<< Chapter < Page Chapter >> Page >

Write a formula for the function graphed in [link] .

A graph of 4sin((pi/5)x-pi/5)+4. Graph has period of 10, amplitude of 4, range of [0,8].

two possibilities: y = 4 sin ( π 5 x π 5 ) + 4 or y = 4 sin ( π 5 x + 4 π 5 ) + 4

Got questions? Get instant answers now!

Graphing variations of y = sin x And y = cos x

Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create graphs from equations.

Instead of focusing on the general form equations

y = A sin ( B x C ) + D  and  y = A cos ( B x C ) + D ,

we will let C = 0 and D = 0 and work with a simplified form of the equations in the following examples.

Given the function y = A sin ( B x ) , sketch its graph.

  1. Identify the amplitude, | A | .
  2. Identify the period, P = 2 π | B | .
  3. Start at the origin, with the function increasing to the right if A is positive or decreasing if A is negative.
  4. At x = π 2 | B | there is a local maximum for A > 0 or a minimum for A < 0 , with y = A .
  5. The curve returns to the x -axis at x = π | B | .
  6. There is a local minimum for A > 0 (maximum for A < 0 ) at x = 3 π 2 | B | with y = A .
  7. The curve returns again to the x -axis at x = π 2 | B | .

Graphing a function and identifying the amplitude and period

Sketch a graph of f ( x ) = 2 sin ( π x 2 ) .

Let’s begin by comparing the equation to the form y = A sin ( B x ) .

  • Step 1. We can see from the equation that A = 2 , so the amplitude is 2.
    | A | = 2
  • Step 2. The equation shows that B = π 2 , so the period is
    P = 2 π π 2    = 2 π 2 π    = 4
  • Step 3. Because A is negative, the graph descends as we move to the right of the origin.
  • Step 4–7. The x -intercepts are at the beginning of one period, x = 0 , the horizontal midpoints are at x = 2 and at the end of one period at x = 4.

The quarter points include the minimum at x = 1 and the maximum at x = 3. A local minimum will occur 2 units below the midline, at x = 1 , and a local maximum will occur at 2 units above the midline, at x = 3. [link] shows the graph of the function.

A graph of -2sin((pi/2)x). Graph has range of [-2,2], period of 4, and amplitude of 2.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Sketch a graph of g ( x ) = 0.8 cos ( 2 x ) . Determine the midline, amplitude, period, and phase shift.

A graph of -0.8cos(2x). Graph has range of [-0.8, 0.8], period of pi, amplitude of 0.8, and is reflected about the x-axis compared to it's parent function cos(x).

midline: y = 0 ; amplitude: | A | = 0.8 ; period: P = 2 π | B | = π ; phase shift: C B = 0 or none

Got questions? Get instant answers now!

Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph.

  1. Express the function in the general form y = A sin ( B x C ) + D  or  y = A cos ( B x C ) + D .
  2. Identify the amplitude, | A | .
  3. Identify the period, P = 2 π | B | .
  4. Identify the phase shift, C B .
  5. Draw the graph of f ( x ) = A sin ( B x ) shifted to the right or left by C B and up or down by D .

Graphing a transformed sinusoid

Sketch a graph of f ( x ) = 3 sin ( π 4 x π 4 ) .

  • Step 1. The function is already written in general form: f ( x ) = 3 sin ( π 4 x π 4 ) . This graph will have the shape of a sine function    , starting at the midline and increasing to the right.
  • Step 2. | A | = | 3 | = 3. The amplitude is 3.
  • Step 3. Since | B | = | π 4 | = π 4 , we determine the period as follows.
    P = 2 π | B | = 2 π π 4 = 2 π 4 π = 8

    The period is 8.

  • Step 4. Since C = π 4 , the phase shift is
    C B = π 4 π 4 = 1.

    The phase shift is 1 unit.

  • Step 5. [link] shows the graph of the function.
    A graph of 3sin(*(pi/4)x-pi/4). Graph has amplitude of 3, period of 8, and a phase shift of 1 to the right.
    A horizontally compressed, vertically stretched, and horizontally shifted sinusoid
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Draw a graph of g ( x ) = 2 cos ( π 3 x + π 6 ) . Determine the midline, amplitude, period, and phase shift.

A graph of -2cos((pi/3)x+(pi/6)). Graph has amplitude of 2, period of 6, and has a phase shift of 0.5 to the left.

midline: y = 0 ; amplitude: | A | = 2 ; period: P = 2 π | B | = 6 ; phase shift: C B = 1 2

Got questions? Get instant answers now!

Questions & Answers

difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Practice Key Terms 5

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask