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MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

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2.5 Linear Functions and Models

Focus on slope and rate from a table.

Before you solve

Ideas in this lesson

Explore the lesson brief and the worked example for Idea 11. Create a free account to continue with the remaining ideas and practice.

01Recognize and simplify linear functionsWorked steps · five questions02Create tables and draw linear graphsWorked steps · five questions03Interpret slopes and intercepts with scale and unitsWorked steps · five questions04Build models from a rate and initial valueWorked steps · five questions05Recover a linear rule from numerical dataWorked steps · five questions06Read a linear model from a graphWorked steps · five questions07Explore slope and intercept familiesWorked steps · five questions08Model grades, ramps, and unit conversionsWorked steps · five questions09Compare motion and solve catch-up problemsWorked steps · five questions10Build and interpret fixed-plus-variable cost modelsWorked steps · five questions11Prove both directions of constant-rate equivalenceWorked steps · five questions
Lesson brief and rules

2.5 Linear Functions and Models

A linear function has constant rate of change m=y2−y1x2−x1m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}.

Worked example 11
01 / Is f(x)=3x−4f(x)=3 x - 4 linear on its natural domain? Simplify before deciding and identify slope and intercept.
02 / Expand or collect terms to get 3x−43 x - 4.
03 / This has the form mx+bmx+b, where m=m=33 and b=b=−4-4.
04 / A constant is allowed: its coefficient of xx is zero.
Worked example 2 · Recognize and simplify linear functions

Is f(x)=x4+34f(x)=\dfrac{x}{4} + \dfrac{3}{4} linear on its natural domain? Simplify before deciding and identify slope and intercept.

Expand or collect terms to get x4+34\dfrac{x}{4} + \dfrac{3}{4}.
This has the form mx+bmx+b, where m=m=14\dfrac{1}{4} and b=b=34\dfrac{3}{4}.
A constant is allowed: its coefficient of xx is zero.

Read the graphs

Read a linear model from a graph

The horizontal axis is hours and the vertical axis is liters. Find the equation and rate represented by the line. Use two marked points, not its apparent angle.
Linear volume model; exact coordinates on scaled axes-1012345051015xyV

The line passes through (0,5)(0,5), (2,11)(2,11).
Slope is 11−52−0=3\frac{11-5}{2-0}=3 liters per hour.
Its vertical intercept is 55, giving V(t)=3t+5V(t)=3t+5.

Answer: V(t)=3t+5V(t)=3t+5; 33 liters per hour.

15 questions with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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