Discussion Forum - Unit 1 - Calculus
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1211
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Mathematics
Date
Nov 24, 2024
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docx
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Uploaded by clerk7133
1.)
Consider the graph of the function y = sin x + cos x.
Describe its overall shape
: The shape emulates a sine wave, oscillating periodically.
Is it periodic
: Yes.
How do you know
: A periodic function repeats the value on steady, regular periods. This
function does so and repeats the period of 2π.
2.)
Using a graphing calculator or other graphing device, estimate the x- and y-values
of the maximum point for the graph (the first such point where x > 0). It may be
helpful to express the x-value as a multiple of π.
The maximum point of the graph is:
(π/4, 1.414).
3.)
Now consider other graphs of the form y = A sin x + B cos x for various values of A
and B.
Sketch the graph when A = 2 and B = 1, and, find the x - and y-values for the maximum
point. (Remember to express the x-value as a multiple of π, if possible.)
:
Has it moved
: Yes, the maximum point has reached (1.107, 2.236). Amplitude has
increased as well, and the graph has moved further to the right.
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4.)
Repeat and sketch the graph for A = 1, B = 2.
Is there any relationship to what you found in part (2)
: The maximum point has changed
like the previous function, in this instance reaching (0.464, 2.236). The amplitude has
increased once again and moves further to the left. The relationship visualized between
the two functions being the functions are the same function, sharing the shape but shift
opposite of each other.
5.)
Explain what you have discovered from completing this activity using details and
examples.
This activity touches on periodic trigonometric functions. It also visualizes how sine and
cosine values can affect the amplitude of graphs.
Similarly, to the previous examples, this graph showcases the relationship between the
two functions: y = sin x + cos x, and y = sin x – cos x. They are the same function, but
they differ in the cosine term. That difference has affected the amplitude and caused the
graph of the second function to shift to the right.