Free Study Materials for Grades 10 to 12

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Linear Functions

This topic forms part of the CAPS-aligned Grade 10 Mathematics curriculum and introduces linear functions used in various mathematical and real-world contexts.

Learning Outcomes

Introduction to Linear Functions

A linear function is a function that forms a straight line when graphed. In Grade 10, students are introduced to the fundamental concepts of linear functions, including their representation in various forms, their properties, and their applications in solving real-world problems.

Standard Form
y = mx + c

where:

Key Concepts and Skills

1

Representation of Linear Functions

Students should be able to represent linear functions in:

  • Equation form: y = mx + c
  • Table form: x-y value relationships
  • Graphical form: Cartesian plane plots
2

Understanding Slope (Gradient)

  • Rate of change of y with respect to x
  • Positive slope: increasing function
  • Negative slope: decreasing function
  • Zero slope: horizontal line
Slope Formula
m = (y₂ - y₁) / (x₂ - x₁)
3

Y-intercept

  • Point where line intersects y-axis
  • Value of y when x = 0
  • Directly from equation: y = mx + c
  • Coordinate: (0, c)

Finding the Equation of a Line

Students should be able to determine the equation of a line given different information:

1

Slope and Y-intercept

Direct substitution into y = mx + c

Example

Find equation with slope 3 and y-intercept -2

Solution

Equation: y = 3x - 2

2

Two Points

Use slope formula and point-slope form

Example

Find equation through points (2, 5) and (4, 9)

Solution
  1. Find slope:
    m = (9 - 5) / (4 - 2) = 4 / 2 = 2
  2. Use point-slope form:
    y - 5 = 2(x - 2)
  3. Simplify:
    y = 2x + 1
3

Slope and One Point

Use point-slope form: y - y₁ = m(x - x₁)

Example

Find equation with slope -2 through point (3, 4)

Solution
  1. Point-slope form:
    y - 4 = -2(x - 3)
  2. Simplify:
    y = -2x + 10

Sketching Linear Graphs

1

Sketching a Linear Function

Problem

Sketch the graph of the linear function y = -3x + 2

Solution
  1. Identify y-intercept:
    c = 2 → Point (0, 2)
  2. Identify slope:
    m = -3 = -3/1

    For every 1 unit increase in x, y decreases by 3 units

  3. Plot second point:
    From (0, 2), move right 1, down 3 → (1, -1)
  4. Draw line:

    Draw straight line through (0, 2) and (1, -1)

Graph Interpretation: The graph shows:
  • Negative slope: line decreases from left to right
  • Y-intercept: line crosses y-axis at 2
  • X-intercept: set y=0 → 0 = -3x + 2 → x = 2/3

Parallel and Perpendicular Lines

P

Parallel Lines

  • Have equal slopes: m₁ = m₂
  • Never intersect
  • Maintain constant distance apart
Example

Is y = 2x + 3 parallel to y = 2x - 1?

Solution

Both have slope 2, so yes, they are parallel

⊥

Perpendicular Lines

  • Slopes are negative reciprocals
  • m₁ × m₂ = -1
  • Intersect at 90° angle
Example

Are y = 2x + 3 and y = -½x + 5 perpendicular?

Solution

Check: 2 × (-½) = -1

Yes, they are perpendicular

Applications of Linear Functions

Real-World Modeling

Problem

A taxi charges a fixed fee of R10 plus R5 per kilometer. Write a linear equation to represent the total cost of a taxi ride and find the cost for 15km.

Solution
  1. Define variables:

    Let x = kilometers, y = total cost (R)

  2. Write equation:
    y = 5x + 10

    Where: m = 5 (rate per km), c = 10 (fixed fee)

  3. Calculate for 15km:
    y = 5(15) + 10 = 75 + 10 = R85
  4. Interpretation:

    The slope (5) represents cost per kilometer

    The y-intercept (10) represents base fee

Distance-Speed-Time Problems

Problem

A car travels at constant speed of 60 km/h. Write an equation for distance traveled over time and find distance after 2.5 hours.

Solution
  1. Define variables:

    Let t = time (hours), d = distance (km)

  2. Write equation:
    d = 60t

    Where: m = 60 (speed), c = 0 (starts at 0)

  3. Calculate for 2.5 hours:
    d = 60 × 2.5 = 150 km
  4. Graph interpretation:

    Graph is straight line through origin with slope 60

Teaching Strategies

1

Visual Aids

  • Graphs and diagrams
  • Interactive software
  • Physical demonstrations
2

Real-World Examples

  • Everyday situations
  • Practical applications
  • Relevant contexts
3

Problem-Solving

  • Varied problems
  • Group activities
  • Step-by-step guidance

Assessment Guidelines

Assessment Methods

  • Tests and Examinations: Concept knowledge and problem-solving
  • Assignments: Real-world application tasks
  • Projects: In-depth investigations and presentations
  • Class Participation: Discussion engagement and explanations
  • Practical Tasks: Graph sketching and data interpretation

Assessment Coverage

  • Representing linear functions in different forms
  • Calculating and interpreting slope and y-intercept
  • Sketching and interpreting linear graphs
  • Finding equations of lines from given information
  • Identifying parallel and perpendicular lines
  • Applying linear functions to solve problems

Example Problems for Assessment

Problem 1

Find equation of line through points (1, 3) and (4, 9)

Problem 2

Sketch y = ½x - 4 and identify intercepts

Problem 3

A printer costs R800 plus R0.50 per page. Write cost equation

Conclusion

Linear functions are a fundamental topic in Grade 10 mathematics, providing a foundation for more advanced concepts in algebra and calculus. By focusing on the key concepts and skills outlined in the CAPS curriculum, educators can ensure that students develop a strong understanding of linear functions and their applications.

Key Takeaway: Effective teaching strategies, diverse assessment methods, and real-world examples are essential for promoting student engagement and success in this important area of mathematics.

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