Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create graphs from equations.
Instead of focusing on the general form equations
we will let
and
and work with a simplified form of the equations in the following examples.
Given the function
sketch its graph.
Identify the amplitude,
Identify the period,
Start at the origin, with the function increasing to the right if
is positive or decreasing if
is negative.
At
there is a local maximum for
or a minimum for
with
The curve returns to the
x -axis at
There is a local minimum for
(maximum for
) at
with
The curve returns again to the
x -axis at
Graphing a function and identifying the amplitude and period
Sketch a graph of
Let’s begin by comparing the equation to the form
Step 1. We can see from the equation that
so the amplitude is 2.
Step 2. The equation shows that
so the period is
Step 3. Because
is negative, the graph descends as we move to the right of the origin.
Step 4–7. The
x -intercepts are at the beginning of one period,
the horizontal midpoints are at
and at the end of one period at
The quarter points include the minimum at
and the maximum at
A local minimum will occur 2 units below the midline, at
and a local maximum will occur at 2 units above the midline, at
[link] shows the graph of the function.
Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph.
Express the function in the general form
Identify the amplitude,
Identify the period,
Identify the phase shift,
Draw the graph of
shifted to the right or left by
and up or down by
Graphing a transformed sinusoid
Sketch a graph of
Step 1. The function is already written in general form:
This graph will have the shape of a
sine function , starting at the midline and increasing to the right.
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy....
I have so many questions in mind ....
all of scientists of the word they just give opinion only.
but they never think true or false ...
i respect all of them...
I believes whole universe depending
on true ...থিউরি
Govinda
hello
Jackson
hi
Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*).
So it would just barely creep over the horizon.
Christopher
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