Everything Maths Grade 10

Functions of the formy=asin+q EMA55

Investigation: The effects ofaandqon a sine graph

In the equation,y=asin+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 =sin 2

2.y 2 =sin 1

3.y 3 =sin

4.y 4 =sin+ 1

5.y 5 =sin+ 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 sin

2.y 7 =sin

3.y 8 =sin

4.y 9 = 2sin

Use your results to deduce the effect ofa.

The effect ofq

The effect ofqis called a vertical shift because the whole sine 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 > 1 , 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 189