Trig functions grade 11 and 12: Determine Equation
Summary
TLDRThis 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
📈 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.
Mindmap
Keywords
💡Graph
💡Vertical Shift
💡Horizontal Shift
💡Stretch
💡Compress
💡Amplitude
💡Resting Position
💡Range
💡Domain
💡Period
💡Transformation
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
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