Trig functions grade 11 and 12: Determine Equation

Summary

This lesson teaches how to determine the equation of a graph by analyzing its transformations. The instructor explains how graphs can be shifted vertically or horizontally, stretched, or compressed. The key to identifying these transformations is to look for the general form of the graph equation and find a known point to determine the shift. The amplitude, range, domain, and period of the graph are also discussed, with the example of a cosine graph being shifted by 30 degrees to the right, resulting in a final equation of the form y = A*cos(B(x - p)) + C.

Takeaways

• 📈 A graph can be manipulated in four ways: vertical shift, vertical stretch/compress, horizontal shift, and horizontal stretch/compress.
• ⬆️ Vertical shifts can move the graph up or down, and horizontal shifts can move it left or right on the x-axis.
• 🔍 To determine the type of shift, look for the general form of the graph equation and identify any transformations.
• 🟢 When identifying horizontal shifts, focus on the point where the graph intersects the x-axis and compare it to the original graph's intersection point.
• ⚖️ The amplitude of a graph represents its maximum distance from the resting position, indicating the height of the wave's peak.
• 📉 The range of a graph is the extent of its y-values, from the lowest to the highest point on the graph.
• 🔢 The domain of a graph refers to the set of all possible x-values, which can be determined by looking at the graph's given range.
• ⏳ The period of a graph is the length of time it takes for the graph to repeat its pattern, which remains constant even with horizontal shifts.
• 🔑 Understanding the normal appearance of basic graphs is crucial for identifying transformations, such as the original coordinates of a cosine graph.
• 📐 The equation of a transformed graph can be determined by identifying the horizontal shift (p), amplitude, range, and period from the graph.

Q & A

What are the four things a graph can do according to the lesson?

A graph can be shifted vertically, stretched or compressed vertically, shifted horizontally, and stretched or compressed horizontally.

How can you determine if a graph has been shifted vertically?

You can determine a vertical shift by looking at the general form of the graph and identifying any changes in the equation that would result in a vertical movement.

What does a plus sign in the equation of a graph indicate in terms of horizontal shift?

A plus sign in the equation does not necessarily mean the graph is shifted to the left. It could be part of a larger expression that results in a negative value, which would shift the graph to the right.

Why is it important to know the coordinates of an original cosine graph?

Knowing the coordinates of an original cosine graph helps in determining how much a graph has been shifted horizontally by comparing it to a known point on the shifted graph.

What is the significance of the point (300, 0) in determining the horizontal shift of the graph?

The point (300, 0) indicates that the graph has been shifted 30 degrees to the right from its original position, where the cosine graph would normally have a point at (270, 0).

What is the amplitude of the graph described in the lesson?

The amplitude of the graph is one, which is the maximum distance from the resting position.

How do you determine the range of the graph from the information provided?

The range of the graph is determined by identifying the lowest and highest y-values on the graph, which in this case are -1 and 1, respectively.

What is the domain of the graph as described in the lesson?

The domain of the graph is from -180 to 360 degrees.

Why does the period of a cosine graph remain the same even when it is shifted horizontally?

The period of a cosine graph, which is 360 degrees for a normal cosine graph, remains unchanged because shifting horizontally does not affect the time it takes for the graph to repeat.

What is the final equation of the graph after determining the horizontal shift?

The final equation of the graph after determining the horizontal shift is y = cos(x - 30), where p is -30.

Outlines

00:00

📈 Understanding Graph Transformations

This lesson focuses on determining the equation of a graph by understanding its transformations. A graph can be shifted vertically or horizontally, stretched, or compressed. The general form of the graph is provided to identify these transformations. For a vertical shift, the sign of 'p' in the equation indicates the direction of the shift, with a positive value shifting the graph to the left and a negative to the right. To find the exact shift, one must compare a known point on the transformed graph with its original position on a standard graph. The amplitude, or the maximum distance from the resting position, is determined by the vertical stretch or compression. The range is the set of all possible y-values, and the domain is the set of all possible x-values. The period of the graph, which is the length of one complete cycle, remains unchanged by horizontal shifts.

Keywords

💡Graph

A graph in the context of the video refers to a visual representation of data or functions, typically used in mathematics to plot points and show relationships. The video discusses how graphs can be transformed through various operations such as shifting and stretching. For instance, the script mentions vertical and horizontal shifts, which are ways to move the graph along the x or y-axis.

💡Vertical Shift

A vertical shift is a transformation that moves a graph up or down along the y-axis. The video explains that this can be indicated by a mathematical expression involving a plus or minus sign, but it cautions that the sign alone is not definitive without knowing the value of 'p', which could be negative, leading to a downward shift.

💡Horizontal Shift

A horizontal shift moves a graph left or right along the x-axis. The video emphasizes the importance of identifying the original position of a graph to determine the amount of shift. It uses the point (300, 0) as an example to illustrate how the graph has been shifted 30 degrees to the right from its original position.

