Download Graphs of Sine and Cosine Functions

Graphs of Sine and Cosine
Functions
Objective: To recognize a sine,
cosine, or tangent graph and to
graph one yourself.
Sin Curve
• Lets us our knowledge of the unit circle to try and
come up with a graph for y = sinx.
Sin Curve
• Lets us our knowledge of the unit circle to try and
come up with a graph for y = sinx.
• The amplitude is the distance from the middle to the
highest or lowest point. You can find the amplitude
by sight or by taking ½ (Max – Min). For a standard
sine curve, the amplitude is 1.
Sin Curve
• Lets us our knowledge of the unit circle to try and
come up with a graph for y = sinx.
• The sine function is periodic, meaning it repeats itself
over and over again over a certain amount of time,
or period. The amount of time it takes to repeat is
the period. To find the period, find the distance from
a peak to a peak. The period of y = sinx is 2p.
Cosine Curve
• Again, lets use our knowledge of the unit circle to
graph the cosine curve.
Cosine Curve
• Again, lets use our knowledge of the unit circle to
graph the cosine curve.
• The period and amplitude are the same as the sine
curve. The period is 2p and the amplitude is 1.
Cosine Curve
• Again, lets use our knowledge of the unit circle to
graph the cosine curve.
• The period and amplitude are the same as the sine
curve. This curve is actually the same exact graph as
the sine curve only shifted over p/2 units.
Negative Sine Curve
• Again, lets use our knowledge of the unit circle to
graph negative curves.
y  sin x
y   sin x
• A negative sine or cosine curve is the exact same,
only a reflection on the y-axis. In other words, upside
down. It will have the same period and amplitude.
Negative Cosine Curve
• Again, lets use our knowledge of the unit circle to
graph negative curves.
• A negative cosine
curve is the exact same,
only a reflection on the y-axis.
In other words, upside down.
It will have the same period
and amplitude.
Sine/Cosine Curves
• Find the period and amplitude for the following
graph.
2p
p
2
p
3p
2
• Sometimes the x-axis scale has integers and
sometimes it has multiples of pi.
Sine/Cosine Curves
• Find the period and amplitude for the following
graph.
2p
p
2
p
3p
2
y  2 sin 2 x
amplitude  2
period  p
• Sometimes the x-axis scale has integers and
sometimes it has multiples of pi.
Sine/Cosine Curves
• Find the period and amplitude for the following
graph.
1
2
3
4
• Sometimes the x-axis scale has integers and
sometimes it has multiples of pi.
Sine/Cosine Curves
• Find the period and amplitude for the following
graph.
y  3 cos px
1
2
3
4
amplitude  3
period  2
• Sometimes the x-axis scale has integers and
sometimes it has multiples of pi.
General Form
• We can have curves with a different period or
amplitude. We need to add something to our
equations. The general form of the equations of the
sine and cosine curves now is:
y = A sin bx
• The amplitude is the |A|.
• The period is 2p/b.
y = A cos bx
General Form
• When the amplitude is increased (|A| > 1) it is called
a vertical stretch. When it is decreased (-1 < A <1) it
is called a vertical shrink.
y = A sin bx
• The amplitude is the |A|.
• The period is 2p/b.
y = A cos bx
Vertical Shrink/Stretch
• Look at the graph to see the vertical shrink/stretch.
Notice that the period is unchanged.
Vertical Shrink/Stretch
• Look at the graph to see the vertical shrink/stretch.
Notice that the period is unchanged.
General Form
• When the period is increased it is called a horizontal
stretch. When it is decreased it is called a horizontal
shrink.
y = A sin bx
• The amplitude is the |A|.
• The period is 2p/b.
y = A cos bx
Horizontal Stretch/Shrink
• Look at this equation and graph. B in this equation is
½, so the period is 2p/.5 or 2p(2) = 4p. The
amplitude is unchanged since A = 1.
Period/Amplitude
• Given the equation y = 3 sin 2x, find the amplitude
and period.
Period/Amplitude
• Given the equation y = 3 sin 2x, find the amplitude
and period.
• The amplitude is 3.
• The period is 2p/2 or p.
Period/Amplitude
• Given the equation y = -2 cospx, find the amplitude
and period.
Period/Amplitude
• Given the equation y = -2 cospx, find the amplitude
and period.
• The amplitude is 2.
• The period is 2p/p = 2.
General Form
• Now, lets look at how to graph these two equations.
y = 2 sin 2x
y = -3 cos px
General Form
• Now, lets look at how to graph these two equations.
y = -3 cos px
y = 2 sin 2x
p
2
p
3p
2
2p
1
2
3
4
General Form
• Now, you try to graph these two equations.
y = -2 sin 4x
y = 3 cos (p/2)x
General Form
• Now, you try to graph these two equations.
y = 3 cos (p/2)x
y = -2 sin 4x
p
4
p
2
3p
4
p
1
2
3
4
Homework
• Pages 490-491
• 1-13 odd, 23, 24, 35-42 all