The Trig Graphs

Build-A-Graph Day (5.5)

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.

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

Whaddya think?

Even, odd, or neither?

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.

Use it

The interval “starts” at x = -1,

so we have a phase shift

of -1. This means –c/b = -1.

Using our value for b,

we get c = π/2.

Test:

0 ≤ π/2 x + π/2 ≤ 2π

- π/2 ≤ π/2 x ≤ 3π/2

-1 ≤ x ≤ 3 It checks.

322

sin5

xy