Find Turning Point by Completing the Square
In this revision note, you will learn the steps to find turning point by completing the square of Quadratic
Graphs.
3 Forms of Quadratic Expressions
A quadratic expression can be written in 3 main forms:
(i) General form: ax2 + bx + c
(ii) Factorised form: (x + p)(x +q)
(iii) “Completed square” form: a(x + h)2 + k
Before you read on, you might want to download this entire revision notes in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
Completing the Square
Apply the following steps to “complete the square”.
|
Step 1: Observe the general form. If the coefficient of x2 is not 1, factorise the coefficient of x2 for the terms with x2 and x only.
|
Example 1
2x2 + 16x – 1 = 2( x2 + 8x ) – 1 |
|
Step 2: Apply the identity: a2 + 2ab + b2 = (a + b)2 a2 + 2ab = (a + b)2 – b2 With reference to Example 1: · a2 is represented by x2, · a is represented by x, · 2ab is represented by 8x, and · b is represented by 4. **In general, b is always half the coefficient of the term with x (not x2). |
2x2 + 16x – 1 = 2( x2 + 8x ) – 1 = 2[(x + 4)2 – 42] – 1 This is the “completed square”. |
|
Step 3: Simplify the constants to obtain the “completed square” form: a(x + h)2 + k |
2x2 + 16x – 1 = 2( x2 + 8x ) – 1 = 2[(x + 4)2 – 42] – 1 = 2(x + 4)2 – 2(16) – 1 = 2(x + 4)2 – 33 |
Find the Turning Point by Completing the Square
Completing the Square Example 1
State the minimum value of = x2 − 4x + 10 and the corresponding value of x.
Solution
= x2 − 4x + 10
= (x − 2) 2 − (−2) 2 + 10
= (x− 2) 2 + 6
Since ( x − 2) 2 ≥ 0 for all real values of x, then ( x − 2) 2 + 6 ≥ 6 for all real values of x.
The minimum value of y is 6, when x = 2. (ans)
*Tip: To find the turning point from the “completed square” form, we let the expression inside the bracket be zero.
Note: Since the graph is a U-shaped graph, the turning point is a minimum point and the coordinates are (2 , 6).
Completing the Square Example 2
State the maximum value of = −2x 2 − 10x + 5 and the corresponding value of x.
Solution
y = −2x2 − 10x + 5
= −2 ( x + 5/2 )2 + 35/2
Since ( x + 5/2 )2 ≥ 0 for all real values of x, then −2 ( x + 5/2 )2 ≤ 0 for all real values of x.
And −2 ( x + 5/2 )2 + 35/2 ≤ 35/2
The maximum value of y is 35/2 , when x = − 5/2 . (ans)
Note: Since the graph is an inverted-U graph, the turning point is a maximum point and the coordinates are (− 5/2 , 35/2 ).
Summary of how to find a turning point by completing the square
In general, the coordinates of the turning point of a quadratic graph after completing the square, y = a(x + h)2 + k is always given by ( −h, k). This information is very useful for graph sketching.
Last Minute Revision for O Level Math?
Check out our exam guide on other topics here!
Before you go, you might want to download this entire revision notes in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
Does your child need help with Mathematics? Find out more about our Math Tuition Class Free Trial.
1) Live Zoom Lessons at Grade Solution Learning Centre
At Grade Solution Learning Centre, we are a team of dedicated educators whose mission is to guide your child to academic success. Here are the services we provide:
– Live Zoom lessons
– EdaptIQ™, a smart learning platform that tracks your child’s progress, strengths and weaknesses through personalised digital worksheets.
– 24/7 Homework Helper Service
We provide all these services above at a very affordable monthly fee to allow as many students as possible to access such learning opportunities. We also offer a free trial class and consultation with our tutors, to help make sure we are the best fit for your child.
We specialise in English, Math, and Science subjects.
You can see our fees and schedules here >>
2) Pre-recorded Online courses on Jimmymaths.com
If you are looking for something that fits your budget, or prefer your child learn at his or her own pace, you can join our pre-recorded online Math courses.
Your child can:
– Learn from recorded videos
– Get access to lots of common exam questions to ensure sufficient practice
– Get unlimited support and homework help
You can see the available courses here >>