Trig functions grade 11 and 12: Horizontal stretch

Summary

In this educational video, the focus is on transforming graphs by stretching or compressing them, specifically using the sine function. The instructor explains how the coefficient in front of 'x' in a sine function affects the period of the graph. For instance, 'sin(2x)' halves the period from 360 degrees to 180 degrees, as each x-value is doubled. The tutorial demonstrates how to adjust the calculator's settings, including the step size, to accurately graph 'sin(2x)'. The instructor also covers key concepts like amplitude, range, domain, and period, providing a clear visual comparison between a standard sine graph and the transformed one.

Takeaways

• 📈 The lesson focuses on transforming graphs by stretching or compressing them, which can also be referred to as compressing.
• 🔄 In previous lessons, the movement of graphs was discussed, such as shifting a graph upwards, moving all points in the same direction.
• 🔍 The new topic is the effect of mathematical operations like '2x' on the graph of a sine function, which stretches or compresses the graph horizontally.
• 📉 When '2x' is applied to a sine function, it compresses the graph horizontally, causing it to complete its cycle in half the original period.
• 📋 The normal period for a sine or cosine graph is 360 degrees, but with '2x', the period becomes 180 degrees, as explained by the rule 360 divided by the coefficient of 'x'.
• 🔢 The step size on the calculator for graphing 'sin(2x)' should be set to 45 degrees, which is 180 divided by 4, to properly display the graph's shape.
• 📊 The amplitude of the 'sin(2x)' graph remains the same as the original sine graph, which is the maximum distance from the resting position (equilibrium line).
• 📐 The range of the 'sin(2x)' graph is between -1 and 1, as these are the minimum and maximum y-values of the sine function.
• 📌 The domain of the 'sin(2x)' graph is from -180 to 360, as specified in the lesson, which indicates the x-values over which the graph is drawn.
• 🔁 The 'sin(2x)' graph completes two cycles within the 360-degree range, which is a key observation for understanding the effect of '2x' on the graph.

Q & A

What is the main focus of the lesson described in the transcript?

The main focus of the lesson is to understand how graphs, specifically sine graphs, are affected when they are stretched or compressed.

What does the term 'stretching' a graph refer to in this context?

In this context, 'stretching' a graph refers to the process of enlarging the graph by altering its equation, which can result in the graph covering more space in fewer cycles.

How does the point (0, 0) on a sine graph move when the graph is stretched?

When a sine graph is stretched, the point (0, 0) would move vertically up or down depending on the nature of the stretch, but the starting point on the x-axis remains the same.

What is the effect of the term '2x' in the equation of a sine graph on the period of the graph?

The term '2x' in the equation of a sine graph halves the period of the graph. If the normal period for a sine graph is 360 degrees, with '2x', the new period becomes 180 degrees.

Why is the step size important when graphing a stretched or compressed sine function on a calculator?

The step size is important because it determines the resolution of the graph. If the step size is too large, the graph may not accurately represent the function's shape, especially after transformations like stretching or compressing.

What is the new step size for a sine graph with the equation 'sin(2x)' as compared to a standard sine graph?

For a standard sine graph, the step size is typically 90 degrees. However, for a graph with the equation 'sin(2x)', the new step size should be 45 degrees to accurately represent the halved period.

How does the amplitude of a sine graph change when the graph is stretched or compressed?

The amplitude of a sine graph does not change when the graph is stretched or compressed horizontally. The amplitude remains the maximum distance from the resting position (equilibrium line) to the peak or trough of the graph.

What is the resting position referred to in the context of a sine graph?

The resting position in the context of a sine graph refers to the equilibrium line, which is the horizontal line where the graph would be at rest, typically the x-axis.

How does the domain of a sine graph change when it is stretched or compressed?

The domain of a sine graph, which represents the set of x-values, does not change when the graph is stretched or compressed. It remains the same as specified in the problem or the original function.

What is the range of a sine graph with the equation 'sin(2x)' if the original range is between -1 and 1?

The range of a sine graph with the equation 'sin(2x)' remains between -1 and 1, as the stretching or compressing affects the period and not the amplitude.

How can you remember the rule for calculating the new period of a sine graph when the equation includes a coefficient in front of 'x'?

You can remember the rule by knowing that the period of a sine or cosine graph is always 360 degrees divided by the number in front of 'x'. So for 'sin(2x)', the new period is 360 degrees divided by 2, which equals 180 degrees.

