Algebra & functions · Grade 8
Linear Functions and Graphs
A linear function describes a relationship with a constant rate of change. Its graph is a straight line.
Learning time: 20–25 minutes · Level: Developing
Learning goals
- Understand the key terms and concepts.
- Apply a reliable method to worked and new problems.
- Check your reasoning and communicate your answer.
1. Input and output
A function assigns exactly one output to each input. In y = 2x + 3, choose an x-value, substitute it, and calculate y. For x = 4, y = 11.
2. Tables and rules
Make an input-output table by substituting several x-values into the same rule. Ordered pairs (x, y) can then be plotted on a coordinate plane.
3. Slope or rate of change
Slope measures vertical change divided by horizontal change: m = (y₂ − y₁)/(x₂ − x₁). A positive slope rises left to right; a negative slope falls.
4. The y-intercept
In y = mx + b, m is the slope and b is the y-intercept, where the line crosses the y-axis. For y = 3x − 2, the slope is 3 and the y-intercept is −2.
5. Graph a linear rule
Plot the y-intercept first. Use the slope as rise/run to find another point, then draw a straight line through the points. Check by substituting a plotted point into the rule.
Interpret slope and intercept in context
A linear function has a constant rate of change. In y = mx + b, m is the slope and b is the y-intercept. The slope tells how much y changes when x increases by 1; the intercept is the output when x = 0. A positive slope rises from left to right, while a negative slope falls.
Worked reasoning: A line passes through (1, 5) and (3, 11). Its slope is (11 − 5) ÷ (3 − 1) = 6 ÷ 2 = 3. Substitute (1, 5) into y = 3x + b: 5 = 3 + b, so b = 2. The rule is y = 3x + 2.
If the situation describes a taxi fare with a fixed starting fee plus a charge for each kilometre, the intercept represents the starting fee and the slope represents the cost per kilometre.
- Use the same order of points in the numerator and denominator when finding slope.
- Do not confuse the y-intercept with the x-intercept.
- Check a graph by substituting one plotted point into its equation.
Worked examples
- For y = 2x + 1, x = 0, 1, 2 gives y = 1, 3, 5.
- Between (1, 4) and (3, 10), slope = (10 − 4)/(3 − 1) = 3.
- In y = −2x + 5, slope = −2 and y-intercept = 5.
Independent practice
Attempt each problem before checking the worked answer.
- Find y when y = 3x + 2 and x = 4. Answer: 14.
- For y = −x + 6, find y when x = 2. Answer: 4.
- Find the slope between (2, 3) and (4, 9). Answer: 3.
- State the slope and y-intercept of y = 5x − 7. Answer: 5 and −7.
- Does y = 4x + 1 rise or fall left to right? Answer: Rise; slope is positive.
- Complete y = 2x for x = −1, 0, 3. Answer: −2, 0, 6.
- Find the y-intercept of y = −3x + 8. Answer: 8.
- A taxi costs $4 plus $2 per kilometre. Write the rule. Answer: C = 2k + 4.
Knowledge check
Choose one answer for each question, then check your answers for explanations.
Printable worksheet and answer key
Use the independent-practice list above as your worksheet. Show working, label units where appropriate, and review the answers after attempting all questions.
Study reminder
Write the rule or method used beside each solution. This helps identify and correct mistakes.