Consider the values of a and b in the equation y=asin[b(x-h)]+k. What do they represent?
B
b
Transform y=sin x until it has double the period and three times the amplitude.
C
c
How did you transform the parent function to obtain y=3sin( 12x). What was changed? What was not?
A
a
See solution.
B
b
Diagram:
C
c
Similarities: Same vertical and horizontal position.
Differences: Different amplitude and period.
Practice makes perfect
a
Let's have a look at the general equation for a sine curve.
y=asin[b(x-h)]+k
The value of a shows the graph's amplitude, and the coefficient b determines the graph's period. If b>1 the period is shortened by a factor of b, while b<1 gives a longer period by a factor of b. We can write this as a formula.
period=2π/b
The parent function has an amplitude of 1 and a period of 2π. Therefore, the amplitude and period of y= 3sin( 12x) must be 1* 3=3 and 2π 1/2=4π respectively.
b
Let's start by sketching the parent function y=sin x.
Let's now give the parent function the same amplitude as y=3sin( 12x). The amplitude is the distance from the graphs midline to its highest or lowest point. The graph's midline is at y=0. Therefore, to give it an amplitude of 3, we have to strecth the graph's peaks and troughs until they reach y=3 and y=- 3, respectively.
Finally, we will double the period of the parent function to reflect the coefficient of x.
The functions have different periods and amplitudes. Therefore, these are the differences of the graphs. The similarities must therefore be what is left that could have been changed when transforming the parent function to y=3sin( 12x). That is they have the same horizontal and vertical shift, which is none.
