Graphs of sine and cosine functions are called sinusoids. Consider the general forms of transformed sine and cosine functions.

Sine Function : Cosine Function : y = a sin b (x - h) + k y = a cos b (x - h) + k

In these functions

∣ a ∣

is the amplitude,

\frac{2 π}{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 = \frac{Maximum value + Minimum value}{2}

Let's substitute the maximum and minimum values into this equation.

k = \frac{Max value + Min value}{2}

SubstituteValues

Substitute values

k = \frac{2 + ( - 2 )}{2}

▼

Evaluate right-hand side

AddNeg

$a + (- b) = a - b$

k = \frac{2 - 2}{2}

SubTerms

Subtract terms

k = \frac{0}{2}

$\frac{0}{a} = 0$

k = 0

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 minimum point, the graph is a cosine curve with a horizontal shift of 4 units to the left. This means that $h = - 4 .$

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

4 .

The period of the graph of sine and cosine functions is

\frac{2 π}{b} .

With this information, we can write an equation in terms of

b .

4 = \frac{2 π}{b}

Let's solve this equation!

4 = \frac{2 π}{b}

▼

Solve for

b

MultEqn

$LHS \cdot b = RHS \cdot b$

4 b = 2 π

DivEqn

$LHS / 4 = RHS / 4$

b = \frac{2 π}{4}

ReduceFrac

$\frac{a}{b} = \frac{a / 2}{b / 2}$

b = \frac{π}{2}

We found that

b = \frac{π}{2} .

Step $4$

Next, let's find the amplitude

∣ a ∣

of the graph. This is half the difference between the maximum and the minimum values.

∣ a ∣ = \frac{Maximum value - Minimum value}{2}

Let's substitute the maximum and minimum values in the above equation and simplify.

∣ a ∣ = \frac{Max value - Min value}{2}

SubstituteValues

Substitute values

∣ a ∣ = \frac{2 - ( - 2 )}{2}

▼

Evaluate right-hand side

SubNeg

$a - (- b) = a + b$

∣ a ∣ = \frac{2 + 2}{2}

AddTerms

Add terms

∣ a ∣ = \frac{4}{2}

CalcQuot

Calculate quotient

∣ a ∣ = 2

The graph is 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 = \frac{π}{2},

h = - 4,

and

k = 0 .

Finally, we can write the function.

y = - 2 cos \frac{π}{2} (x - (- 4)) + 0 ⇕ y = - 2 cos \frac{π}{2} (x + 4)

Step 1

Step 2

Step 3

Step 4

Equation of the Sinusoid

Step $1$

Step $2$

Step $3$

Step $4$

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