Trig functions grade 11 and 12: Horizontal shift
Summary
TLDRThis educational video tutorial focuses on the horizontal shifting of trigonometric graphs, specifically using the sine function as an example. The instructor explains the concept of shifting, distinguishing it from stretching, and guides viewers on how to use a calculator to plot the graph of 'sin(x) - 30' over the interval 0 to 360 degrees. The tutorial covers the process of setting up the calculator for the graph, determining the step size based on the period of the graph, and plotting the points. It also touches on the importance of identifying key features like amplitude, range, domain, and period of the graph, and how these are affected by horizontal shifts.
Takeaways
- 📈 The lesson focuses on horizontal shifts in trigonometric graphs, specifically how they move in the horizontal direction.
- 🔄 Horizontal shifting involves moving the entire graph to the left or right, unlike vertical stretching which affects the graph's amplitude.
- ✏️ The instructor demonstrates how to use a calculator to graph the function 'sin(x) - 30' over the interval of 0 to 360 degrees.
- 📱 The calculator is set to 'table' mode, and the equation 'sin(x) - 30' is entered to generate the necessary x and y values for plotting.
- 📐 The step size for the calculator is determined by dividing the period of the graph (360 degrees) by 4, resulting in a step size of 90 degrees.
- 📉 The amplitude of the graph remains unchanged at 1, as horizontal shifts do not affect the graph's height from its resting position.
- 📋 The range of the graph is from -1 to 1, reflecting the minimum and maximum y-values of the sine function.
- 🗺️ The domain is specified as from 0 to 360 degrees, which is the interval over which the graph is drawn.
- ⏳ The period of the sine graph remains 360 degrees, as horizontal shifts do not alter the graph's periodicity.
- 🔍 The instructor points out the limitations of using a calculator for graphing, such as not being able to accurately determine turning points due to the shift.
Q & A
What is the primary focus of the lesson described in the transcript?
-The primary focus of the lesson is to understand how trigonometric graphs can be shifted horizontally, specifically by exploring the effect of a horizontal shift on the sine graph.
What is the difference between shifting and stretching in the context of trigonometric graphs?
-Shifting refers to moving the graph horizontally or vertically without changing its shape or size, while stretching involves altering the period or amplitude of the graph, which changes its size and appearance.
How does the instructor suggest using a calculator to graph the function sin(x) - 30 over the interval 0 to 360?
-The instructor suggests using the calculator's table function to find the values of the function over the interval 0 to 360, with a step size of 90 (which is 360/4, as the period of the sine graph is 360 degrees).
What is the significance of the step size chosen for the calculator in this lesson?
-The step size of 90 is significant because it corresponds to one-fourth of the period of the sine graph, ensuring that the graph captures all the important points within the given interval.
Why does the instructor emphasize that the highest y-value on the graph should be 1, even though the calculator might show a different maximum?
-The instructor emphasizes that the highest y-value should be 1 because the sine function has a range from -1 to 1, and the calculator's maximum value might not always reflect the true range of the function over the interval considered.
What are the amplitude, range, and period of the graph of the function sin(x) - 30?
-The amplitude of the graph is 1, as it's the maximum distance from the resting position. The range is from -1 to 1, reflecting the y-values of the sine function. The period remains 360 degrees, as the function has only been shifted, not stretched or compressed.
How does the horizontal shift of -30 degrees affect the turning points of the sine graph?
-A horizontal shift of -30 degrees to the right moves the turning points of the sine graph from (90, 0) and (270, 0) to (120, 0) and (300, 0) respectively.
What is the domain of the graph of the function sin(x) - 30 as described in the lesson?
-The domain of the graph is from 0 to 360, as specified in the instructions for drawing the graph.
Why does the instructor caution against relying solely on the calculator for graphing trigonometric functions?
-The instructor cautions against relying solely on the calculator because it may not provide all the necessary points, such as the shifted turning points, which require understanding the transformation of the graph.
How does the instructor suggest completing the graph of sin(x) - 30 when the calculator cannot fill in all the gaps?
