Introduction

Building Linear Functions is an important Grade 8 math skill because students are moving from simple answers toward explaining how the math works.

In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with building linear functions.

What Is Building Linear Functions?

Building Linear Functions means choosing a model, naming what each number means, and explaining the strategy.

The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.

Understanding Building Linear Functions

Before solving, students should slow down and decide what each number, shape, unit, or label represents.

  • Read the question carefully and identify what is being asked.
  • Choose a model, equation, table, or diagram that matches the situation.
  • Solve one step at a time and keep units or labels attached.
  • Use the answer explanation to check that the result makes sense.

Visual Models

Visual Model 1

Question: A spring stretches when weights are attached. The table shows the length \(L\) (in cm) for a spring with \(w\) weights attached. Which equation describes the relationship?

Number of Weights (\(w\))0246
Length (\(L\), cm)10141822
  • A. \(L=2w+10\)
  • B. \(L=10w+2\)
  • C. \(L=4w+10\)
  • D. \(L=w+10\)

Why it works: Initial length is \(10\) cm (when \(w=0\)). Change from 2 to 4 weights: length goes from 14 to 18 cm, a change of 4 cm for 2 weights, so slope is \(2\). Equation: \(L=2w+10\).

Answer: \(L=2w+10\)

Visual Model 2

Question: A town's population \(P\) (in thousands) at year \(t\) (years since 2010) follows this data: What is the rate of change of the population per year?

Year since 2010 (\(t\))0369
Population (\(P\), thousands)45515763
  • A. 2 thousand per year
  • B. 3 thousand per year
  • C. 6 thousand per year
  • D. 9 thousand per year

Why it works: Slope is \(\frac{\Delta P}{\Delta t}=\frac{6}{3}=2\) thousand per year from \(t=0\) to \(t=3\) (or any consecutive pair).

Answer: 2 thousand per year

Worked Examples

Example 1

Question: The graph shows a linear function passing through points \((0, 3)\) and \((2, 7)\). What is the slope of the line?

Example 1

  • A. \(2\)
  • B. \(3\)
  • C. \(\frac{1}{2}\)
  • D. \(\frac{2}{7}\)
  1. Slope is \(m=\frac{y_2-y_1}{x_2-x_1}=\frac{7-3}{2-0}=\frac{4}{2}=2\).

Answer: \(2\)

Example 2

Question: The graph shows a line passing through \((0, 4)\) and \((3, 10)\). What is the equation of the line?

Example 2

  • A. \(y=2x+4\)
  • B. \(y=3x+4\)
  • C. \(y=2x+3\)
  • D. \(y=4x+3\)
  1. \(y\)-intercept is 4.
  2. Slope is \(\frac{10-4}{3-0}=\frac{6}{3}=2\).
  3. Equation: \(y=2x+4\).

Answer: \(y=2x+4\)

Example 3

Question: The table shows a linear function. Find the missing \(y\) value.

\(x\)1234
\(y\)711?19
  • A. \(13\)
  • B. \(15\)
  • C. \(14\)
  • D. \(16\)
  1. Slope is \(\frac{11-7}{2-1}=4\).
  2. When \(x=3\): \(y=11+4=15\).

Answer: \(15\)

Real-World Word Problems

Problem 1

Question: A gym charges a \($25\) sign-up fee plus \($15\) per month. Which function gives the total cost \(C\) after \(m\) months?

  • A. \(C=15m+25\)
  • B. \(C=25m+15\)
  • C. \(C=40m\)
  • D. \(C=15m-25\)

Why it works: The sign-up fee \($25\) is a one-time cost (the \(y\)-intercept, the value when \(m=0\)). The monthly rate \($15\) is the slope (rate of change per month). Therefore, \(C=15m+25\). The slope tells how much additional cost per month; the intercept tells the starting cost.

Answer: \(C=15m+25\)

Problem 2

Question: A music streaming service charges \($8\) for premium features plus \($12\) per month for access. Which equation represents the total cost \(T\) for \(m\) months of service?

  • A. \(T=12m+8\)
  • B. \(T=8m+12\)
  • C. \(T=20m\)
  • D. \(T=8m-12\)

Why it works: One-time premium feature fee is \($8\) (intercept). Monthly service cost is \($12\) (slope). Total is \(T=12m+8\).

