34
MATH 12 PRINCIPLES E1 NOTES
GRAPHING y = cosx and y = sin x
Complete the table of values, then plot and join the points on the graph below to create the graphs of and
x(Degrees) / x(Radians) / /0
30
45
60
90
120
135
150
180
210
225
240
270
300
315
330
360
/
Features of the graphs of and
c) Justify the answers for a and b using the circle definitions for and .
d) On average, the value of and are both ______. In other words, the average value of y on the graphs of ______and ______is ______.
e) Both the sine and cosine graphs vary between their average heights and their maximum and minimum heights. The difference between the average and maximum height is called the Amplitude of the graph.
f) The graphs of and both repeat every ______radians
(or ______degrees). Why does this repetition occur?
Because of this repeating pattern, we say that the graphs of both and are periodic graphs with period = ______radians (or ______degrees)
Amplitudey-intercept
x-intercepts (zeros)
Period
Domain
Range
Compare the graphs of and .
How are they the same? How are they different?
How could you transform the graph of to create the graph of ?
We describe both sine and cosine as sinusoidal functions.
MATH 12 PRINCIPLES E2 NOTES
VERTICAL TRANSFORMATIONS OF SINUSOIDAL GRAPHS
PART I: VARYING THE AMPLITUDE
1) Fill in the table and use the values to graph
(radians) / / /0
Period
amplitude
2) Fill in the table and use the values to graph
(radians) / / /0
Period
amplitude
PART II: VERTICAL TRANSLATIONS
(FINDING THE MEDIAN VALUE)
1) Fill in the table and use the values to graph
(radians) / / /0
Period
Amplitude
Vertical translation
2) Fill in the table and use the values to graph
0
Period
amplitude
Vertical translation
SUMMARY OF VERTICAL TRANSFORMATIONS TO SINUSOIDAL GRAPHS:
or
ymax ______
Vertical translation ______yavg ______
Amplitude ______ymin ______
Period ______
Practice: List the vertical translation and amplitude for each graph, calculate ymax and ymin values, then sketch each graph from
1) amplitude _____ ymax ___
vertical translation _____ yavg ___
ymin ___
2) amplitude _____ ymax ___
vertical translation _____ yavg ___
ymin ___
3) amplitude _____ ymax ___
vertical translation _____ yavg ___
ymin ___
4) amplitude _____ ymax ___
vertical translation _____ yavg ___
ymin ___
5) Find an equation to describe each following graph:
Homework: AW p. 247#19eol, 14, 15, 18
MATH 12 PRINCIPLES E3 NOTES
HORIZONTAL TRANSFORMATIONS OF SINUSOIDAL GRAPHS
I. HORIZONTAL TRANSLATIONS (PHASE SHIFTS)
1) a) GraphNOTE: For the horizontal scale:
units = ____spaces (half a period)
(quarter period) / 2) a) Graph
b) Graph
______of ____ units _____
/ b) Graph
______of ____ units _____
units = ____spaces
units = ____spaces
c) Graph
______of ____ units _____
units = ____spaces
units = ____spaces
/ c) Graph
______of ____ units _____
units = ____spaces
units = ____spaces
d)
Horizontal Translation ______(Phase Shift)
Yavg point at ( ) / d)
Horizontal Translation ______(Phase Shift)
graph passes through the ______point ( )
Combining Horizontal Translations with Vertical Transformations
Amplitude ___
Horizontal Translation___ /
2) Graph
Amplitude ___
Horizontal Translation___ /
3) Graph -1
Amplitude ___
Vertical Translation ___
Ymin___ Ymax ___
Horizontal Translation___ /
4) Graph
Amplitude ___
Vertical Translation ___
Ymin___ Ymax ___
Horizontal Translation___ /
SUMMARY:
For the graphs of or
The Amplitude = ____ Ymax = ___
Vertical translation = ____ Yavg = ___
Period = ____ Ymin = ___
Horizontal Translation = ____
For a cosine graph, the horizontal translation indicates the horizontal coordinate of a ______point. For a sine graph, the horizontal translation indicates the horizontal coordinate of an ______point where the slope is ______.
