonzero input as the initial value. Restrictions and interpretation: A contextual domain may restrict a linear function to nonnegative or integer inputs. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$f(t) = f(t₀) + m(t − t₀)$

Worked examples

Try itA machine produces at a constant rate of 25 units per hour. Its output counter reads 73 at hour 3. What will the counter read at hour 8?Show answer

The elapsed time is 8 − 3 = 5 hours. Additional output is 25 × 5 = 125. Add to the observed 73: 198 units. Therefore the requested answer is 198.

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