Outlines

00:00

📈 Introduction to Stretching and Compressing Graphs

This paragraph introduces the concept of stretching and compressing graphs, which is a shift from the previous lessons that focused on graph translations. The instructor explains that stretching or compressing can alter the appearance of a graph, such as a sine graph, by enlarging or reducing its scale. The example given is the transformation of a graph by the function 'y = sin(x)' into 'y = sin(2x)', which results in a graph that completes its cycle in half the period of the original, thus changing the graph's frequency. The instructor emphasizes the importance of understanding how the coefficient in front of 'x' affects the period of the graph, which is crucial for setting the correct step size on a calculator when graphing. The new period is calculated as 360 (the standard period for sine and cosine graphs) divided by the coefficient of 'x'. The instructor also demonstrates how to input the equation into a calculator, set the correct table mode, and determine the appropriate step size to accurately graph the function 'y = sin(2x)'.

05:02

🔍 Analyzing the Effects of '2x' on a Sine Graph

In this paragraph, the focus is on the specific effects of the '2x' term in the sine function. The instructor clarifies that doubling 'x' does not double the graph's period but rather halves it. This is a common misconception that the instructor addresses by explaining that the 'x' term complicates the graph's behavior. The new period for 'y = sin(2x)' is 180 degrees, which is half of the standard 360-degree period for a sine graph. The instructor provides a visual demonstration of the graph's transformation, showing how the graph completes two cycles within the usual 360-degree span. The amplitude, range, and domain of the graph are also discussed, with the amplitude remaining at one and the range spanning from -1 to 1. The domain is specified as the interval from -180 to 360 degrees. The instructor concludes by reiterating the importance of understanding the impact of the 'x' term on the graph's period and shape.

Keywords

💡Stretch

In the context of the video, 'stretch' refers to the transformation of a graph by increasing its scale or period. This is a key concept when discussing how the value in front of 'x' in a trigonometric function affects the graph's period. For instance, when the instructor mentions 'stretching' a graph, they are referring to the process of altering the graph's period from the standard 360 degrees to a different value, such as 180 degrees when the function is 'sin(2x)'.

💡Compress

Similar to 'stretch,' 'compress' is used to describe the inverse transformation where the graph is made to have a smaller period than the standard 360 degrees. This is also related to the coefficient in front of 'x' in a trigonometric function. The video explains that 'compressing' a graph is a result of increasing the coefficient, which in turn decreases the period over which the graph completes a cycle.

💡Graph Transformation

Graph transformation is a central theme of the video, where the instructor discusses how mathematical functions are visually altered. This includes stretching or compressing, which changes the graph's period, and shifting, which changes the graph's position. The video specifically focuses on the effect of the '2x' term in the function 'sin(2x)', which is a type of transformation.

💡Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in the video's discussion. The instructor uses these functions to demonstrate how graph transformations work. For example, the standard sine function 'sin(x)' has a period of 360 degrees, but when modified to 'sin(2x)', the period is halved to 180 degrees.

💡Period

The 'period' of a trigonometric function is the interval over which the function completes one full cycle. In the video, the instructor explains how the period changes with the transformation of the graph. A key point is that the period of a sine or cosine graph is affected by the coefficient of 'x', as seen with 'sin(2x)' where the period becomes 180 degrees.

💡Amplitude

Amplitude refers to the maximum distance from the resting position (equilibrium line) to the peak or trough of a wave. In the video, the instructor discusses how the amplitude of a sine graph remains unchanged when the graph is stretched or compressed horizontally, staying at a value of one.

💡Domain

The 'domain' in the context of the video refers to the set of all possible input values (x-values) for a function. The instructor specifies the domain as the range of x-values over which the graph is drawn, which in the example from the script is from -180 to 360 degrees.

💡Calculator

The use of a calculator is emphasized in the video as a tool for graphing and understanding transformations. The instructor demonstrates how to input equations into the calculator, set the table mode, and adjust the step size to accurately represent the graph's transformations.

💡Step

In the context of graphing on a calculator, 'step' refers to the increment of the x-values used to calculate corresponding y-values. The video explains the importance of choosing the correct step size to accurately represent the graph's shape, especially when the graph's period is altered through transformations.

💡Resting Position

The 'resting position' is the equilibrium line or the horizontal line that serves as the baseline for the amplitude measurement. In the video, the instructor uses the resting position to determine the amplitude of the sine graph, which is the maximum distance from this line to the peak or trough of the wave.

Highlights

Introduction to stretching and compressing graphs, which can also be referred to as compressing.

Explanation of how moving a graph upwards affects all points on the graph.

Transition to studying the effects of stretching or compressing a graph.

Illustration of how stretching a graph results in a larger cycle in the same space.

Example of compressing a graph to show a smaller cycle within the same space.

Emphasis on using a calculator to explore these graph transformations.

Instruction to set the calculator to table mode for graphing.

Importance of using the given start and end points for graphing.

Explanation of the significance of the '2x' term in altering the graph's period.

