Linear Functions

Master straight line graphs and their properties

CAPS Grade 10 Mathematics

This topic forms part of the CAPS-aligned Grade 10 Mathematics curriculum and introduces linear functions used in various mathematical and real-world contexts. Use the interactive graph below to visualize how changing slope (m) and y-intercept (c) affects the line.

Linear Graph: y = mx + c

Graph Controls

Adjust Parameters

Current Function: y = 2x + 1
Hover over graph to see coordinates: x = 0, y = 1

Learning Outcomes

  • Represent linear functions in equation, table, and graphical forms
  • Understand and calculate slope (gradient) and y-intercept
  • Sketch and interpret graphs of linear functions
  • Find equations of lines given different information
  • Identify parallel and perpendicular lines
  • Apply linear functions to solve real-world problems

Introduction to Linear Functions

A linear function forms a straight line when graphed.

Standard Form
y = mx + c

Quiz 1 - Standard Form

In y = mx + c, what does 'c' represent?

A) Slope
B) y-intercept
C) x-intercept
D) Variable

Key Concepts and Skills

1

Representation

  • Equation: y = mx + c
  • Table: x-y value pairs
  • Graph: straight line
2

Slope (Gradient)

m = (y2 - y1) / (x2 - x1)
  • Positive: increasing
  • Negative: decreasing
  • Zero: horizontal
3

Y-intercept

  • Point (0, c)
  • Where line crosses y-axis

Quiz 2 - Slope

Find slope between (2,5) and (4,9)

A) 1
B) 2
C) 3
D) 4

Finding the Equation of a Line

Method 1: Slope & Y-intercept

Direct substitution into y = mx + c

Example

Slope 3, y-intercept -2

y = 3x - 2

Method 2: Two Points

Example

Through (2,5) and (4,9)

m = (9-5)/(4-2) = 2
y - 5 = 2(x - 2)
y = 2x + 1

Method 3: Slope & One Point

Example

Slope -2 through (3,4)

y - 4 = -2(x - 3)
y = -2x + 10

Sketching Linear Graphs

Sketching y = -3x + 2

Problem

Sketch y = -3x + 2

Solution
  1. Y-intercept: (0,2)
  2. Slope -3 = -3/1 → down 3, right 1
  3. Second point: (1,-1)
  4. X-intercept: 0 = -3x + 2 → x = 2/3

Parallel and Perpendicular Lines

P

Parallel Lines

Equal slopes: m1 = m2

y = 2x + 3 and y = 2x - 1

Parallel (both slope 2)

⊥

Perpendicular Lines

Slopes multiply to -1: m1 × m2 = -1

y = 2x + 3 and y = -½x + 5

Perpendicular (2 × -½ = -1)

Quiz 3 - Parallel/Perpendicular

Are y = 3x + 2 and y = 3x - 4 parallel?

A) Yes
B) No
C) Perpendicular
D) Same line

Applications of Linear Functions

Taxi Fare Problem

Problem

Taxi charges R10 fixed fee plus R5 per km. Cost for 15km?

y = 5x + 10
y = 5(15) + 10 = R85

Distance-Speed-Time

Problem

Car travels at 60 km/h. Distance after 2.5 hours?

d = 60t
d = 60 × 2.5 = 150 km

Practice & Assess

Test your knowledge with these interactive games.

Match - Slope Signs

m > 0
Increasing line
m < 0
Decreasing line
m = 0
Horizontal line
m undefined
Vertical line

Fill - Linear Equation

y = __x + c

Summary of Key Concepts

Standard Form: y = mx + c
Slope Formula: m = (y2 - y1)/(x2 - x1)
Parallel: m1 = m2
Perpendicular: m1 × m2 = -1

Key Terms

LinearSlopeGradientY-interceptX-interceptParallelPerpendicularIncreasingDecreasingHorizontalVertical
Hyperbolic FunctionsParabolic Functions