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Use Algebra to Identify the Turning Point of a Quadratic

In this worksheet, students will practise completing the square to find the turning point (vertex) of a quadratic graph.

'Use Algebra to Identify the Turning Point of a Quadratic' worksheet

Key stage:  KS 4

Year:  GCSE

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, OCR, Pearson Edexcel,

Curriculum topic:   Algebra, Graphs of Equations and Functions

Curriculum subtopic:   Graphs Graphs of Equations and Functions

Popular topics:   Square Numbers worksheets

Difficulty level:  

Worksheet Overview

Quadratics can be dealt with in two ways - algebraically and graphically.

A lot of students see these as two separate topics, however, they do cross over and you can use one technique when dealing with the other.


The significant points of a quadratic graph

All quadratic graphs have the same form, they look something like this:


A quadratic graph


For all quadratic graphs, there are three things you need to be able to find using an algebraic method - the roots, the y-intercept and the vertex.


A quadratic graph


In this worksheet, we’re going to look at how we find the vertex (the turning point) algebraically.



How to find the coordinates of the vertex (turning point) for the quadratic y = x2 + 3x - 4

For this, we have to use one of the techniques for dealing with quadratics, completing the square.


Step 1: Complete the square for the quadratic

Using the technique of completing the square, we can see that:

y = x2 + 3x – 4 → y = (x + 1.5)2 – 6.25


Step 2: Find the x  coordinate of the turning point

This relies on you knowing something about square numbers – they can never be negative.

The smallest a squared number can be is 0.


When we are looking for the vertex, we are looking for where the value of y is as low as possible (the minimum) - this happens when the squared bracket is zero.

So what value of x will make the squared bracket zero?

(x + 1.5)2 =  0

x + 1.5 =  0

x = -1.5


Step 3: Find the y  coordinate of the turning point

We’ve already worked out that the minimum point is at x = -1.5.

To find the value of y, all we have to do is to plug x = -1.5 into the completed square  y = (x + 1.5)2 – 6.25

If we do this, we get y = (-1.5 + 1.5)2 – 6.25

The bracket now becomes zero and leaves us with y = -6.25


This means that the coordinates of the vertex are (-1.5,-6.25)


Is the vertex always the lowest point?

No. If you have a graph of the form y = x2 + ax + b you will get a minimum, however if you have a graph of the form y = -x2 + ax + b, you will get a maximum.


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