Trig functions grade 11 and 12: Horizontal shift

Summary

This 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

00:00

📈 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.

05:02

📚 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.

Keywords

💡Trigonometric graphs

Trigonometric graphs refer to the graphical representations of functions that relate angles to their sine, cosine, or tangent values. In the video, the focus is on how these graphs can be manipulated, specifically in terms of horizontal shifting. The script discusses how understanding these graphs is crucial for predicting the behavior of functions and their transformations.

💡Vertical shifting

Vertical shifting is a transformation that moves a graph up or down along the y-axis. The script mentions that this has been a topic in previous lessons, indicating that it's a foundational concept for understanding more complex transformations of trigonometric graphs.

💡Vertical stretching

Vertical stretching is another transformation that affects the graph by either compressing or expanding it vertically. It's mentioned that this will be explored in upcoming videos, suggesting that it's a separate concept from horizontal shifting, which is the main focus of the current lesson.

💡Horizontal shifting

Horizontal shifting is the process of moving a graph left or right along the x-axis. The video script uses the analogy of moving a point to demonstrate this concept. It's the primary focus of the lesson, with the script detailing how to calculate and apply this shift to trigonometric functions.

💡Sine function (sin x)

The sine function is a fundamental trigonometric function that relates an angle to the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle. In the script, the sine function is used as the basis for demonstrating how horizontal shifting affects the graph of trigonometric functions.

💡Amplitude

Amplitude in trigonometry refers to the maximum distance from the resting position (the horizontal axis) to the peak of a wave. The script explains that for the sine function, the amplitude is one, indicating the maximum value the function reaches.

💡Range

Range in the context of functions refers to the set of possible output values. The script specifies the range of the sine function as from -1 to 1, which represents the minimum and maximum y-values the function can take.

💡Domain

Domain is the set of all possible input values (x-values) for a function. The script clarifies that the domain for the example given is from 0 to 360, which is the interval over which the sine function is being analyzed and graphed.

💡Period

The period of a trigonometric function is the length of the repeating interval over which the function repeats its values. The script explains that for the sine function, the normal period is 360 degrees, which is unchanged by horizontal shifting.

💡Calculator

The calculator is used as a tool in the video to help students graph trigonometric functions by generating a table of x and y values. The script provides instructions on how to use a calculator to find these values, emphasizing its utility for visualizing and understanding graph transformations.

💡Step size

Step size refers to the interval between successive x-values in a table or graph. The script specifies that the step size should be equal to the period of the graph divided by 4, which in this case is 90, ensuring that the graph accurately represents the function's behavior over the given interval.

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