Cambridge International Mathematics

Part 5: Graphs of the form y=a(x¡h)^2 +k, a 6 =0

What to do:

1 Without the assistance of technology, sketch the graphs of y=2x^2 and y=2(x¡1)^2 +3

on the same set of axes. State the coordinates of the vertices and comment on the shape of the two

graphs.

2 Use agraphing packageorgraphics calculatorto check your graphs in step 1.

3 Repeat steps 1 and 2 for:

a y=¡x^2 and y=¡(x+2)^2 +3 b y=^12 x^2 and y=^12 (x¡2)^2 ¡ 4

4 Copy and complete:

² The graph of y=a(x¡h)^2 +k has the same shape and opens in the same direction as the

graph of ......

² The graph of y=a(x¡h)^2 +k is a .......... of the graph of y=ax^2 through a translation

of ......

You should have discovered the following important facts:

² Graphs of the form y=x^2 +k have exactly the same shape as the graph of y=x^2.

Every point on the graph ofy=x^2 is translated

μ

0

k

¶

to give the graph of y=x^2 +k:

² Graphs of the form y=(x¡h)^2 have exactly the same shape as the graph of y=x^2.

Every point on the graph of y=x^2 is translated

μ

h

0

¶

to give the graph of y=(x¡h)^2.

² Graphs of the form y=(x¡h)^2 +k have the same shape as the graph of y=x^2 and can be

obtained from y=x^2 by atranslationof

μ

h

k

¶

. Thevertexis at(h,k).

² If a> 0 , y=ax^2 opens upwards i.e.,

If a< 0 , y=ax^2 opens downwards i.e.,

If a<¡ 1 or a> 1 then y=ax^2 is ‘thinner’ than y=x^2.

If ¡ 1 <a< 1 , a 6 =0 then y=ax^2 is ‘wider’ than y=x^2.

²

y=a(x¡h)^2 +k

a> 0

a< 0

vertical shift ofkunits:

if k> 0 it goes up

if k< 0 it goes down

horizontal shift ofhunits:

if h> 0 it goes right

if h< 0 it goes left

a<¡ 1 or a> 1 , thinner than y=x^2

¡ 1 <a< 1 , a 6 =0, wider than y=x^2

434 Quadratic equations and functions (Chapter 21)

IGCSE01

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Y:\HAESE\IGCSE01\IG01_21\434IGCSE01_21.CDR Monday, 10 November 2008 12:28:32 PM PETER