MathematicsFoundation20 min read

Graphs of Functions

Plotting a curve, and reading answers off it

This topic appears in:

9th Class Mathematics · ch.10O Level Mathematics (4024) · ch.2
01

A graph is a picture of every solution at once

An equation such as y = 2x − 1 has infinitely many solutions: (0, −1), (1, 1), (2, 3) and so on for ever. Plotting them all gives a line, and that line is the solution set. Every point on it satisfies the equation, and no point off it does.

That is why graphs are useful in an exam: once the curve is drawn, questions that would each need their own algebra can be answered by reading off the paper.

  • Value of y for a given x — go up from the x-axis and across.
  • Value of x for a given y — go across from the y-axis and down.
  • Roots of the equation — where the curve crosses the x-axis, because y = 0 there.
  • Maximum or minimum — the highest or lowest point of the curve.
  • Solution of two equations together — the point where the two graphs cross.
02

Drawing a graph that earns its marks

The marks in this chapter are given for method as much as for the picture, so the routine matters.

Build a table of values with at least five x values, including negatives. Work out each y carefully — a single arithmetic slip drags the whole curve out of shape. Choose a scale that uses most of the grid, mark it on both axes, plot the points, and join them with a smooth curve rather than a chain of straight segments. Label the graph with its equation.

Two habits that cost marks every year

First, joining plotted points with a ruler when the function is a curve — a parabola has no straight parts and a segmented "curve" is marked wrong. Second, squeezing the graph into a corner of the grid; a scale that fills the page is easier to read from, and reading off is where the remaining marks are.

03

The shapes worth recognising

You can tell most of what a graph looks like before plotting a single point, just from the highest power of x. That check catches a mis-plotted point immediately.

EquationHighest powerShapeKey feature
y = mx + c1straight linegradient m, y-intercept c
y = ax² + bx + c2parabolaopens up if a > 0, down if a < 0; one turning point
y = ax³ + …3cubicends go opposite ways; up to 3 roots
y = k/x−1hyperbolatwo branches; both axes are asymptotes
y = aˣexponentialrising curvepasses through (0,1); never touches the x-axis

Set the quadratic and drag a through zero. The parabola flips from opening upward to opening downward the instant a changes sign — which is the single fastest check on any quadratic sketch.

04

Solving equations from a graph

This is the part of the chapter that appears in the long-question section. The technique is always the same: rearrange the equation you have been asked to solve so that one side is the curve you have already drawn, then draw the other side as a second graph and read off where they cross.

Worked example

The graph of y = x² − 3x has been drawn for −1 ≤ x ≤ 4. Use it to solve x² − 3x = 2 and then x² − 4x + 1 = 0.

  1. For the first equation, draw the horizontal line y = 2.The left-hand side is already the curve you have, so you only need the right-hand side as a second graph.
  2. Read the two x values where the line cuts the curve: approximately x = −0.56 and x = 3.56.A quadratic crossed by a horizontal line gives two solutions, so quoting only one loses a mark.
  3. For the second, rearrange so the drawn curve appears: x² − 4x + 1 = 0 becomes x² − 3x = x − 1.Add x and subtract 1 on both sides. The aim is to leave x² − 3x untouched on the left.
  4. Draw the straight line y = x − 1 on the same axes and read the crossings: about x = 0.27 and x = 3.73.Two points of intersection, two solutions. Graphical answers are accepted to one decimal place.

x ≈ −0.6 and 3.6; then x ≈ 0.3 and 3.7

05

Simultaneous equations, graphically

Two straight lines drawn on the same axes cross at exactly one point, and its coordinates are the solution of the two equations taken together. The graph also explains the two special cases that algebra reports as odd-looking nonsense.

If the two lines are parallel, they never meet, and the equations have no solution. If they are the same line, every point on it works, and there are infinitely many solutions. When elimination produces "0 = 5" or "0 = 0", these are the two situations you are looking at.

Before you leave this chapter

  1. Table of values with at least five points, including negative x; then plot and join smoothly.
  2. The shape follows from the highest power — use it to check your plot before you trust it.
  3. Roots are where the curve meets the x-axis; the y-intercept is the value at x = 0.
  4. To solve a new equation from a drawn curve, rearrange until one side is that curve, and draw the other side.
  5. Parallel lines → no solution; identical lines → infinitely many.

