1. “Teach A Level Maths” Vol. 1: AS Core Modules
4: Translations and Completing the Square
© Christine Crisp

2. Module C1
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3. The graph of forms a curve called a parabola
This point
is called the vertex

4. Adding a constant translates up the y-axis
e.g.
The vertex is now ( 0, 3)
has added 3 to the y-values

5. Adding 3 to x gives
We get
Adding 3 to x moves the curve 3 to the left.
This may seem surprising but on the x-axis, y = 0
so,

6. We can write this in vector form as: translation
Translating in both directions
e.g.
We can write this in vector form as:
translation

7. SUMMARY
The curve
is a translation of by
The vertex is given by

8. Exercises: Sketch the following translations of1. 2. 3.

9. 4 Sketch the curve found by translating
by What is its equation?
5 Sketch the curve found by translating
by What is its equation?

10. A quadratic function which is written in the form
is said to be in its completed square form.
We often multiply out the brackets as follows:
e.g.
This means multiply ( x – 5 ) by itself

11. The completed square form of a quadratic function
writes the equation so we can see the translation from
gives the vertex

12. e.g. Consider translated by 2 to the left and 3 up.
We can write this in vector form as:
translation
The equation of the curve is
Completed square form
Check: The vertex is ( -2, 3)

13. Subtract 16 to get rid of (-4)2 Check by multiplying out!
To write a quadratic function in completed square form:
- 8 7
e.g.
- 4
- 16 +7
Subtract 16 to get rid of (-4)2
Half the coefficient of x
But,
So,
Check by multiplying out!

14. SUMMARY
To write a quadratic function in completed square form:
e.g.
Draw a pair of brackets containing x with a square outside.
Insert the sign of b and half the value of b.
Square half of b and subtract it.
Add c.
Collect terms.

15. Exercises
Complete the square for the following quadratics: 1. 2. 3.

16. 4. 5. 6.
Hint: Start by taking 2 out as though it were a common factor

17. 4. 5. 6.

19. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

20. SUMMARY
The curve
is a translation of by
The vertex is given by

21. We can write this in vector form as: translation
Translating in both directions
e.g.
We can write this in vector form as:
translation

22. SUMMARY To write a quadratic function in completed square form: e.g.
Draw a pair of brackets containing x with a square outside.
Insert the sign of b and half the value of b.
Square the value used and subtract it.
Add c.
Collect terms.
e.g.
To write a quadratic function in completed square form:

23. SUMMARY
Completing the Square
e.g.
e.g.