Mathematics · Lesson 11

Graphs of linear & quadratic functions

JAMBWAECNECO

What the syllabus asks

JAMB: graphs of linear and quadratic functions, graphical solutions, gradients, linear inequalities

WAEC: graphs of linear functions (table of values, gradient, intercepts, y = mx + c); graphical solution of simultaneous linear equations; graphs of quadratic functions (table of values, choice of scale, smooth curve); reading roots, maximum and minimum values, axis of symmetry; solving related equations by drawing a straight line; gradient of a curve at a point by a tangent; linear inequalities in…

NECO: graphs of linear functions (table of values, gradient, intercepts, y = mx + c); graphical solution of simultaneous linear equations; graphs of quadratic functions (table of values, choice of scale, smooth curve); reading roots, maximum and minimum values, axis of symmetry; solving related equations by drawing a straight line; gradient of a curve at a point by a tangent; linear inequalities in…

What you will learn

Prepare tables of values and plot linear and quadratic graphs with suitable scales.
Find the gradient and intercepts of a straight line and write its equation.
Solve simultaneous linear equations graphically.
Read roots, minimum or maximum values and the axis of symmetry from a quadratic graph.
Use a drawn graph to solve related equations by adding a straight line.
Estimate the gradient of a curve at a point by drawing a tangent.
Show regions defined by linear inequalities.

Key points

Complete tables carefully; negative numbers squared are positive.
Use the given scale; plot accurately; ruler for lines, smooth curve for quadratics.
Gradient = rise ÷ run in axis units; y = mx + c gives gradient m and intercept c.
Parallel lines have equal gradients; perpendicular gradients multiply to −1.
Intersection of two lines solves the simultaneous equations.
Quadratic: ∪ if a > 0 (minimum), ∩ if a < 0 (maximum); axis x = −b/(2a).
Roots are read where the curve crosses the x-axis; solve f(x) = k with the line y = k.
To solve a related equation, rearrange so the drawn quadratic equals a straight line; draw it and read intersections.
Gradient of a curve at a point = gradient of the tangent; zero at a turning point.
Inequality regions: solid or broken boundary, test (0, 0), shade as instructed.

Start here: Tables of values and plotting

A graph is a picture of a relationship. Graph questions are very common in WAEC and NECO Paper 2 (usually one full question worth 12–15 marks), so careful technique earns many marks.

Steps for any graph question:

Copy and complete the table of values, substituting each x carefully (use brackets for negatives).
Choose the scale given in the question (e.g. 2 cm to 1 unit on the x-axis, 2 cm to 5 units on the y-axis). If none is given, choose one that uses most of the graph paper.
Draw and label both axes, numbering them evenly.
Plot each point accurately with a small cross or dot.
Join the points: a ruler for a straight line; a smooth freehand curve for a quadratic (never join with straight segments).
Label the graph with its equation.

Worked example 11.1

Complete a table of values for y = 2x − 1 for −2 ≤ x ≤ 3.

x	−2	−1	0	1	2	3
y	−5	−3	−1	1	3	5

For x = −2: y = 2(−2) − 1 = −5. Each step of 1 in x raises y by 2.
The y-values go up evenly, which confirms a straight line.

Worked example 11.2 (quadratic table)

Complete the table for y = x² − 2x − 3 for −2 ≤ x ≤ 4.

x	−2	−1	0	1	2	3	4
x²	4	1	0	1	4	9	16
−2x	4	2	0	−2	−4	−6	−8
−3	−3	−3	−3	−3	−3	−3	−3
y	5	0	−3	−4	−3	0	5

Splitting into rows for each term reduces sign errors.
The y-values are symmetric about x = 1, a useful check.

⚠️ Common mistake: (−2)² written as −4. A negative number squared is positive.

Try it yourself. Find y for y = 3 − x² at x = −3.

Show solution

3 − 9 = −6.

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Practice questions

1. Complete y = x² − 4x + 3 at x = −1, 0, 2, 5.

Show worked answer

8, 3, −1, 8.

2. Find the gradient and y-intercept of 4x + 2y = 10.

Show worked answer

y = −2x + 5: m = −2, c = 5.

3. Find the equation of the line through (1, 3) and (3, 7).

Show worked answer

m = 2: y = 2x + 1.

4. State the roots and minimum point of y = x² − 6x + 8.

Show worked answer

Roots 2 and 4; minimum (3, −1).

5. Given the graph of y = x² − 6x + 8, which line solves x² − 5x + 2 = 0?

Show worked answer

x² − 6x + 8 = −x + 6: y = 6 − x.

Key words

Coordinates: An ordered pair (x, y) locating a point.
Table of values: A table giving y for chosen x-values, used to plot a graph.
Linear function: A function whose graph is a straight line, y = mx + c.
Gradient (slope): Rise ÷ run; how steeply a line rises.
Intercept: Where a graph crosses an axis.
Quadratic function: A function y = ax² + bx + c with a ≠ 0.
Parabola: The U-shaped (or ∩-shaped) curve of a quadratic.
Root: A value of x where y = 0.
Turning point: The lowest (minimum) or highest (maximum) point of a parabola.
Axis of symmetry: The vertical line through the turning point.
Tangent: A straight line touching a curve at one point.
Scale: The number of units represented by each grid division.

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