Everything Maths Grade 10

Functions of the formy=acos+q EMA59

Investigation: The effects ofaandqon a cosine graph

In the equation,y=acos+q,aandqare constants and have different effects on the graph.

On the same set of axes, plot the following graphs for 0 ° 360 °:

1.y 1 =cos 2

2.y 2 =cos 1

3.y 3 =cos

4.y 4 =cos+ 1

5.y 5 =cos+ 2

Use your results to deduce the effect ofq.

On the same set of axes, plot the following graphs for 0 ° 360 °:

1.y 6 = 2 cos

2.y 7 =cos

3.y 8 =cos

4.y 9 = 2cos

Use your results to deduce the effect ofa.

The effect ofq

The effect ofqis called a vertical shift because the whole cosine graph shifts up or down byqunits.

• Forq > 0 , the graph is shifted vertically upwards byqunits.


• Forq < 0 , the graph is shifted vertically downwards byqunits.

The effect ofa

The value ofaaffects the amplitude of the graph; the height of the peaks and the depth of the troughs.

• Fora > 0 , there is a vertical stretch and the amplitude increases.
  For 0 < a < 1 , the amplitude decreases.


• Fora < 0 , there is a reflection about thex-axis.
  For 1 < a < 0 , there is a reflection about thex-axis and the amplitude decreases.
  Fora < 1 , there is a reflection about thex-axis and the amplitude increases.

Note that amplitude is always positive.

Chapter 6. Functions 193