Try placing one more minimum by repeating the given pattern.
Cosine Function: y=-2.5cos4x+5.5 Sine Function: y=2.5sin4(x-π/8)+5.5
Practice makes perfect
We are told that during one cycle a sinusoid has a minimum at ( π2,3) and a maximum at ( π4,8). Let's draw the points in a coordinate plane!
Let's place a third point so we can take a better look at the complete cycle. Since sinusoids alternate between their maximum and minimum value, we will place a minimum at (0,3).
We want to write a sinusoid that passes through these points. This means that we need to write a sine or a cosine function.
y=asinb(x-h)+k [0.5em]
or [0.5em]
y=acosb(x-h)+k
Since the y-intercept of the function occurs at a minimum of the function, we can try to match a cosine function.
Finding a Cosine Function
Let's begin by finding the vertical shift k. This corresponds to the mean of the maximum and minimum values.
k=8+ 3/2=5.5
Next, we will find the amplitude and period. To find the amplitude |a| we calculate half the difference between the maximum and minimum values.
|a| = 8- 3/2=2.5
In order to find the period we notice that the horizontal length of a cycle is π2.
Knowing the period, we can find b using the following relation.
Period=2π/b
Let's find b!
Finally, we notice that the y-intercept occurs at the minimum of the function. This means that the sinusoid is a cosine function with no phase shift, and the a value is negative.
y=-2.5cos4x+5.5
Finding a Sine Function
We found that the sinusoid can be modeled using a cosine function. Let's see what happens if we were to write it as a sine function.
We notice that the sine function does not match the points of the requested graph. Let's see what happens if we translate it π8 units to the right.
We found that the sine function matches the sinusoid after translating it π8 units to the right. Knowing this, we can write the requested sine function.
y=2.5sin4(x- π/8)+5.5
Using a Calculator
We will now verify if our functions pass through the given points. We begin by pressing the Y= button and typing each function in two different rows. We can write π directly by pressing 2nd= and ∧.
We now push GRAPH to graph them.
Notice that it appears that only one graph is shown — this is because our functions are overlapping. Our two functions are equivalent.