💡Stretch

Stretching a graph refers to scaling its dimensions, either vertically or horizontally, to make it longer or shorter. The video uses arrows on both sides of the graph to illustrate vertical stretching, indicating an increase in the amplitude of the graph's wave.

💡Compress

Compression is the opposite of stretching; it reduces the scale of a graph. The video does not explicitly use this term, but it is implied when discussing transformations that could make the graph's wave smaller, such as reducing the amplitude.

💡Amplitude

Amplitude is the maximum distance from the resting position to the peak or trough of a wave. In the video, the amplitude is determined by the maximum distance from the resting position, which is identified as one unit in the example given.

💡Resting Position

The resting position is the baseline or equilibrium level of a graph, from which deviations (like peaks and troughs) are measured. The video explains that the amplitude is the distance from this resting position to the highest or lowest point of the graph.

💡Range

Range in the context of the video refers to the set of values that the dependent variable (usually the y-values) can take within a function. The video identifies the range as going from -1 to 1, indicating the highest and lowest points the graph reaches.

💡Domain

Domain is the set of all possible input values (usually the x-values) for a function. The video discusses determining the domain by looking at the given x-values on the graph, which in the example provided, spans from -180 to 360.

💡Period

The period of a graph is the length of one complete cycle of a repeating pattern, typically measured in degrees for trigonometric functions. The video explains that moving a cosine graph to the right does not affect its period, which remains at 360 degrees for a standard cosine function.

💡Transformation

Transformation in the video refers to the various operations that can be applied to a graph to change its shape, position, or size. This includes shifts, stretches, and compressions, which are all ways to transform the graph to fit a particular equation or to visualize different data patterns.

Highlights

Graphs can be shifted vertically or horizontally, stretched, or compressed.

Vertical shifts can be upwards or downwards, affecting the graph's position.

Horizontal shifts slide the graph to the left or right.

Stretching or compressing a graph affects its scale in the vertical or horizontal direction.

The general form of a graph's equation is provided to determine its transformations.

A plus sign in the equation does not necessarily mean a leftward shift due to the variable 'p'.

Identifying a known point on the graph is crucial for determining its horizontal shift.

The coordinates of a point where the graph intersects the x-axis can indicate the horizontal shift.

The normal cosine graph's intersection with the x-axis at 270 degrees is a reference point.

If a graph's intersection is at 300 degrees, it indicates a 30-degree rightward shift.

The amplitude of a graph is the maximum distance from its resting position.

The resting position is the horizontal line where the graph oscillates.

The range of a graph is the extent of its y-values from lowest to highest.

The domain of a graph is the set of x-values for which the function is defined.

The period of a graph is the length it takes to repeat, typically 360 degrees for a cosine graph.

Shifting a cosine graph horizontally does not affect its period.

Transcripts

hello everyone in this lesson we are

going to determine the equation of a

graph

now a graph can do four things

it can be

shifted vertically so shift

vertically so for example if you shift

it up then it does that

you can

stretch a graph vertically stretch or

compress okay so that can do something

like this

notice how the arrow is on both sides

meaning it goes up and down

you could shift a graph horizontally and

that means something like that so you

just slide it over or you could stretch

or compress

horizontally and so that's when it would

do something like that so it would go in

both directions now

how do we know which one's happening to

this graph well it's quite easy because

what they do is they'll always give you

the general form of that particular

graph so all that we can see in that

graph is this part over here

now that part is no not there is a

horizontal shift

so we need to see how this graph has

been shifted

now please something that students

always do they say oh there's a plus

over there that means the graph is

shifted left

yes a plus does mean shifted left guys

but

we don't know what p is p might end up

being a negative value so then that

would cause that whole expression to be

negative so don't pay too much attention

to that positive over there what you

need to do is find a point on the graph

where you know the coordinates okay so

this point over here in green that has

the coordinates of 300 and zero

now you need to know what an original

cos graph

what the coordinates of an original

clause graph would be at that point

so that's why it's very important that

you know what

the normal graphs look like so if you

look at a cause graph at that point

where it cuts the x-axis for the second

time the coordinate there is usually 270

and 0. this coordinate is 300 and 0. so

that means that that graph has been

moved over by 30 degrees so if you move

a graph 30 degrees to the right

then that means it would have to be

something like

that okay so p is -30

and so that is the final equation of

that graph then the next question is the

amplitude so remember the amplitude is

the maximum distance from the resting

position and in this video the resting

position in this question sorry the the

resting position is there and if we look

at that maximum distance it is one so

the amplitude is one the range well

that's your y value so we say y is an

element

and then we can see that the lowest

value is over here which is that minus

one and the highest value here is at one

so we can say that the range goes from

minus one up to one

the domain

so with the domain you simply just look

at what they've given you so let me just

erase this line over here so we can see

that it goes from minus 180 up to 360

and so the domain will be

from -360

up sorry minus 180 up to 360.

then the last one is the period now the

period is how long does the graph take

to repeat which is 360 for a normal

cause graph

if you move a cause graph over to the

right you're not gonna affect the period

it's still gonna take 360 degrees

to repeat