Sine Function Properties: Range and Period

The graph of the function f(x)=\sin(x^{\circ}) has a minimum and a maximum value of y. The minimum value is -1; the maximum value is +1. The period of this curve is 360.

Vertical Stretch, af(x); Vertical Translation: f(x)+d

Now when this graph is stretched and translated vertically, the minimum and maximum values change.

The amount of vertical stretch will tell you how far on the y axis it is from ‘the bottom of the trough to the top of the crest’. For example when we multiply \sin(x^{\circ}) by 5, we get the¬†curve y=5\sin (x^{\circ}) ¬†which has a minimum y value at -5; maximum at 5 – the distance on the y axis between top and bottom is 10.

Now let’s translate this curve 1 unit upwards – we get the curve y=5 \sin (x^{\circ})+1. Now the min is -4; the max is +6 – because the whole curve has been moved up 1 unit.

Horizontal Stretch f(bx) and the period of the curve

Now a sinusoidal function of the form f(x)=a\sin(b(x-c)^{\circ})+d that has b=1 has period 360 – it has not undergone a horizontal stretch. When b=2, it undergoes a horizontal stretch of scale factor \frac{1}{2} (a compression scale factor 2). Consider the graph y=5\sin (2x^{\circ})+1. The parameter b=2, therefore the period is 360 \times \frac{1}{2}=180.

You may find it helpful to remember the formula:

    \[ \text{period} = \frac{360}{b}\]

or

    \[b = \frac{\text{360}}{\text{period}}\]

In conclusion, the graph y=5 \sin(2x^{\circ})+1 has range [-4, 6] and has period 180.

Short Comprehension Exercises


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