Graphing Trig Functions
Jump to: Using
the Special Angles
Transforming
Graphs
5-Point
Method
- We retain our theme that Algebra is Transformations:
Turning difficult-looking problems into ones we have an idea
of
how to solve. In the case of graphing the trig functions, we
can
use our knowledge of:
- Tables of Values, as when graphing any equation, and
- Trig function values of the Special
Angles.
- With this starting point, we hope to discover patterns and
similarities within and between the functions, leading to a better
understanding and perhaps even some graphing short cuts.
Graphing
using the Special Angles
-
Our text book describes and
demonstrates building an appropriate table of values using the special
angles, so we'll not reiterate that here. Rather, we can
explore interactively where those values come from in the unit circle
and how they help form the graphs of the trig functions - using what we
already know.
- GraphSine
allows us to trace a point around the Unit Circle and note how
the y-coordinate
(vertical
distance of the point from 0) or "opposite side" of its reference
triangle links to the graph of the Sine function.
- GraphCosine
likewise matches the x-coordinate
(horizontal distance from 0) or "adjacent side" of its reference
triangle to the graph of the Cosine function.
- In both of these function graphs, notice that the
horizontal axis shows
ANGLES and the vertical axis shows the value of the trig
function at those angles (the ratio: a pure, dimensionless
NUMBER). It is not the same as for
the unit circle graphs.
Transformations
(Shifting the Graphs)
- As with the polynomial functions studied in Algebra, we
can
translate (shift) and stretch/squeeze the graphs of the trig functions.
The general forms of the equations we will use are
- A represents Amplitude
and will affect the "height and depth" of the curves,
- B is related to frequency and will affect the Period of
the curves,
- C/B is the Phase
Shift
(horizontal shift),
- D is the Vertical
Shift.
- With each of the links below, move the sliders
to
adjust the values of A, B, C, and D to explore how they affect the
position and shape of the 6 trig function graphs.
- As you experiment, look for patterns and consistencies.
What always seems to happen at the "starting point" of the
function? What always seems to happen at full-period
intervals? At other intervals? How high or low do the graphs
go? And so on.
Graphing
(5-Points)
- After experimenting with the above links, we should have
discovered some useful patterns in the graphs of the trig functions.
- Relative to the shifted origin:
- Each function "starts" at a particular value and heads
either up or down to the right,
- That initial position/direction repeats every Period,
- Half- and quarter-period points are also consistent and
readily identifiable.
- To take advantage of those patterns:
- Calculate phase and vertical shifts to plot the
shifted origin,
- Calculate the Period and the quarter-period interval
between the key
points,
- Plot the key points and sketch the graph.
- The links below allow us to set the parameters for a trig
equation and then walk step-by-step through the 5-point graphing
process. For best benefit, try to predict the results of each step
before moving on.
Remember, practice makes permanent. Practice and learn now,
so the quizzes and tests are easy.
