Ch 7 · 2.10–2.11 Graphs of Functions and Sketching Curves · IGCSE Mathematics 0580
Topic 7 of 9 in Algebra and Graphs — IGCSE Mathematics 0580 — 58 lessons.
Introduction
Each family of functions has a recognisable shape: a straight line, a U-shaped parabola, an S-shaped cubic, a reciprocal curve that never touches its asymptotes, an exponential that shoots upwards. Knowing the shapes lets you sketch quickly; plotting accurately lets you solve equations that algebra cannot. (Syllabus sections 2.10 and 2.11.)
Key ideas — Plotting from a table of values
Work out y for each x in the table (use the calculator's table mode), plot points accurately, and join with a smooth curve (a ruled line only for linear graphs). For y = a/x, leave a gap at x = 0.
🧑🏫 Teacher's explanation
Think of a function like a recipe: you put in an input (x) and get out a specific output (y). A table of values is just a pre-planned list of ingredients to help you see the pattern before you draw it. If you plot these points on a graph, the 'shape' of the recipe emerges.
📖 Definitions
🔍 Worked example — Easy
Complete the table of values for y = 2x + 3 for x = -1, 0, 1, 2 and sketch the graph.
🔍 Worked example — Standard
Complete the table of values for y = 4/x for x = -2, -1, 0.5, 1, 2 and sketch the graph.
🔍 Worked example — Exam-style
A curve is defined by y = x^2 - 4. Use a table of values for x = -3, -2, -1, 0, 1, 2, 3 to sketch the graph.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Solving equations graphically
The solutions of f(x) = 0 are where y = f(x) crosses the x-axis. To solve f(x) = g(x), draw both graphs and read the x-coordinates of the intersections. To solve, say, x^2 - 3x - 6 = 0 using a graph of y = x^2 - 2x - 8, rearrange so that the left side is x^2 - 2x - 8: x^2 - 2x - 8 = x - 2, then draw y = x - 2.
🧑🏫 Teacher's explanation
Solving an equation graphically is like finding where two paths cross on a map. When we say f(x) = 0, we are looking for the points where the height (the y-value) is exactly zero. On a graph, the height is zero whenever the curve touches or crosses the horizontal x-axis. These points are called the roots of the equation.
📖 Definitions
🔍 Worked example — Easy
Find the roots of y = x - 3 by considering the equation x - 3 = 0.
🔍 Worked example — Standard
Solve x^2 = 2x + 3 graphically by sketching y = x^2 and y = 2x + 3.
🔍 Worked example — Exam-style
A curve is defined by y = x^2 - 4x + 3. Find the solutions to x^2 - 4x + 3 = 2x - 5 using a graph.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Gradients by tangents
Draw a tangent touching the curve at the point, then calculate its gradient from two points on the ruled line. For an exponential growth graph this gives the rate of growth at that moment.
🧑🏫 Teacher's explanation
Imagine you are driving a car along a winding mountain road. If you look at your speedometer at one exact moment, it tells you your instantaneous speed. However, your speedometer doesn't show the whole trip; it only shows how fast you are going at that specific split second. In mathematics, a curve is like that winding road. Because the direction is constantly changing, we cannot use a single formula like m = y_2 - y_1/x_2 - x_1 for the whole curve.
📖 Definitions
🔍 Worked example — Easy
Estimate the gradient of the curve y = x^2 at the point (2, 4) by drawing a tangent.
🔍 Worked example — Standard
A curve is defined by y = x^3. A tangent is drawn at x = 1. If the tangent passes through (1, 1) and (0, -2), find its gradient.
🔍 Worked example — Exam-style
An exponential growth function is modeled by y = 2^x. Estimate the gradient of the curve at x = 3 using a tangent that passes through (3, 8) and (4, 16).
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) Sketch the tangent to the curve at the point where x = 1. (b) Calculate the gradient of this tangent. [3 marks]
Key ideas — The shapes to sketch
🧑🏫 Teacher's explanation
Sketching a graph is like drawing a simplified map of a landscape. You don't need to plot every single point; you just need to know the 'landmarks'—where the graph hits the axes (roots), where it turns around (turning points), and where it gets stuck and can't cross a certain line (asymptotes). Think of a reciprocal graph like a slide that gets flatter and flatter as you move away; it never quite touches the ground.
📖 Definitions
🔍 Worked example — Easy
Sketch the graph of y = 3x + 4.
🔍 Worked example — Standard
Sketch the graph of y = 2/x + 1.
🔍 Worked example — Exam-style
Sketch the graph of y = x^2 - 4x + 3 by completing the square.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) Find the value of the constant k, where k > 0. (b) Sketch the graph, clearly labeling the turning point and the y-intercept. [4 marks]
Key ideas — Turning point of a quadratic
Complete the square: y = x^2 - 2x - 8 = (x - 1)^2 - 9, so the minimum point is (1, -9) and the line of symmetry is x = 1. Factorising gives the roots: (x - 4)(x + 2) = 0, so x = -2 and x = 4.
🧑🏫 Teacher's explanation
Imagine you are walking through a valley. The lowest point you reach before you start climbing back up is the 'turning point'. In algebra, a quadratic function y = ax^2 + bx + c creates a U-shaped curve called a parabola. If the curve opens upwards, this turning point is a minimum; if it opens downwards, it is a maximum.
📖 Definitions
🔍 Worked example — Easy
Find the turning point of the quadratic y = x^2 + 6x + 5 by completing the square.
🔍 Worked example — Standard
A curve has the equation y = 2x^2 - 8x + 3. Find the coordinates of its minimum point.
🔍 Worked example — Exam-style
Given y = -(x - 4)^2 + 7, find the roots of the equation (where y = 0).
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) Express y in the form (x - h)^2 + k, where h and k are constants. [3] (b) State the coordinates of the minimum point of the curve. [1] (c) Find the coordinates of the points where the curve intersects the x-axis. [2] [6 marks]
Worked examples
Use the graph of y = x^2 - 2x - 8 to solve x^2 - 3x - 6 = 0. Draw y = x - 2; the intersections are at x ≈ -1.4 and x ≈ 4.4 (exact values -1.37 and 4.37).
🔍 Worked example — Easy
Find the x-intercepts of the graph y = x^2 - 5x + 6 by solving the equation x^2 - 5x + 6 = 0 algebraically.
🔍 Worked example — Standard
Sketch the graph of y = 3/x + 2. Identify the vertical and horizontal asymptotes.
🔍 Worked example — Exam-style
A curve is defined by y = 2 × 3^x - 12. Use the graph to find the x-intercept and the y-intercept.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) (x + 2)^2 - 9: minimum (-2, -9); roots x = -5 and x = 1; y-intercept -5. (2) x = 0 and y = -2. (3) x^2 - 4x = 2x - 3, so draw y = 2x - 3. (4) increasing curve through (0, 2) with asymptote y = 1.