The general form of transformed sine and cosine functions are y=asin b(x-h)+k and y=a cos b(x-h)+k, respectively. Determine whether the graph is modeled by a sine or a cosine function, and then find the values of a, b, h, and k.
y=2cos 3x
Practice makes perfect
Graphs of sine and cosine functions are called sinusoids. Consider the general forms of transformed sine and cosine functions.
Sine Function:& y= asin b(x- h)+ k
Cosine Function:& y= acos b(x- h)+ k
In these functions | a| is the amplitude, 2Ď€ b is the period ( b>0), h is the horizontal shift, and k is the vertical shift. One method to write a sine or cosine function that models a sinusoid is to find these values. Let's now consider the given graph.
To write a function for this sinusoid, we will follow four steps.
Find the maximum and minimum values and use them to identify any vertical shift k.
Determine whether the graph should be modeled by a sine or a cosine function, and identify any horizontal shift h.
Find the period and the value of b.
Find the amplitude and the value of a.
Let's do it!
Step 1
From the graph, we can see that the maximum value is 2 and the minimum value is - 2.
The vertical shift k is the mean of the maximum and minimum values.
k=Maximum value+Minimum value/2
Let's substitute the maximum and minimum values into this equation.
We found that the vertical shift k is 0.
This means that there is actually no vertical shift!
Step 2
We can see that the graph crosses the y-axis at y=2.
Because the function passes through the y-axis at its maximum point, the graph is a cosine curve with no horizontal shift.
This means that h= 0.
Step 3
The shortest repeating portion of a sinusoid is called a cycle. The horizontal length of each cycle is called its period. Let's find the period of the given graph.
We can see that the period is 2Ď€3. The period of the graph of sine and cosine functions is 2Ď€ b. With this information, we can write an equation in terms of b.
2Ď€/3=2Ď€/b
Let's solve this equation!
Next, let's find the amplitude | a| of the graph. This is half the difference between the maximum and the minimum values.
| a|=Maximum value-Minimum value/2
Let's substitute the maximum and minimum values in the above equation and simplify.
The graph is not a reflection, so a>0. Therefore, a= 2.
Equation of the Sinusoid
Because of the location of the graph's intersection with the y-axis, we concluded that the sinusoid shown should be modeled by a cosine function. We also found that a= 2, b= 3, h= 0, and k= 0. Finally, we can write the function.
y= 2cos 3(x- 0)+ 0
⇕
y=2cos 3x
