DSE Maths | Functions and Graphs: A Tutor Guide

DSE Maths | Functions and Graphs: A Tutor Guide | Vertex and Intercepts First

2026-11-03

Tutor lesson-prep guides · Secondary Maths

Most students plot a quadratic by substituting x from minus three upwards and only notice halfway that the curve is not symmetric. This lesson reverses the order: find the axis of symmetry and vertex, then both intercepts, then add just two points, using the real numbers of 12 questions, 30 marks and 45 minutes.

Objectives — Students can: (1) get the axis x equals 3, the vertex at (3, -4) and the roots 1 and 5 for y equals x squared minus 6x plus 5; (2) state the four plotting steps of axis, vertex, intercepts and two extra points; (3) read the signs of a, b and c from a graph; (4) work out 3.75 minutes and 2.5 marks per question from 12 questions, 30 marks and 45 minutes, and finish the exit check in five minutes.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Completing the square	What do you add to x squared minus 6x to complete the square?	Add 9, giving (x minus 3) squared minus 9
Discriminant	What does b squared minus 4ac equal to 16 tell you?	Two real roots, because it is positive
Intercepts	What is x where a curve crosses the y axis?	x equals 0, so the point is (0, c)
Timing	With 30 marks in 45 minutes, how many marks per minute?	About 0.67

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-10 min	Plotting contrast	Ask for thirteen substitutions from minus three	See that the curve is not symmetric
10-22 min	Four steps	Axis, vertex, intercepts, two points	Get the vertex (3, -4) unaided
22-34 min	Graph to equation	Three graphs, decide the signs of a and c	Give the reason for each sign
34-44 min	Timing arithmetic	12 questions, 30 marks, 45 minutes	Get 3.75 minutes per question
44-54 min	Independent practice	Three questions in ten minutes	Mark answers and label the error type
54-60 min	Exit check	Four questions spoken, notes closed	Pass only when all four are right

03 Teaching script (read it out)

Scene 1: Find the vertex instead of substituting points

For y equals x squared minus 6x plus 5 the axis is x equals negative b over 2a, which is 6 over 2, so 3. Substituting 3 gives 9 minus 18 plus 5, which is minus 4, so the vertex is (3, -4). With the vertex and the axis, the two halves are mirror images and only two or three more points are needed. Plotting twenty points earns no extra marks and certainly costs time.

Ask for the axis value and why point-by-point plotting is unnecessary.

Scene 2: Intercepts are the cheapest two points

Put x equals 0 and the curve meets the y axis at (0, 5). Set y equals 0 and the discriminant is 36 minus 20, which is 16, so the roots are 6 plus or minus 4 over 2, giving 1 and 5. Four points, vertex included, fix the shape. Require the student to write these four coordinates beside the question before drawing anything.

Ask for the discriminant orally and then both roots.

Scene 3: Reading the equation off the graph is faster

An upward opening means a is positive; a positive y intercept means c is positive; an axis to the right of the y axis means b and a have opposite signs. Three rules finish a multiple-choice item in thirty seconds with no algebra. Give three graphs and ask only for the signs with one reason each, training judgement before calculation.

Ask for the sign of a when the curve opens downwards and for b when the axis sits to the left.

04 Question types and worked steps

Type	Example	Steps	Answer
Plotting	Given y equals ax squared plus bx plus c	Axis, vertex, intercepts, two points	All four coordinates shown
Vertex	Find the maximum or minimum	Complete the square or use negative b over 2a, then substitute	Written as a coordinate pair
Graph to equation	Signs of the coefficients	Opening, y intercept, axis position	One reason per sign
Application	Maximum profit or height	Model, find the vertex, answer with units	Units and one explanation

Sanity check (teach this): Order check: 45 divided by 12 is 3.75 minutes per question, and 0.375 means the division is inverted; 30 divided by 45 is about 0.67 marks per minute, and 6.7 means a misplaced decimal. Vertex check: the axis at 3 sits exactly between the roots 1 and 5, so a midpoint that disagrees with the axis means the discriminant or the formula was copied wrongly.

05 Independent practice (3 questions)

Question	Answer	What to watch
Minutes per question for 12 questions in 45 minutes	3.75 minutes	Division the right way round
Marks per minute for 30 marks in 45 minutes	0.67 marks	Decimal point in place
Vertex of y equals x squared minus 6x plus 5	(3, -4)	Written as a pair, not just the y value

06 Common mistakes and what to say

Mistake	Student writes	Say this
Substituting point by point	“I went from minus three to three and plotted thirteen points.”	“Thirteen points never told you where the curve turns. Find the axis x equals 3 and the vertex, and two points fix the shape.”
Giving only the y value of the vertex	“The vertex is minus four.”	“A vertex is a coordinate, so write (3, -4). Minus four alone does not say where on the x axis it sits.”
Calling a positive discriminant no solution	“I got 16, so there are no real roots.”	“16 is greater than zero, so there are two distinct real roots. No real roots means a negative discriminant, and the two cases cannot be swapped.”
Ignoring the axis when reading signs	“It opens upwards so a and b are both positive.”	“b depends on the axis: if the axis is to the right, b and a have opposite signs, so b is negative.”

07 Homework (paper, section and question numbers)

Maths chapter index — Revise quadratics, then do practice questions 1 and 2
Quadratic equations and the discriminant — Question 3 needs the discriminant before anything else
Graph transformations — Questions 4 and 5 compare vertex positions after a shift

08 Exit check (last 5 minutes)

Question	Answer	Decision
Minutes per question for 12 questions in 45 minutes	3.75 minutes	Error: inverted the division
Marks per minute for 30 marks in 45 minutes	0.67	Error: decimal point moved
How a vertex is written	(3, -4)	Error: y value only
First step in plotting	Find the axis of symmetry	Error: substituted points immediately

09 Tutor FAQ

My student insists on plotting every point. How do I convince them?

Time both methods on the same question: thirteen substitutions take about four minutes, the four steps take one and never break symmetry. Let them compare their own times.

Is completing the square really necessary?

Yes, because some questions give letters instead of numbers. Then only completing the square or the formula works, and the formula is error-prone when a is not 1.

Why do graph questions lose marks so often?

Usually a dropped minus sign or a missing bracket rather than weak understanding. Insist that every coordinate is written with brackets and a comma and half of those losses disappear.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.