Algebra
Graphs
Year 10 · Year 11
- ✓Plot and interpret straight-line graphs, including finding the gradient and y-intercept.
- ✓Plot and recognise the shapes of quadratic, cubic, reciprocal, and exponential graphs.
- ✓Identify key features of quadratic graphs, such as roots, turning points, and lines of symmetry.
- ✓Interpret and draw real-life graphs, including distance-time and speed-time graphs.
- ✓Solve problems by drawing and interpreting graphs.
Key concepts
A straight-line graph is represented by the equation y = mx + c, where 'm' is the gradient and 'c' is the y-intercept. The gradient describes the steepness and direction of the line. A positive gradient means the line slopes upwards from left to right, a negative gradient means it slopes downwards. The y-intercept is the point where the line crosses the y-axis. Parallel lines have the same gradient. Perpendicular lines have gradients whose product is -1.
A quadratic graph is represented by an equation of the form y = ax² + bx + c (where a ≠ 0). Its shape is a parabola. If 'a' is positive, the parabola is U-shaped (minimum point). If 'a' is negative, it's an inverted U-shape (maximum point). Key features include the roots (where the graph crosses the x-axis, i.e., y=0), the turning point (minimum or maximum point), and the line of symmetry (a vertical line passing through the turning point).
A cubic graph is represented by an equation of the form y = ax³ + bx² + cx + d (where a ≠ 0). The general shape of a cubic graph has one or two turning points, or a point of inflection where the gradient changes from increasing to decreasing (or vice versa) without a turning point.
A reciprocal graph is represented by an equation of the form y = k/x (where k is a constant). These graphs have two separate branches and never touch the x-axis or y-axis. These axes are called asymptotes.
An exponential graph is represented by an equation of the form y = k^x or y = a^x + b (where k > 0 and k ≠ 1). These graphs show rapid growth or decay. They have a horizontal asymptote, which is a line the graph approaches but never touches. For y = k^x, the x-axis (y=0) is the asymptote. For y = a^x + b, the line y=b is the asymptote.
Real-life graphs represent relationships between two quantities in practical contexts. Common examples include distance-time graphs and speed-time graphs. On a distance-time graph, the gradient represents speed. On a speed-time graph, the gradient represents acceleration, and the area under the graph represents the distance travelled. Conversion graphs allow conversion between different units or currencies.
Key facts to remember
- 1The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept.
- 2Parallel lines have the same gradient. Perpendicular lines have gradients m₁ and m₂ such that m₁m₂ = -1.
- 3Quadratic graphs (y = ax² + bx + c) are parabolas. 'a' > 0 gives a U-shape (minimum), 'a' < 0 gives an inverted U-shape (maximum).
- 4The roots of a graph are the x-values where the graph crosses the x-axis (y=0).
- 5Reciprocal graphs (y = k/x) have asymptotes at x=0 and y=0.
- 6Exponential graphs (y = k^x) show rapid growth or decay and have a horizontal asymptote.
- 7On a distance-time graph, the gradient represents speed.
- 8On a speed-time graph, the gradient represents acceleration, and the area under the graph represents distance travelled.
Worked examples
Example 1
a) Plot the graph of y = 2x - 1 for x values from -2 to 3. b) Find the gradient and y-intercept of the line.
Answer
a) Graph plotted as described. b) Gradient = 2, y-intercept = -1.
Always use a ruler for straight-line graphs.
Example 2
a) Complete the table of values for y = x² - 2x - 3. b) Plot the graph of y = x² - 2x - 3 for x values from -2 to 4. c) Use your graph to find the roots of the equation x² - 2x - 3 = 0. d) Write down the coordinates of the turning point.
Answer
a) Table completed. b) Graph plotted. c) Roots: x = -1, x = 3. d) Turning point: (1, -4).
Ensure your curve is smooth and passes through all plotted points.
Example 3
The graph shows a journey. Section A: (0,0) to (1, 60) - straight line. Section B: (1, 60) to (2, 60) - horizontal line. Section C: (2, 60) to (3.5, 0) - straight line downwards. Describe what is happening in each section (A, B, C) and calculate the average speed for the entire journey.
Answer
Section A: Travelling at a constant speed of 60 km/h for 1 hour. Section B: Stopped for 1 hour. Section C: Returning to the start at a constant speed of 40 km/h for 1.5 hours. Average speed for the entire journey ≈ 34.3 km/h.
Remember that speed is the magnitude of the gradient on a distance-time graph. A negative gradient means returning towards the origin.
Common mistakes
- ✗Confusing the gradient and y-intercept in y = mx + c.
- ✗Drawing straight lines for quadratic or cubic graphs instead of smooth curves.
- ✗Not using a consistent scale on both axes or labelling axes correctly.
- ✗Misinterpreting the meaning of a horizontal line on a distance-time graph (it means stationary, not constant speed).
- ✗Incorrectly calculating the gradient, especially with negative coordinates.
Exam tips
- ★Always use a sharp pencil and a ruler for plotting graphs.
- ★Label your axes clearly and use a consistent scale.
- ★For quadratic and cubic graphs, plot enough points to ensure you can draw a smooth, accurate curve.
- ★Read the question carefully to determine the required range of x-values and the level of accuracy for answers.
- ★When interpreting real-life graphs, pay close attention to the units on the axes.
Ready to practise?
Try a problem on this topic
Snap a photo or type a question — get step-by-step working instantly.