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 $360 \times \frac{1}{2}=180$ .