Practice questions

6 questions · 20 marks · full working on every one

Try each one on paper first, then open the working. The marks are shown where they are actually awarded, because that is where they are actually lost.

Short questions

3 · 6 marks

Two marks each, in the style of the short-question section of the paper. Answer in two or three lines.

SQ1[2 marks]
State the gradient and the y-intercept of the line 2y = 6x − 8.
Model answer

Divide by 2 to reach y = 3x − 4. Gradient 3, y-intercept −4.

Examiner tip. Rearrange into y = mx + c before reading anything off. Quoting 6 and −8 from the unrearranged form is the standard error.

SQ2[2 marks]
How can the roots of y = x² − 5x + 6 be found from its graph?
Model answer

The roots are the x-coordinates of the points where the curve crosses the x-axis, because y = 0 at every point on that axis. Here they are x = 2 and x = 3.

Examiner tip. Say why: "because y = 0 on the x-axis". The reason is what separates a 2-mark answer from a 1-mark one.

SQ3[2 marks]
Two simultaneous linear equations are graphed and the lines turn out to be parallel. What does this tell you?
Model answer

The lines never intersect, so the pair of equations has no solution. Algebraically this shows up as a contradiction such as 0 = 5 during elimination.

Examiner tip. Linking the graphical picture to what the algebra does is often worth the second mark.

Solved numericals

2 · 8 marks

Full working, one step per line, with the marks shown where they are awarded.

N1[4 marks]
Complete a table of values for y = x² − 2x − 3 for x = −2, −1, 0, 1, 2, 3, 4, and state the coordinates of the turning point and the roots.
Full working
  1. y values: 5, 0, −3, −4, −3, 0, 5one wrong value is tolerated; two is not[1]
  2. Roots at x = −1 and x = 3the two x values where y = 0[1]
  3. Turning point at x = 1 by symmetry, midway between the roots[1]
  4. Minimum point (1, −4); it is a minimum because the coefficient of x² is positivethe reason is required for the mark[1]

Roots at x = −1 and x = 3; minimum at (1, −4).

Examiner tip. The turning point of a parabola always sits exactly halfway between the roots. Averaging them is faster and safer than reading the vertex off a hand-drawn curve.

N2[4 marks]
Solve graphically: y = x + 2 and y = −2x + 8.
Full working
  1. First line through (0, 2) with gradient 1two correct points are enough to draw a line[1]
  2. Second line through (0, 8) with gradient −2[1]
  3. Lines cross at (2, 4)[1]
  4. Check by substitution: 4 = 2 + 2 ✓ and 4 = −4 + 8 ✓a substitution check is expected in a graphical solution[1]

x = 2, y = 4

Examiner tip. Always substitute your reading back into both equations. A graphical answer read half a square out is caught immediately and can still be corrected.

Long questions

1 · 6 marks

Theory and numerical together, as they appear in the long-question section.

LQ1[6 marks]
The graph of y = x³ − 3x is to be drawn for −2 ≤ x ≤ 2.
  1. Complete the table of values for x = −2, −1, 0, 1, 2.
  2. Describe the shape of the graph and state the number of times it crosses the x-axis.
  3. Explain how the graph could be used to solve x³ − 3x = 1.
Mark scheme
  1. y values: −2, 2, 0, −2, 2e.g. at x = −1: (−1)³ − 3(−1) = −1 + 3 = 2[1]
  2. A cubic curve rising, turning down, then rising againtwo turning points[1]
  3. The ends go in opposite directions because the highest power is odd[1]
  4. It crosses the x-axis three timesat x = 0, √3 and −√3[1]
  5. Draw the horizontal line y = 1 on the same axesthe left-hand side is already the drawn curve[1]
  6. The x-coordinates of the three intersections are the solutionsthe number of solutions must be stated as three[1]

(a) −2, 2, 0, −2, 2 (b) a cubic with two turning points, crossing the x-axis three times (c) draw y = 1 and read the x values where it meets the curve

Examiner tip. Cubics with three real roots are the usual choice for this question because they make the "how many solutions" part worth asking. Count intersections, do not assume there are two.