How To Graph Trigonometric Functions | Trigonometry

Summary

This educational video script delves into the intricacies of graphing trigonometric functions, focusing on the sine and cosine functions. It explains how the amplitude affects the vertical stretch or compression of the graph, and how the period is calculated using the formula 2π divided by the coefficient of x. The script also covers phase shifts, detailing how to determine them and their impact on the graph's starting point. Practical examples, such as graphing 'y = 2 sin(x)' and 'y = -3 cos(x)', are provided to illustrate these concepts, making the complex nature of trigonometric graphing more accessible.

Takeaways

• 🌊 The sine function, \( \sin x \), graphs as a sinusoidal wave with one period ending at \( 2\pi \).
• 🔄 Adding a negative sign in front of the sine function flips the wave over the x-axis, inverting its direction.
• 📈 The cosine function, \( \cos x \), starts at the top for one period, unlike sine which starts at the center.
• 🔄 Negative cosine function graphs start at the bottom and follow a similar pattern to negative sine but inverted vertically.
• 📊 Graphing multiple cycles involves breaking each cycle into four key points to plot the wave accurately.
• 🔝 The amplitude of a sine wave is determined by the coefficient 'a' in front of the sine function, indicating the wave's peak height from the midline.
• 🔄 Doubling the amplitude, as in \( 2\sin x \), stretches the graph vertically, with the wave varying from -2 to 2.
• 🔢 The amplitude of a cosine function is the absolute value of the coefficient in front of cosine, as in \( -3\cos x \) where the amplitude is 3.
• ⏱ The period of a sine or cosine function is found by dividing \( 2\pi \) by the coefficient 'b' in front of 'x', affecting the wave's horizontal stretch.
• 🔄 A vertical shift in the graph is indicated by a constant added to the function, such as in \( \sin x + 3 \), which raises the entire wave by 3 units.
• 🔄 Phase shifts in trigonometric functions are calculated by setting the inside of the function to zero and solving for 'x', determining where the wave starts on the x-axis.

Q & A

What is the basic shape of the sine function, sine x?

The sine function, sine x, is a sinusoidal function that forms a sine wave. It oscillates between -1 and 1, starting at the origin, going up to 1, back down to -1, and then back to the origin, completing one full cycle at 2π.

How does the graph of sine x change if a negative sign is placed in front of it?

Placing a negative sign in front of the sine function, as in -sine x, flips the graph over the x-axis. Instead of starting at the origin and going up, it starts at the origin and goes down, creating a wave that oscillates between -1 and 1 in the opposite direction.

What is the starting point of the cosine function, cosine x, on the graph?

The cosine function, cosine x, starts at the top of its cycle, which is the value 1 on the y-axis, as opposed to the sine function that starts at the center (origin).

How does the graph of cosine x differ when a negative sign is included, as in -cosine x?

For -cosine x, the graph starts at the bottom of the wave, going up to the middle, then to the top, back to the middle, and then back down, completing one cycle.

What are the four useful points to consider when graphing one cycle of the sine wave?

The four useful points to consider when graphing one cycle of the sine wave are π/2, π, 3π/2, and 2π. These points help in breaking up the cycle into four intervals for easier graphing.

What is the amplitude of the sine wave and how is it represented in the function's formula?

The amplitude of the sine wave is the distance from the midline of the wave to its peak (or trough). It is represented by the coefficient 'a' in the function's formula, as in a sin(bx + c) + d, where 'a' is the amplitude.

How does the amplitude affect the graph of the sine function when it is doubled, as in 2 sine x?

Doubling the amplitude, as in 2 sine x, causes the graph to stretch vertically. The wave will now oscillate between -2 and 2 instead of the usual -1 and 1 range.

What is the domain and range of the cosine graph for the function y = -3 cosine x?

The domain of the cosine graph for y = -3 cosine x is all real numbers, as it is for all sine and cosine functions. The range is from the lowest y-value, which is -3, to the highest y-value, which is 3.

How does the period of the sine function change with the value of 'b' in the function a sin(bx)?

