How do you find equation from sinusoidal graph?

Mathematics
Tutor: None Selected Time limit: 1 Day

The graph is sinusoidal that runs 50 meters horizontally and 30m vertically. The high point is at and starts at 30m, the low point is -3m. I don't understand how to find the period as the graph does not show a repeating process. Or if I need it?..

Nov 16th, 2014

Introduction

You know how to graph the functions  and . Now you’ll learn how to graph a whole “family” of sine and cosine functions. These functions have the form  or , wherea and b are constants.

Periodic Functions

We used the variable  previously to show an angle in standard position, and we also referred to the sine and cosine functions as  and . Often the sine and cosine functions are used in applications that have nothing to do with triangles or angles, and the letter x is used instead of  for the input (as well as to label the horizontal axis). So from this point forward, we’ll refer to these same functions as  and . This change does not affect the graphs; they remain the same.

You know that the graphs of the sine and cosine functions have a pattern of hills and valleys that repeat. The length of this repeating pattern is . That is, the graph of  (or) on the interval  looks like the graph on the interval  or  or . This pattern continues in both directions forever.

The graph below shows four repetitions of a pattern of length. Each one contains exactly one complete copy of the “hill and valley” pattern.

If a function has a repeating pattern like sine or cosine, it is called a periodic function. The period is the length of the smallest interval that contains exactly one copy of the repeating pattern. So the period of  or  is . Any part of the graph that shows this pattern over one period is called a cycle. For example, the graph of  on the interval  is one cycle.

You know from graphing quadratic functions of the form  that as you changed the value of a you changed the “width” of the graph. Now we’ll look at functions of the form  and see how changes to b will affect the graph. For example, is  periodic, and if so, what is the period? Here is a table with some inputs and outputs for this function.

x (in radians)

2x (in radians)

sin2x

0

0

0

1

0

0

As the values of x go from 0 to , the values of  go from 0 to . We can see from the graph that the function  is a periodic function, and goes through one full cycle on the interval [0, ], so its period is . If you substituted values of x from  to , the values of  would go from  to , and  would go through another complete cycle of the sine function.

Notice that has two cycles on the interval [0, 2], which is the interval needs to complete one full cycle.

What is the smallest positive value for where  is at its minimum?

A) 

B) 

C) 

D) 

Show/Hide Answer

What is the period of the function? Here is a table with some inputs and outputs for this function.

x(in radians)

3x (in radians)

sin3x

0

0

0

1

0

0

As the values of x go from 0 to, the values of  go from 0 to. We can see from the graph that  goes through one full cycle on the interval , so its period is .

Notice that has three cycles on the interval [0, 2], which is the interval needs to complete one full cycle.

What is the period of the function ? Here is a table with some inputs and outputs for this function.

x (in radians)

x (in radians)

sin

0

0

0

1

0

0

As the values of x go from 0 to , the values of  go from 0 to .

We can see from the graph that  goes through one full cycle on the interval , so its period is.

Notice that has half of one full cycle on the interval [0, 2], which is the interval needs to complete one full cycle.

Let’s put these results into a table. For the first three functions we have rewritten their periods with the numeratorso that the pattern becomes clear. Can you see a relationship between the function and the denominator in the periods?

Function

Period

In each case, the period could be found by dividing  by the coefficient of x. In general, the period of  is , and the period of  is . Since the period is the length of an interval, it must always be a positive number. Since it is possible for b to be a negative number, we must use  in the formula to be sure the period, , is always a positive number.

The period of  or  is .

You can think of the different values of b as having an “accordion” (or a spring) effect on the graphs of sine and cosine. A large value of b squeezes them in and a small value of b stretches them out.

There is another way to describe this effect. In the interval  goes through one cycle while  goes through two cycles. If you go back and check all of the examples above, you will see that  has  cycles in the interval . Likewise,  has  cycles in the interval .


Nov 16th, 2014

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