Graphs Of Linear Functions Quizzes
Introduction
Graphs of linear functions are a fundamental topic in GCSE Maths. A linear function is a function in which the graph forms a straight line. Understanding linear graphs allows students to visualise relationships between variables, interpret real-life data, and solve equations graphically. Mastery of linear graphs is essential for algebra, coordinate geometry, and problem-solving.
For example, the linear function $$y = 2x + 3$$ produces a straight line on a coordinate plane with a gradient of 2 and a y-intercept of 3. Graphing linear functions helps students identify key features such as slope, intercepts, and direction of the line.
Core Concepts
Linear Functions
A linear function can be written in the form:
$$y = mx + c$$
- m = gradient (slope) → change in y ÷ change in x
- c = y-intercept → point where the line crosses the y-axis
Gradient (Slope)
The gradient measures the steepness of the line:
Formula: $$m = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1}$$
Example:
- Points: (1,2) and (3,6)
- Gradient: $$m = (6-2)/(3-1) = 4/2 = 2$$
Y-Intercept
The y-intercept is the value of y when x = 0:
- In $$y = 2x + 3$$ → y-intercept = 3
X-Intercept
The x-intercept is the value of x when y = 0:
Example:
- $$0 = 2x + 3 → x = -3/2$$
- X-intercept = -1.5
Plotting Linear Graphs
- Identify y-intercept (c)
- Use gradient (m) to find another point
- Plot points on the coordinate plane
- Draw a straight line through the points
Example: $$y = 2x + 1$$
- y-intercept = 1 → point (0,1)
- Gradient = 2 → rise = 2, run = 1 → point (1,3)
- Draw straight line through (0,1) and (1,3)
Finding the Equation from a Graph
Steps:
- Find the y-intercept from where the line crosses the y-axis
- Calculate gradient from two points: $$m = (y_2 - y_1)/(x_2 - x_1)$$
- Form equation: $$y = mx + c$$
Example:
- Points: (0,2) and (2,6)
- Gradient: $$m = (6-2)/(2-0) = 4/2 = 2$$
- Equation: $$y = 2x + 2$$
Parallel and Perpendicular Lines
- Parallel lines: same gradient → $$y = mx + c_1$$ and $$y = mx + c_2$$
- Perpendicular lines: gradients are negative reciprocals → $$m_1 × m_2 = -1$$
Real-Life Applications
- Distance-time graphs → speed = gradient
- Cost vs quantity → unit cost and fixed cost
- Temperature over time → trends and prediction
- Finance → income vs expenditure
- Science experiments → linear relationships
Worked Examples
Example 1 (Foundation): Gradient and intercepts
Graph: $$y = 3x + 2$$
- y-intercept = 2 → point (0,2)
- Gradient = 3 → rise 3, run 1 → point (1,5)
- X-intercept: 0 = 3x + 2 → x = -2/3
Example 2 (Foundation): Equation from graph
Points on line: (0,1) and (2,5)
- Gradient: $$m = (5-1)/(2-0) = 4/2 = 2$$
- Y-intercept = 1
- Equation: $$y = 2x + 1$$
Example 3 (Higher): Parallel lines
Line 1: $$y = 2x + 3$$, Line 2 parallel through y = -1 → $$y = 2x - 1$$
Example 4 (Higher): Perpendicular lines
Line 1: $$y = 2x + 3$$, find line perpendicular through (0,0)
- Gradient of perpendicular: $$m_2 = -1/2$$
- Equation: $$y = -\frac{1}{2}x$$
Example 5 (Higher): Distance-time application
Distance travelled: d = 5t + 10, t in hours, d in km
- Y-intercept = 10 → initial distance 10 km
- Gradient = 5 → speed = 5 km/h
- Graph: straight line starting at (0,10) with slope 5
Example 6 (Higher): Solving linear equation from graph
Line passes through (0,3) and (4,11)
- Gradient: $$m = (11-3)/(4-0) = 8/4 = 2$$
- Equation: $$y = 2x + 3$$
Example 7 (Higher): Intersection point
Lines: $$y = 2x + 3$$ and $$y = -x + 6$$
- Set equal: 2x + 3 = -x + 6 → 3x = 3 → x = 1
- y = 2×1 + 3 = 5
- Intersection: (1,5)
Example 8 (Higher): Real-life cost problem
Fixed cost = £50, variable cost = £3 per unit, total cost C = 3x + 50
- Graph: C-intercept = 50
- Gradient = 3 → cost increases £3 per unit
- Units produced = 20 → cost = 3×20 + 50 = £110
Example 9 (Higher): Negative gradient
Equation: $$y = -2x + 10$$
- Y-intercept = 10
- Gradient = -2 → line slopes downwards
Example 10 (Higher): X-intercept
Equation: $$y = 3x - 9$$
- X-intercept: 0 = 3x - 9 → x = 3
Common Mistakes
- Confusing gradient with y-intercept
- Plotting points incorrectly
- Mixing up parallel and perpendicular lines
- Not reversing slope sign for perpendicular lines
- Errors in reading intersection points
Tips to avoid errors:
- Always identify y-intercept and gradient first
- Use two points to check gradient
- For parallel lines, keep gradient same; for perpendicular, use negative reciprocal
- Label axes clearly
- Check intersection by substitution
Applications
- Distance-time and speed graphs
- Cost and revenue models
- Temperature vs time trends
- Physics: force, pressure, or motion graphs
- Exam problems: predicting, interpreting, and solving linear relationships
Strategies & Tips
- Always determine slope and intercept before plotting
- Use formula $$y = mx + c$$ to predict points
- Check gradients with two points
- Label points and axes for clarity
- Practice real-life linear function problems
Summary / Call-to-Action
Graphs of linear functions are essential in understanding algebraic relationships. By mastering gradient, intercepts, plotting, and interpreting lines, students can solve equations graphically and apply them in real-life contexts. Regular practice ensures confidence in plotting, analysing, and using linear functions effectively.
Next Steps:
- Attempt linear function quizzes to reinforce learning
- Practice plotting lines from equations and finding gradients/intercepts
- Apply knowledge to real-life scenarios such as distance, cost, and temperature
- Challenge yourself with parallel, perpendicular, and intersection problems
Consistent practice makes graphing linear functions intuitive and accurate in all GCSE Maths problems.