The period of the sine function is determined by the value of 'b' and is calculated as 2π/b. If 'b' is increased, the period decreases, causing the graph to complete more cycles in the same horizontal distance.

What is a phase shift in trigonometric functions and how do you calculate it?

A phase shift is a horizontal shift to the left or right of the graph of a trigonometric function. It is calculated by setting the inside of the function equal to zero and solving for 'x', which gives the phase shift value. For example, in the function a sin(bx + c), the phase shift is -c/b.

Outlines

00:00

📈 Introduction to Graphing Sine and Cosine Functions

This paragraph introduces the concept of graphing trigonometric functions, specifically focusing on the sine function. The sine function is described as a sinusoidal wave, with one cycle depicted from 0 to 2π. The effect of placing a negative sign in front of the sine function is explained, which results in the wave flipping over the x-axis. The paragraph also briefly discusses the cosine function, contrasting its starting point with that of the sine function. The focus is on graphing one period of the wave, which is broken down into four key points for clarity. The importance of plotting these points before graphing the entire wave is emphasized.

05:01

🔍 Exploring Amplitude and Graphing Techniques

The paragraph delves into the concept of amplitude in the context of sine waves, using the generic formula a * sin(bx + c) + d to explain that 'a' represents the amplitude. It illustrates how the amplitude affects the vertical stretch or compression of the sine wave, using examples to show how different amplitudes change the graph. The amplitude is defined as the distance from the midline to the highest or lowest point of the wave. The process of graphing the cosine function with a negative amplitude is also discussed, with a step-by-step guide on how to plot one period of the graph, taking into account the starting point, amplitude, and the typical wave pattern.

10:05

🔗 Understanding Period and Vertical Shifts

This section explains the concept of the period in trigonometric functions, demonstrating how the period is affected by the value of 'b' in the function a * sin(b * x). It uses examples to show how increasing or decreasing 'b' impacts the horizontal compression or stretching of the graph. The paragraph also introduces the idea of vertical shifts, explaining how to graph functions with a vertical shift by plotting the new center line and adjusting the amplitude accordingly. Examples of graphing sine and cosine functions with different amplitudes and vertical shifts are provided, emphasizing the method of plotting key points to construct the graph.

15:06

📉 Graphing with Phase Shifts and Vertical Shifts

The paragraph discusses the impact of phase shifts on the graph of sine and cosine functions. It explains how to calculate the phase shift by setting the inside of the function to zero and solving for 'x'. The phase shift determines where the wave starts on the x-axis. The process of graphing a sine function with a phase shift is detailed, including plotting the phase shift point and then adding one period to find additional points on the graph. The amplitude and period of the function are also considered in the graphing process. An example of graphing a sine function with both a phase shift and a vertical shift is provided, illustrating the steps to plot the graph correctly.

20:07

🌟 Advanced Graphing with Multiple Shifts and Amplitude Changes

This final paragraph covers more complex graphing scenarios involving both phase and vertical shifts, as well as changes in amplitude. It provides a method for graphing functions with these elements, starting with plotting the vertical shift and then determining the phase shift. The amplitude and period are recalculated based on the function's parameters. The paragraph walks through an example of graphing a sine function with a phase shift and vertical shift, explaining how to find the range of the graph and plot the sine wave accordingly. It concludes with a comprehensive example that ties together all the concepts discussed in the previous paragraphs.

Keywords

💡Sine Function

The sine function, often denoted as sin(x), is a fundamental mathematical function that describes the periodic oscillation of a point on a unit circle as it rotates with respect to the x-axis. In the context of the video, the sine function is used to illustrate a sinusoidal wave, which is a smooth, repetitive wave form. The video explains that plotting one cycle of the sine wave involves focusing on key points like pi/2, pi, 3pi/2, and 2pi, which help in graphing the wave from its starting point at the origin, going up to the peak, down to the trough, and back to the origin.

