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POD #1
Complete these limit statements. For limit statements, go to the graphs, not the unit circle.
For which of these does
direction not matter?
In other words, which ones
have the same limit at the
given value of x, from either
side? ________sinlim
________cotlim
________tanlim
_______sinlim
1
1
4
0
x
x
x
x
x
x
x
POD #1
Complete these limit statements. For limit statements, go to the graphs, not the unit circle.
For which of these does
direction not matter?
Sine and tangent
2sinlim
cotlim
1tanlim
0sinlim
1
1
4
0
x
x
x
x
x
x
x
POD #2– if we have time
By tables, on calculators, graph one of these functions. You can do it on the calculator. Compare it to its parent function. Then one person from each table go to the board to sketch the graph and its parent.
Then we’ll compare them to their parents.
xxg
xxf
cos2)(
sin3)(
xxq
xxp
2
1cos)(
2sin)(
1cos)(
3sin)(
xxt
xxs
What we’ve done…
… is transform the graphs of two basic trig functions, sine and cosine.
Ultimately, we’re going to graph
and each change in the equation from the parent corresponds to a change in the graph from the parent. As you might expect, the change in the graph is opposite the change in the equation.
dcbxay
dcbxay
)cos(
)sin(
Amplitude
Amplitude refers to what aspect of the graph? How do we determine amplitude from a graph? Which graphs besides these have amplitude?
From the equation, amplitude is determined by |a|.
Amplitude is a scale change of which variable? How is this “opposite” from the equation?
What happens when a is negative?
dcbxay
dcbxay
)cos(
)sin(
Period
Period refers to what aspect of the graph? How do we determine period from a graph?
From the equation, period is determined by 2π/|b|.
Period is a scale change of which variable? How is this “opposite” from the equation?
What happens when b is negative?
dcbxay
dcbxay
)cos(
)sin(
Vertical shift
Vertical shift refers to what aspect of the graph? How do we determine vertical shift from a graph?
From the equation, vertical shift is determined by d.
Vertical shift is a translation of which variable? How is this “opposite” from the equation?
What happens when d is negative?
dcbxay
dcbxay
)cos(
)sin(
Phase shift
Horizontal (phase) shift refers to what aspect of the graph?
From the equation, phase shift is determined by –c/b.
THIS PHASE SHIFT IS SEEN IN MOVEMENT FROM THE Y-AXIS.
Phase shift is a translation of which variable?
What happens when c/b changes sign?
dcbxay
dcbxay
)cos(
)sin(
Cycles
These are VERY helpful when graphing trig functions.
You can determine the interval for a full cycle of a trig function by solving for x:
0 ≤ bx+c ≤ 2π
This will give us an interval starting when sin x = 0 and cos x = 1, the same values we have on the y-axis for these parent functions.
dcbxay
dcbxay
)cos(
)sin(
Use it
Find the amplitude, period, and vertical and phase shifts for .
What is an interval for a complete cycle?
1)2/2sin(2)( xxr
Use it
Find the amplitude, period, and vertical and phase shifts for
What is the interval for a complete cycle?
Amplitude: 2 *How does the
Period: π interval for the
Vertical shift: 1 cycle compare
Phase shift: -π/4 with the values for
Cycle: [-π/4,3π4]* period and phase
shift? (Foot stomp)
1)2/2sin(2)( xxr
Use it
Using this information, graph the function.
In this graph, the x interval is π/4, and the y interval is 1.
I also graphed y = 1 to show the vertical shift.
Note how the cycle starts at
–π/4 (where sine is 0) and runs a length of π. 1)2/2sin(2)( xxr
Use it
Find the equation from the
graph. (We simply
work backwards.)
In this graph both the x
and y intervals are 1.
Which parent function would you use?
How could you use the other one?
Use it
Find the equation from the
graph. (We simply
work backwards.)
In this graph both the x
and y intervals are 1.
What are the amplitude and period?
What are the shifts?
dcbxay
dcbxay
)cos(
)sin(
Use it
Find the equation from the
graph. (We simply
work backwards.)
In this graph both the x
and y intervals are 1.
What are the amplitude and period? Amplitude is
10/2 and period is 4. Vertical shift is 3.
dcbxay
dcbxay
)cos(
)sin(
Use it
So, a = 5, b = π/2, and d = 3.
If we used cosine, we’d have no phase shift,
c = 0. 32
cos5
xy
Use it
If we used sine, we would
have a phase shift. We
can determine the phase
shift by seeing where the graph crosses the “center,” y = 3.