Loading the lesson
Loading the lesson
Search AllRounder.ai
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Learn it your way
A short class with Sarah and her students. The other ways are under More.
Sign up free to listen. Transcripts are free to read.
Today, we're going to talk about how transformations affect trigonometric graphs. Let’s start with the general form of a trigonometric function: y = A sin(Bx + C) + D. Can anyone tell me what 'A' represents?
Is 'A' the amplitude? It shows how tall the waves are?
Exactly! Remember, the amplitude is the height of the wave from the center line. Now, what about 'B'?
That affects the period of the graph, right?
Correct! The period can be calculated by the formula Period = 360° / |B|. Great job! Now, if we change 'C', what happens?
That's the phase shift! It moves the graph left or right.
Right again! Lastly, 'D' is important for vertical shifts. It moves the graph up or down. Let’s move on to some examples to see these transformations in action.
Let’s analyze the graph of y = 3 sin(x). What does the '3' tell us about this graph?
The amplitude is 3, so the wave will reach 3 units high and -3 units low!
Exactly! Now, what if I have the function y = sin(2x)? What can you tell me about its period?
The period is 180°, since it’s 360° divided by 2!
Well done! Now, let's consider y = sin(x - 90°). What does the '-90°' do?
That shifts the graph to the right by 90°.
Great job! These transformations help us understand how to manipulate and draw these graphs accurately.
Now, let’s apply what we learned to sketch the graph of y = -2 cos(3x) together. What can we determine about the amplitude and period?
The amplitude is 2, but because of the negative sign, the graph is inverted.
Exactly! What about the period?
The period would be 120°, since it’s 360° over 3!
You’ve got it! When sketching, remember to account for the vertical inversion and make sure to label the key points appropriately.
Amplitude tall, with height to call; Period quick, makes cycles tick.
Once in a math land, there lived a wave named Sine. It wanted to explore. By adding amplitude, it grew tall. Adjusted period, it learned to roll. Together with C and D, its transformations were a delightful tale.
A - Amplitude, P - Period, PS - Phase Shift, V - Vertical Shift: Remember 'A-P-P-V' for the key parameters of transformations!
APPS: Remember Amplitude, Period, Phase Shift, and Vertical Shift for analyzing transformations.
Term
Amplitude
Click to flip
Term
Period
Click to flip
Term
Phase Shift
Click to flip
Term
Vertical Shift
Click to flip
The maximum height of a wave from its center line in a trigonometric graph.
The length of one complete cycle of a wave; calculated as 360° / |B| in the function.
A horizontal shift of the graph due to the value of C in the function.
The movement of the entire graph up or down determined by the value of D.
This section covers how to analyze transformations of trigonometric graphs, focusing on changes to amplitude, period, phase shift, and vertical shift.
In this section, we explore the transformations of trigonometric graphs, specifically how the general form of trigonometric functions can be altered through modifications of amplitude, period, phase shift, and vertical shift. These transformations help in understanding how to graph and interpret variations in sinusoidal functions effectively.
In this section, we delve into the transformations of key trigonometric functions expressed in the general form:
y = A sin(Bx + C) + D
where:
For example:
We are given the graph of a sine function with:
Step 1: Find the amplitude (A).
Step 2: Find the midline.
Step 3: Identify D.
Since the midline represents the vertical shift,
✅ Therefore, the equation of the function is:
Understanding these transformations is essential for both theoretical studies and practical applications, allowing us to model various phenomena, make predictions, and analyze real-world data.