Answer: \(T=12m+8\)

Common Mistakes

  • Rushing before identifying what the numbers represent.
  • Choosing an operation that does not match the situation.
  • Dropping labels, units, or context from the answer.
  • Skipping the estimate or reasonableness check.

Strategy Tips

  • Underline the question being asked.
  • Use a model before jumping to computation.
  • Write an equation that matches the story or picture.
  • Explain the final answer in a sentence.

Practice Questions

Question 1

A taxi charges \($3\) to get in the car plus \($0.50\) per mile driven. If \(d\) represents the distance in miles and \(C\) represents the cost in dollars, which equation models this situation?

  • A. \(C=0.50d+3\)
  • B. \(C=3d+0.50\)
  • C. \(C=3.50d\)
  • D. \(C=0.50d-3\)

Question 2

A line passes through the points \((1, 5)\) and \((4, 11)\). What is the slope of the line?

  • A. \(2\)
  • B. \(3\)
  • C. \(\frac{2}{3}\)
  • D. \(6\)

Question 3

A line passes through \((2, 9)\) and \((5, 18)\). Which equation represents this line?

  • A. \(y=3x+3\)
  • B. \(y=3x+9\)
  • C. \(y=2x+5\)
  • D. \(y=x+7\)

Question 4

The equation \(y=12x+50\) represents a driver's earnings in dollars, where \(x\) is the number of hours worked. What does the slope 12 represent?

  • A. Hourly wage of \($12\)
  • B. Total earnings of \($12\)
  • C. Initial debt of \($12\)
  • D. Number of hours worked

Question 5

The equation \(C=25t+80\) represents the total cost of a phone plan, where \(t\) is the number of months. What does the 80 represent?

  • A. Monthly cost of \($25\)
  • B. Equipment fee of \($80\)
  • C. Number of months
  • D. Total cost after 1 month

Question 6

Which table represents the linear function \(y=4x-2\)?

  • A. \begin{tabular}{|c|c|}\hline \(x\) & \(y\)
    \hline 0 & \(-2\)
    \hline 1 & \(2\)
    \hline 2 & \(6\)
    \hline \end{tabular}
  • B. \begin{tabular}{|c|c|}\hline \(x\) & \(y\)
    \hline 0 & \(4\)
    \hline 1 & \(5\)
    \hline 2 & \(6\)
    \hline \end{tabular}
  • C. \begin{tabular}{|c|c|}\hline \(x\) & \(y\)
    \hline 0 & \(2\)
    \hline 1 & \(6\)
    \hline 2 & \(10\)
    \hline \end{tabular}
  • D. \begin{tabular}{|c|c|}\hline \(x\) & \(y\)
    \hline 0 & \(0\)
    \hline 1 & \(4\)
    \hline 2 & \(8\)
    \hline \end{tabular}
Full Answer Explanations Click to show all answers and explanations

Question 1

Answer: \(C=0.50d+3\)

Initial fee is \($3\) (intercept), rate per mile is \($0.50\) (slope). Equation: \(C=0.50d+3\).

Question 2

Answer: \(2\)

Slope is \(m=\frac{11-5}{4-1}=\frac{6}{3}=2\).

Question 3

Answer: \(y=3x+3\)

Slope: \(m=\frac{18-9}{5-2}=\frac{9}{3}=3\). Using point \((2,9)\): \(9=3(2)+b \Rightarrow b=3\). So \(y=3x+3\).

Question 4

Answer: Hourly wage of \($12\)

In \(y=mx+b\), the slope \(m\) is the rate of change per unit of \(x\). Here, for each additional hour worked, earnings increase by \($12\), so the slope represents the hourly wage.

Question 5

Answer: Equipment fee of \($80\)

The constant term \(80\) is the \(y\)-intercept, which is the initial cost when \(t=0\) (before any monthly charges). This one-time equipment fee is paid at the start.

Question 6

Answer: Table A

When \(x=0\): \(y=4(0)-2=-2\). When \(x=1\): \(y=4(1)-2=2\). When \(x=2\): \(y=4(2)-2=6\). This matches option A.

Timed practice quizzes

Related Quizzes

After studying Building Linear Functions, open these two timed quizzes in a new tab for focused extra practice.

Connection to Standards

This lesson supports Grade 8 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.

Summary

Building Linear Functions becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.

GOLDEN RULE

Understand the model before choosing the operation.