💡Negative Sine Function

The negative sine function, represented as -sin(x), is a variation of the sine function where the wave is reflected over the x-axis. This means that instead of starting at the origin and moving upwards, the wave starts at the origin and moves downwards, creating a mirror image of the standard sine wave. The video script uses this concept to demonstrate how the shape of the sine wave changes when a negative sign is introduced, emphasizing the importance of understanding the effects of algebraic manipulation on the graph of trigonometric functions.

💡Cosine Function

The cosine function, or cos(x), is another trigonometric function that, like the sine function, describes the position of a point on the unit circle as it rotates. Unlike the sine function, which starts at the origin, the cosine function starts at the top of the unit circle. The video script explains that graphing the cosine function involves plotting points at intervals of pi/2, pi, 3pi/2, and 2pi, similar to the sine function, but with the wave starting at the peak instead of the center.

💡Amplitude

Amplitude in the context of trigonometric functions refers to the distance from the midline of the wave to its peak (or trough). It is a measure of the wave's height and is represented by the coefficient 'a' in the equation 'a * sin(b * x + c) + d'. The video script discusses how the amplitude affects the graph of the sine and cosine functions, explaining that a larger amplitude results in a wave that extends further from the midline, both upwards and downwards.

💡Period

The period of a trigonometric function is the length of one complete cycle of the wave. It is determined by the coefficient 'b' in the equation 'a * sin(b * x + c) + d' and is calculated as 2π divided by 'b'. The video script uses the period to explain how the frequency of the wave changes; a smaller period means the wave completes more cycles within the same interval, resulting in a more compressed graph horizontally.

💡Phase Shift

A phase shift in trigonometric functions refers to the horizontal displacement of the wave from its standard position. It is represented by the coefficient 'c' in the equation 'a * sin(b * x + c) + d'. The video script demonstrates how to calculate the phase shift by setting the inside of the function to zero and solving for 'x'. This shift can either delay or advance the start of the wave on the x-axis, and the video provides examples of how to graph functions with phase shifts.

💡Vertical Shift

A vertical shift is the upward or downward movement of the entire graph of a trigonometric function. It is represented by the coefficient 'd' in the equation 'a * sin(b * x + c) + d'. The video script explains that a vertical shift affects the midline of the wave, changing the starting point of the wave from the x-axis to a higher or lower value. The video provides examples of how to graph functions with vertical shifts, including how to adjust the amplitude and period accordingly.

💡Domain and Range

In the context of trigonometric functions, the domain refers to the set of all possible input values (x-values), while the range refers to the set of all possible output values (y-values). The video script clarifies that the domain for sine and cosine graphs is all real numbers, as these functions are defined for any real number input. The range, however, is determined by the amplitude and any vertical shifts, with the lowest y-value being the amplitude below the midline and the highest y-value being the amplitude above the midline.

💡Graphing

Graphing in the context of the video refers to the process of visually representing trigonometric functions using a coordinate system. The video script provides a detailed step-by-step guide on how to graph sine and cosine functions, including identifying key points, calculating amplitude and period, and adjusting for phase and vertical shifts. The process involves plotting these points and connecting them to form the sinusoidal wave, which is essential for understanding the behavior of these functions.

💡Sinusoidal Wave

A sinusoidal wave is a smooth, repetitive wave form that is described by trigonometric functions such as sine and cosine. The video script emphasizes the characteristics of a sinusoidal wave, including its periodic nature and the way it oscillates between a maximum and minimum value. The video uses the sinusoidal wave to teach the viewer how to graph and understand the properties of trigonometric functions, highlighting the importance of this wave form in mathematics and its applications in various fields.

Highlights

Sine function is a sinusoidal wave, oscillating between positive and negative values.

Negative sine function inverts the wave, starting downwards from the origin.

The sine wave is periodic, repeating its pattern infinitely.

Cosine function starts at the peak, unlike sine which starts at the center.

Negative cosine function starts at the bottom and mirrors the positive cosine wave.

Graphing one period of a trigonometric function involves breaking it into four key points.

Amplitude determines the vertical stretch or compression of a sine wave.

The amplitude is the absolute value of the coefficient in front of the sine or cosine function.

The period of a sine or cosine function is calculated as 2π divided by the coefficient of x.

