Sinusoidal Function Transformations
1. Below is the graph of a sinusoidal function, with angles measured in degrees.
1 a)
What is the domain of the function?
1 b)
What is the range of the function? Then what is the amplitude?
1 c)
What is the period of the function?[br][Don't forget the units.]
1 d)
What is the axis of the curve of the function?[br](i.e. the horizontal line halfway between the max and the min)
1 e)
What is the phase shift of the function? Did you compare to the sine or cosine function?[br][Make sure to use "right" or "left" in your answer.]
1 f)
State the equation of the function in the form [math]f\left(x\right)=a\sin\left(k\left(x-d\right)\right)+c[/math] or [math]f\left(x\right)=a\cos\left(k\left(x-d\right)\right)+c[/math].
2. Describing the properties of a sinusoidal function given its equation.
[size=200][size=100]The height of the tide in a small beach town is measured along a seawall. The equation [code][/code][size=150][code][/code][math]h\left(t\right)=2\cos\left(60\left(t+3\right)\right)+5[/math] [/size]describes the height of the tide[size=150] [math]h\left(t\right)[/math] [/size]in meters at time[size=150] [math]t[/math] [/size]in hours.[/size][/size]
2 a)
What is the range and domain of the function?
2 b)
What is the amplitude of the function?
2 c)
What is the period of the function?
2 d)
What is the phase shift of the function?
3. Describing the average monthly temperature using sinusoidal functions.
The average monthly temperature (in [math]^{\circ}[/math]C) in London, ON can be described by a sinusoidal function. The temperature fluctuates between [math]-10^\circ[/math]C and [math]26^\circ[/math]C. Representing January as [math]t=1[/math], February as [math]t=2[/math], and so on; the phase shift of the function is 4 months right. The average temperature is hottest in July.
3 a)
What is the equation of the sinusoidal function that models the average monthly temperature in London, ON? State it as [math]T\left(m\right)[/math] where [math]m[/math] is the month as described above.
4. a) Follow the step below to model the support railing (diagonal beams) with the sine curve.
4. b)
Which variables ([math]a,k,x,d,c[/math]) do the sliders 1 to 4 correspond with in the following equation of a sine function: [math]f\left(x\right)=a\sin\left(k\left(x-d\right)\right)+c[/math]?
4. c)
Which property of a sine curve (e.g. amplitude, period, axis of curve, phase shift) does each slider affect?