-The instructor suggests using the knowledge of the sine function's properties, such as its turning points and period, to fill in the missing gaps and complete the graph accurately.
Outlines
📈 Understanding Horizontal Shifts in Trigonometric Graphs
This paragraph introduces the concept of horizontal shifts in trigonometric graphs, contrasting it with vertical stretching. The speaker explains that shifting involves moving the entire graph in a certain direction, akin to moving a point from one location to another. The focus is on the horizontal movement, which is different from vertical stretching, which will be covered in subsequent videos. The audience is guided to experiment with the effect of a -30 degree shift on a sine graph over the interval of 0 to 360 degrees using a calculator. The process involves setting the calculator to 'table' mode, inputting the equation sin(x - 30), and adjusting the starting and ending points as well as the step size, which should be one-fourth of the graph's period. The speaker emphasizes that the period remains unchanged at 360 degrees despite the shift, and the step size is calculated accordingly. The audience is then instructed to plot the graph on paper, marking the highest and lowest y-values and using the step size on the x-axis to fill in the values from the calculator. The paragraph concludes with a discussion on the limitations of the calculator method for plotting, especially when it comes to identifying turning points that are shifted due to the horizontal movement.
📚 Exploring Amplitude, Range, Domain, and Period of Trigonometric Graphs
In the second paragraph, the focus shifts to explaining the amplitude, range, domain, and period of trigonometric graphs. The amplitude is defined as the distance from the resting position of the graph, which is identified as the maximum distance from the midline, and in this case, it is one. The range is described as the set of y-values the graph takes, which extends from -1 to 1 for the given graph. The domain is clarified as the set of x-values provided by the problem, which is from 0 to 360 degrees in this instance. The period of the graph, which is the length of one complete cycle, is discussed in relation to the standard periods of sine and cosine graphs, which are 360 degrees. The speaker notes that while the period can be altered by stretching or compressing the graph, in this particular case, the graph has only been shifted horizontally by 30 degrees, so the period remains at 360 degrees. The paragraph aims to familiarize the audience with these fundamental concepts of trigonometric graph analysis.
Mindmap
Keywords
💡Trigonometric graphs
💡Vertical shifting
💡Vertical stretching
💡Horizontal shifting
💡Sine function (sin x)
💡Amplitude
💡Range
💡Domain
💡Period
💡Calculator
💡Step size
Highlights
Introduction to horizontal shifting of trigonometric graphs.
Difference between shifting and stretching of graphs explained.
Vertical shifting moves a point, while stretching changes the shape of the graph.
The example uses the sine function with a horizontal shift of 30 degrees.
Instructions on using a calculator to plot sine graphs.
Graphing the sine function over the interval of 0 to 360 degrees.
Explanation of how to use the step size based on the period of the sine graph.
Clarification that the period of a sine graph remains 360 degrees even after shifting.
Graphing points using specific x-values and y-values derived from a calculator.
Plotting the sine graph using x-axis steps of 90 degrees.
Manual adjustment of points not given by the calculator due to the 30-degree shift.
Explanation of amplitude, range, domain, and period for the sine graph.
Amplitude is defined as the distance from the resting position, which is 1 for this graph.
The range of the sine graph is from -1 to 1 on the y-axis.
The period remains 360 degrees, despite the horizontal shift of the graph.
Transcripts
hello everyone
in the previous lessons we've been
looking at vertical shifting and
vertical
stretching for a for trigonometric
graphs but now we're going to look at
the way that trigonometric graphs can
also move in a horizontal direction so
we're going to start off with looking at
how they can shift now remember shift
means if you take this point of view you
can move it
to this point over here
okay so we've moved it in that direction
over there
stretching is something totally
different that's something we're going
to look at in the next few videos
stretching is if you have a piece of
string like that for example and you
pull either side and so that the new
piece of string looks something like
that
shifting however is taking a single
point and then moving it so that it
lands up over there
so what i want us to do for the
beginning just so we can start
experimenting with what this minus 30 is
going to do to the graph is i want us to
draw this graph over the interval of 0
to 360. so we just use the calculator
for that and so by now you guys know how
to do this but just remember we're going
to go mode then we're going to go to
table on the normal casio calculators
the more basic ones i think you would
push option number three but nonetheless
you're looking for the option that says
table
you would then type in the equation so
this is sin x minus 30. once you've
typed that in you just push equals g of
x is any other graph which we don't have
any other one so we'll just push equals
the starting point must always be the
start that they gave you so they want
you to draw the graph from zero so we'll
say zero and you must say equals the
ending point will be 360.