Graphing multiple periods involves extending the key points accordingly.

Vertical shifts in a trigonometric graph are represented by adding or subtracting a constant.

Phase shifts occur when there is a horizontal translation of the graph, calculated by setting the inside of the function to zero.

The domain of sine and cosine functions is all real numbers, while the range depends on the amplitude.

Graphing a function with a phase shift requires plotting from the phase shift point rather than the origin.

Combining vertical and horizontal shifts with amplitude adjustments results in a comprehensive graph of a trigonometric function.

Transcripts

now let's talk about graphing

trigonometric functions

let's start with

the sine function

sine x

sine x is basically a sinusoidal

function it's a sine wave

and that's how it looks like

at least that's one period

this ends at two pi that's one cycle of

the wave

now let's say if you put a negative in

front of the sine function

it's going to flip over the x-axis so

instead of going up initially it's going

to start from the origin it's going to

go down and then back up

and then back down

so that's the shape of sine and negative

sign now keep in mind

this wave keeps on going forever in both

directions

but

for the course of this lesson i'm going

to focus on graphing one period

which is basically one cycle of the wave

now what about the graphs of cosine x

and negative cosine

x cosine

starts at the top

whereas sine starts at the center

so that's one period of the cosine wave

let me do that a little bit better

but it can continue going on forever

negative cosine starts at the bottom

it goes up

to the middle and then goes up and then

back down

so that's the graph of one period of

negative cosine

so this is one cycle

now let's go back to the sine graph

let's draw two cycles

of this graph

so one cycle

you need to

break it up into four

useful points

one cycle is two pi

you want to break that up into four

points such as pi over two pi and three

pi over two

now if we want another period let's add

two pi to it

so we want to go to four pi in between

two pi and four pi is three pi and

between two pi and three pi

it's five pi over two

you add these two then divide by two

if we add three pi and four pi that's

seven pi

and then divided by two we get seven

over two

sine starts at the center

and then it's going to go up

back to the middle

down

back to the middle so that's one cycle

of the sine wave

and then it's going to go back up

back to the middle

down and then back to the middle that's

why it's helpful to plot the points

first

before

putting everything else

if you break up each cycle into five key

points

which equates to four intervals

it's going to be easier to graph the

sine wave

let's do the same for cosine

let's graph

two periods of the cosine wave

so one period is going to be two pi

two periods

four pi

for each period or each cycle break it

up into five

points which is four intervals

the first point by the way is the origin

it's zero

so these points will be the same as

the graphics sign

now we know that cosine starts at the

top

it's going to go back to the middle

and then to the bottom

back to the middle

and then to the top and it's going to

alternate

so it's going to look something like

this

so that's one cycle

and here's the second cycle

so that's two cycles of the cosine wave

now let's talk about the amplitude of

the sine wave

the generic formula is a

sine

bx plus c

plus d

now we're going to focus on a

a the number in front of sine is the

amplitude

so in this case the amplitude

is equal to 1.

so when you graph the sine wave

you plot your four points of interest

for one full cycle

the amplitude is going to be one

so it's going to vary from one to

negative one

so we know sine starts at the center

it's going to go to the top

back to the middle and then to the

bottom and then back to the middle

so it's going to look like this

and we know the period is 2 pi

now what if we wanted to graph

two sine x

so if we increase the amplitude this

graph is going to stretch

vertically

so it's going to vary from 2 to negative

2. by the way this is the amplitude

it's the distance between

the midline of the sine wave

and

the highest point

now let's plot one period

so this is going to be 2 pi

so once again sine is going to start

at the middle then it's going to go up

back to the middle and then down and

then back to the middle

so it's going to look like that

and so the amplitude tells you how much

it's going to stretch or compress

vertically

consider the equation

y is equal to negative 3 cosine x

what is the amplitude

of this function

the amplitude is always a positive

number so you ignore the negative sign

and it's going to be 3.