Rule for calculating the new period when a coefficient is in front of 'x' in a trigonometric function.

Clarification that '2x' halves the period of the graph, contrary to initial expectations.

Demonstration of how to adjust the step size on the calculator for the new period.

Tabulation of values using the adjusted step size for the transformed graph.

Visualization of the transformed graph by plotting the calculated points.

Comparison of a normal sine graph with the transformed graph to show the difference in cycle completion.

Discussion on the amplitude of the graph, which is the maximum distance from the resting position.

Definition of the range of the graph as the set of y-values between the minimum and maximum.

Explanation of the domain of the graph, which are the given x-values.

Final note on the period of the sine function when '2x' is present, emphasizing the halving effect.

Conclusion and appreciation for watching the lesson.

Transcripts

hello everyone welcome to this lesson so

in this lesson we're moving on to

something slightly different now we're

going to be stretching the graphs okay

but when i say stretch it could also

mean compress

so in the previous videos we've been

looking at questions where you have

graphs for example a syn graph that then

gets moved upwards so for example this

point moves over here

and this point moves up and this point

moves up

and so all of the points were moved up

and so the graph did that

but what we're going to start looking at

now is what happens when we compress or

stretch a graph so for example if we

take this original graph that we see

here and we

we enlarge it by stretching it like that

for example

or we could compress it so it would look

something like this

see so in the space of

the white graph doing one cycle the blue

one has done two so that's what we're

going to look at in this

when the this lesson and the next couple

of lessons so as always you're just

going to do this on the calculator and

so let's get started so we're going to

bring out the calculator and then just

always remember to put it into table

mode you then type in the equation

remember your start must always be the

start that they've given you which is

minus 180 the ending point is what

they've given us in this question which

is 360. now the step is very important

so we know that the step of a normal syn

graph or sorry the period of a normal

syn graph under usual conditions is 360.

but now

this 2x over here we need to know what

that does to the graph so x's are

they complicated okay you can even think

about it in if you wanted to uh just

came to mind now uh having

ex-girlfriends having ex-boyfriends

that's complicated if you have a graph

that is

for example the sun of x minus 30 that

actually causes the graph to go 30

degrees to the right not to the left as

we would expect if we have a graph that

is now the sun of 2x well logically the

every person would expect that that

doubles the graph but that's not true

because remember x's are complicated and

so

2x is actually going to cause the graph

to

half and so instead of having so if your

original graph looks like that the 2x

graph is now going to be completely

half of that okay so every point is

going to halve so if your period of a

normal centigraph is 360 well this new

graph is going to be completing in 180

degrees so here's a little rule that you

can remember you can remember that the

period

is always going to be equal to 360 for a

sin and cos graph divided by whatever

number is in front of the x and so in

this case it's a 2 and so our new period

is going to be 180. why do i need to

know that because when you're choosing

the step on your calculator well we know

that step is always 180 over 4. that'll

always be the scenario

and so the step on this

graph is going to be 45 degrees if you

kept it at 90 you're going to struggle

on the calculator because you're going

to have two little values and your

graphs just not going to have the shape

that it's supposed to have so let's go

ahead we'll make our step equal to 45

and there we have our values okay so i'm

going to tabulate those values quickly

and then we'll draw the graph and so

there we have our table with all our

values and now we can draw our graph i'm

going to now plot all the points

and so there we have it

so

quite an interesting graph so what i

want to show you quickly is a normal sun

graph i'm just going to draw it between

0 and 360.

and so there we can see that

that pink graph has completed one cycle

between 360 degrees whereas the green

graph has completed so up to up to

this point over here is one cycle and up

to there's another cycle so it completes

two cycles in 360 degrees and so always

remember that x's are complicated and so

when you say 2x it actually has the

effect of halving okay so every single

coordinate halved okay so if question b

it says what does the 2x do to the graph

it halves the graph what compresses it

in the horizontal direction

then it says determine the amplitude so

amplitude is always the maximum distance

from your resting position

and so the resting position is this

equilibrium line over here the the

x-axis and so that maximum distance well

that's one so your amplitude is one the

range well that's your y values and so

we see the lowest value is minus one and

the highest value is one and so we can

say that y is an element going between

minus one and one

the domain well domain is the x values

and that's what they gave us so that's

from minus 180 up to 360.

as with the tan graph or with the time

graph we would have had to

have excluded the asymptotes but in this

one there are no asymptotes

and then the period now here's where

things are a little bit interesting we

know that a normal period is 360 for a

sin and cos but when you have a sin 2x

well you've halved everything and so

this graph would now compete not compete

complete every

180 degrees and we can see that it

completes one cycle between here so this

point here and this point here that is

one cycle and look how long it took

180 degrees thank you for watching