WHAT WAS THE QUESTION?            DERIVE INVESTIGATIONAL ACTIVITY

Completing the square

As you have seen in a previous activity, the standard form of a quadratic function is
f(x) = ax2 + bx + c , where a, b, c are given numbers.

An alternative way of writing a quadratic function is in the so-called "completed square" form. This is
f(x) = A(x + B)2 + C , where A, B, C are numbers not necessarily the same as a, b, c.

"Completed square" quadratics can be converted algebraically into the original form by expanding the bracket and simplifying.

Example:

f(x) = 2(x - 3)2 + 1         [A = 2 , B = -3 , C = 1]

= 2(x - 3)(x - 3) + 1

= 2(x2 - 3x - 3x + 9) + 1

= 2(x2 - 6x + 9) + 1

= 2x2 - 12x + 18 + 1

= 2x2 - 12x + 19         [a = 2 , b = -12 , c = 19]

EXERCISE 1

Express the following in the form ax2 + bx + c :

1) (x + 2)2 + 5          Ans:

2) (x - 7)2 + 2          Ans:

3) 4(x - 5)2 - 11          Ans:

4) 12 - 3(x + 4)2          Ans:

5) 100 - (x + 1)2          Ans:

6)          Ans:

7)         Ans:

8)         Ans:

9)         Ans:

10)         Ans:

EXERCISE 2

Express the following in the form ax2 + bx + c :

1)                         Ans: x2 + 4x + 1

2)                         Ans: x2 - 2x + 15

3)                         Ans: x2 - 18x - 50

4)                         Ans: 2x2 + 8x + 3

5)                         Ans: 2x2 - 10x - 11

6)                         Ans: -2x2 + 6x + 7

7)                         Ans: -3x2 - 12x + 1

8)                         Ans: 2x2 + 3x + 8

9)                         Ans: 2x2 - 5x - 20

10)                         Ans: 5x2 + x - 2

EXERCISE 3

If you are given a quadratic function in its original form f(x) = ax2 + bx + c , explain the rules for working out A, B, C in the "completed square" form.

EXERCISE 4

For each of the "completed square" quadratics in Exercise 1, plot on the same axes the graph of y = A(x + B)2 + C and the graph y = x2 . Explain what the values of A, B, C tell you about the graph.

 

Hints on using DERIVE

 

For Exercise 1

You should do these pen-and-paper first as shown in the example. It is good algebra practice!

To check your answers using DERIVE, try the following:

<A>uthor (x + 2)^2 + 5 <enter>

<E>xpand

<S>implify

 

For Exercise 2

You could just <A>uthor your own guess at what the question was (based on any patterns you have spotted while doing Exercise 1) and then <E>xpand and <S>implify it to see if you get the answer shown. Then improve your guess if necessary.

Alternatively, it might be quicker to <A>uthor the general form A(x + B)^2 + C and use <M>anage <S>ubstitute to replace A, B, C by what you think they are. Then <E>xpand and <S>implify as before to check.

Use <R>emove to tidy the screen between questions.

 

For Exercise 4

Split the screen vertically so that the Plot window appears beside the Algebra window.

First <A>uthor and <P>lot the graph x^2

Then <A>uthor and <P>lot the graph (x + 2)^2 + 5

Compare the graphs to see if you can spot the significance of B = 2 and C = 5.

Remember that F9 zooms in and F10 zooms out.

When you are ready, use the <D>elete <L>ast option since you want the graph x^2 to remain shown for the next one.

David Bowers,
Mathematics Workshop,
Suffolk College.