the amplitude is the absolute value of a

the number in front of cosine

now let's go ahead and graph it

let's plot one period

so let's break it up into five points

or four intervals

now the amplitude

is three so we need to

vary the sine graph or rather the cosine

graph from negative three to three

so cosine typically starts at the top

but we have negative cosine so it's

going to start from the bottom

then it's going to go to the middle

to the top

back to the middle and then back to the

bottom

so that's how we can graph

one cycle of negative three cosine x

now keep in mind this graph can keep on

going forever

in both directions

so let's say if we want to write the

domain and range

of this cosine graph

the domain for sine and cosine graphs

will always be the same it's all real

numbers

the range is based on the amplitude the

lowest y value is negative three the

highest y value is three

so that's how you can write the domain

and range of this particular cosine

graph

now let's talk about finding the period

so given this

sine function a sine b x we know a

represents the amplitude

now b

is not the period itself

but it's used to find the period

the period is two pi

divided by b

so in the case of sine x

b was equal to one so the period was two

pi

divided by one

now let's go ahead and graph these two

functions

sine x

and sine 2x

let's see

what effect b has on a graph

now we know the general shape of sine x

it has a period of 2 pi

and for the most part it looks like this

now if b is equal to 2 in this example

the period is going to be 2 pi divided

by b

so the period is pi

so therefore it's going to do one full

cycle

in less time so to speak

so what happens is the graph

it shrinks

horizontally

so one full cycle occurs

in one pi

two cycles occur in two pi

here's another example go ahead and

graph

this function two sine

one half x

so first we need to find the amplitude

the amplitude is the number in front of

sine that's two

the period

is two pi over b where b is the number

in front of x so in this case is one

half

two pi divided by one half is four pi

so this one is going to stretch

horizontally

the amplitude is 2

and the period is 4 pi

but we need to break it up into four

intervals

that's 1 pi 2 pi

3 pi and 4 pi

sine starts at the center then it goes

up

back to the middle

down and then back to the middle

so we're going to have a graph that

looks like that

so if you have a fraction what's going

to happen is it's going to stretch

horizontally

let's try another example

let's graph

4 cosine

pi x

so first identify the amplitude and the

period

the amplitude is simply 4 in this

example

and a period is 2 pi over b

in this case b

is the number in front of x so b is pi

2 pi divided by pi is 2.

