How To Graph Trigonometric Functions | Trigonometry
Summary
TLDRThis 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 2Ο 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
- π The sine function, \( \sin x \), graphs as a sinusoidal wave with one period ending at \( 2\pi \).
- π Adding a negative sign in front of the sine function flips the wave over the x-axis, inverting its direction.
- π The cosine function, \( \cos x \), starts at the top for one period, unlike sine which starts at the center.
- π Negative cosine function graphs start at the bottom and follow a similar pattern to negative sine but inverted vertically.
- π Graphing multiple cycles involves breaking each cycle into four key points to plot the wave accurately.
- π 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.
- π Doubling the amplitude, as in \( 2\sin x \), stretches the graph vertically, with the wave varying from -2 to 2.
- π’ 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.
- β± 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.
- π 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.
- π 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 2Ο.
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 Ο/2, Ο, 3Ο/2, and 2Ο. 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 2Ο/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
π 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 2Ο. 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.
π 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.
π 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.
π 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.
π 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.
Mindmap
Keywords
π‘Sine Function
π‘Negative Sine Function
π‘Cosine Function
π‘Amplitude
π‘Period
π‘Phase Shift
π‘Vertical Shift
π‘Domain and Range
π‘Graphing
π‘Sinusoidal Wave
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 2Ο 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
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