and then your step now this is an
interesting one your step must always be
equal to the following step must always
be equal to your period of your graph
divided by 4. now we know that a normal
syn graph has a period of 360. if you
shift that graph upwards let's say
you've got a sun graph and you shift it
upwards well all that's going to happen
is your new graph's just going to do
something like that but you're not going
to change the period you only change the
period if you start compressing it so
you make it look like that or if you
have to stretch it but we are just
sliding it along so our period will
still be 360 and so we can still say 360
over 4 and so our step will be 90. so we
can fill in the 90 over there and there
we have it it then gives us all the x
values and all the y values so now we
can draw the graph so the step that you
used
on the calculator that's also what you
want to use on your x axis so we used a
step of 90. most times in a test they'll
give you a piece of
block paper to draw these
then on the y-axis well we know that
if we look on the calculator the highest
y value is 0.867 but we should also
remember that a normal syn graph goes up
to one okay and we're gonna have to show
that
so let's just let ours go up to one and
then it also goes to negative
one
now the reason i'm not going on the left
here is because they've told us that we
can go from zero so now we just go fill
in the values so if you look on your
calculator when x is zero then y is
negative zero comma five so that's that
point there then when x is 90 y zero
comma eight six six so let's just fill
that in on the y axis here as zero comma
eight six six like that
then at one eighty it is zero comma five
there 270 and negative zero comma eight
six six that's something over here so
you can just fill that in on your y-axis
as negative zero comma eight six six
and then lastly 360 is negative zero
comma five there are other ways to do
these questions i usually don't
use a calculator but i've noticed that
many students do like to use the
calculator and so for that reason i've
just decided to show it using a
calculator
and so this point is going to go
somewhere over here
then what you can do is try complete
that as neatly as possible so it's a syn
graph so it will do something like
that let me try that again there we go
so we've got some type of shape over
there but now we need to try fill in a
few of the missing gaps that the
calculator can't help us with and this
is why i don't really like the
calculator method due to the fact that
we've actually moved up by 30 degrees so
we know that a normal syn graph
has a turning point at 90 and 1. now
what does minus 30 do to a graph think
of your parabolas hyperbolas well it
moves at 30 degrees to the right and so
that's going to become 120 and 1. so
this point over here
should just be labeled
120 and 1.
now a cent graph also turns at 270
degrees and minus 1. but if you had to
shift that up by 30 degrees that becomes
300 and minus 1. so that's why this
point over here will be 300 and minus 1.
you see the calculator doesn't give us
that
just due to that 30 degree over there
and so there we have it we've got our
graph what i now want to do is talk
about amplitude range
domain
and period so i'm going to do this for
every single graph that we draw just so
that you really get used to the idea of
what it means so amplitude is the
distance from the resting position so
our resting position is this point over
here so if you had to look at that or
maximum distance there that will just be
one okay so the amplitude is one the
range is the y value so you always go y
as an element and then we're just gonna
go from minus one up to one because you
can see that the lowest value is minus
one and the highest value is one the
domain has nothing to do with the graph
it's got to do with what they gave you
so that is the domain so the domain will
be x as an element from 0 up to 360. the
period well now that depends on the
graph so we've learned that a syn graph
and a cause graph their normal period is
360. in grade 10 you couldn't really
change that but in grade 11 you can
because you can cause the graph to
stretch out or you can cause the graph
to compress but this graph has not been
stretched or compressed it's only been
moved by 30 degrees so the period which
is how long the graph takes to repeat
will still be
360 degrees
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