so that's the period in this example

so let's go ahead and make a graph

so the amplitude is four

it's going to vary from four and

negative four

the period is two

so two should be about here

and we need to break it into four parts

so this is one

one half

and then between one and two you add

them up one plus two is three then

you average it or you divide it by two

so it's three over two

so those are the four points of interest

cosine starts at the top then it's going

to go to the middle

and then back to the bottom

to the middle and to the top

so we're going to have a graph that

looks like this

that's one cycle

and if we wish to extend it to draw

another cycle this is going to be three

uh next one is 2.5 or five over two

and then three plus four is seven but

then divided by two so three point five

is seven over two

the next point is going to be at the

middle and then back to the bottom

back to the middle and then to the top

and that's it

so that's how you can graph 4 cosine pi

x

so when you find your period make sure

you put that first on the x axis and

then break it into four intervals

now what is the domain and range of this

function

as you recall the domain for any sine or

cosine wave is all real numbers

the range

is from negative four to four

it's from the lowest y value to the

highest y value

now let's talk about what to do

when there's a vertical shift

let's say if we wish to graph sine

x plus three

so the vertical shift is three

the amplitude

is one

so what you want to do first

is you want to plot the vertical shift

so at 3 i'm going to draw a horizontal

line

that's going to be the new center of the

graph

the amplitude is 1

so

sine

is going to vary one unit higher than

the midline and one unit lower than it

so it's going to vary between two and

four

now we're still going to plot just one

period

so let's write our four key points

sine starts at the top

and then it goes to the middle

actually i take that back sign starts at

the middle

and then it goes to the top and then

back to the middle

to the bottom and then back to the

middle

so this would be one sine wave

so that's how you can graph sine x plus

string

let's try another example

let's graph

two periods

of two cosine

x

minus one

so this is going to be one cycle

and two cycle

but let's start with the first cycle

so the midline is at negative one

now the amplitude is two so we got to go

up two units

and down two units

now cosine

will start at the top

and then it's going to go to the middle

back to the bottom

and vice versa

now we need to plot one more cycle

so this is pi

and this is three pi

so it's going to go back to the middle

and then to the bottom

back to the middle

and to the top

so that's how we can graph

two cosine periods

now what is the range for this graph

notice the lowest y values at negative

three but the highest is at one

so the range

is from negative three to one

let's go ahead and graph this one

negative three sine x

plus four

so feel free to pause the video actually

let's also let's change it a bit

let's make it one-third

x plus four

the majority of the graph will be above

the x-axis

so let's draw the center line at four

first

the amplitude is three so we're gonna

have to go up three four plus three is

seven

and then down three starting from four

four minus three is one

so the range

is going to be from one to seven

now let's find the period we know the

period is two pi divided by b

and b is one third

so it's two pi divided by one third

so it's equal to six pi

and let's break it into four points half

of six pi is three pi

half of three pi

is three pi over two

and if you multiply this number by three

it will give us to this point which is

nine pi over two

now we know that sign starts at the

center

positive sign will go up initially but

negative sign

will go down

and then it's going to go back to the

middle

and then to the top at 7

and then back to the middle

so that's how you can plot

negative three

sine one third x plus four

now let's talk about how to graph this

function

sine x

minus pi divided by two

how can we do so

so considering the generic formula a

sine bx plus c

plus d

anytime there's a c value

there's a phase shift

which means that the graph is going to

shift either to the right or to the left

and so you want to find the phase shift

because sine won't start at the origin

in this case

so to find the phase shift

set the inside equal to zero and solve

for x so when you set b x plus c equals

to zero

x is going to equal negative c divided

by b

and this is your phase shift that's

where it starts on the x-axis

so let's set x minus pi over two

equal to zero so we can see x

is at pi over two so that's where the

sine wave is going to start

now let's go ahead and graph it

the amplitude is one and the period is

two pi over one so it's two pi

but first plot

pi over two

because that's where the phase shift is

and then what you want to do is

add one period to the phase shift

so you're adding two pi to pi over two

two pi is the same as four pi over two

so this will give you

five pi over two

so this is going to be three pi over two

and you want to break it into five key

points

this is one pi over two in between one

and three is two two pi over two is pi

in between three pi over two and five pi

over two we have four pi over two which

reduces to two pi

now the amplitude is 1 so it's going to

vary from 1

and negative 1.

now sine starts at the middle but we're

not going to start at the origin

in this example we're going to start at

the phase shift which is pi over 2.

positive sign

is going to go up negative sign is going

to go down first so negative sign will

be like this

positive sign will have that shape

and then at 2 pi it's going to have a y

value of negative 1 and that 5 pi over 2

is going to be back

on the x axis

so that's how you can plot this

particular sine wave

with a phase shift

now let's try another example let's say

if you want to plot 2

sine x

minus pi over 4

plus

three

so we have a vertical shift of three an

amplitude of two

the number in front of x is one so two

pi over one is two pi the period is

still two pi

but we do have a phase shift

so if we set the inside equal to zero

the phase shift is positive pi over four

the majority of the graph will be above

the x axis so

we're going to plot it up there

so let's plot the

midline first

or the center line

which is at three

the amplitude is two

so we need to travel through two units

above the center line which will take us

to five three plus two is five and then

two units down three minus two is one

so the graph is going to vary from one

to five and that's the range

of this sine function

now the phase shift is going to start at

pi

over four that's where the sine wave is

going to start

and if we add one period to that the

period is two pi

two pi over one is the same as a pi over

four we need to get common denominators

so if we add these two numbers this will

give us nine pi over four

so that's where the first period will

end

the midpoint between one and nine is

five

and the midpoint between one and five is

three

and between five and nine is seven

now we can graph it

so let's start with the phase shift sine

is going to start at the middle

and then it's going to go up

back to the middle

and then down and then back to the

middle

so this is the graph of just one period

you