Find two sine equations, one with a positive horizontal translation and one with a negative horizontal translation, to describe the graph above.
equations
Vertical Translation ____
Amplitude ____
Positive Horizontal Translation ______
Negative Horizontal Translation______
Find two cosine equations, one with a positive horizontal translation and one with a negative horizontal translation, to describe the graph.
Positive Horizontal Translation ______
Negative Horizontal Translation______
PART II: HORIZONTAL COMPRESSIONS AND EXPANSIONS OF SINUSOIDAL GRAPHS (VARYING THE PERIOD)
1) Review:
To transform to , what transformation occurs?
2) The graph of is shown below.
a) On the same grid, graph by applying the transformation described in 1.
b) What is the period of ?
c) There are two ways to transform a graph. One way is to leave the axes scales the same and change the shape of the curve. A second way is to keep the shape the same and change the ______. On the graph below, label the x-axis scale so the curve describes .
3) Graph , using 12 spaces for one period.
Period=_____
4) Graph , using 12 spaces to represent one full period.
5) Graph , using 12 spaces to represent one full period.
PART III: COMBINING TRANSFORMATIONS OF SINE AND COSINE GRAPHS
The graphs of or
will have the following features:
amplitude =____
vertical translation = ___
period =____
horizontal translation =____
1) Graph
Steps: 1) List the 4 features 2) From the vertical translation and the amplitude, set the vertical scale:
amplitude =____ Ymin =______Ymax =____
vertical translation = ___ 3) Use the period (=____spaces) to set the horizontal scale:
period =____ spaces
4) Use the horizontal translation to find a starting point for the graph.
horizontal translation =____ start point ( )
5) Graph the function, using 12 spaces for one complete period.
2) Graph
amplitude =____ Ymax =____ Ymin =____
vertical translation = ___
period =____
horizontal translation =____
3) Graph
amplitude =____ Ymax =____ Ymin =____
vertical translation = ___
period =____
horizontal translation =____
4) Graph
Recall: the horizontal translation is the value added or subtracted from x.amplitude =____ Ymax =____ Ymin =____
vertical translation = ___
period =____
horizontal translation =____
5) Find two sine and two cosine equation to describe the following graph:
amplitude =____vertical translation = ___
period =____ / Positive horizontal translation / Negative horizontal translation
Sine
cosine
Homework: AW p. 254#1,7-9,11,12,16ac,17cd,19all
MATH 12 PRINCIPLES E4 NOTES
SINUSOIDAL GRAPHS WITH RATIONAL PERIODS
1) Find a sine equation to describe each graph:
Period:____ Period:______
Equation:______Equation:______
2) Graphperiod = =
/ 3) Graph
Amplitude = _____ Ymax =______Ymin = _____
Vertical Translation = ___
Period = ____
Horizontal Translation = ___
SUMMARY:
For a sine or cosine graph with amplitude a, vertical translation d, period P, and horizontal translation c , the equations are:
______
Eg. Graph:
a)/ b)
Find a cosine equation with a positive horizontal translation to describe the following graph:
Homework: AW p. 265#1-3,5, 7, 15, 16
MATH 12 PRINCIPLES E5 NOTES
HORIZONTAL AND VERTICAL REFLECTIONS OF SINUSOIDAL GRAPHS
1) Graph , , and on the same grid.
/ If , then______, and
Which two graphs are the same?
Functions for which are called even functions because even powers of x have this property
2) Graph , , and on the same grid.
/ If , then______, and
Which two graphs are the same?
Functions for which are called odd functions because odd powers of x have this property.
3) The function is graphed below. On the same grid, graph , and .
/ Which two graphs are the same?The function is an ______function.
4) The function is graphed below. On the same grid, graph , and
/ Which two graphs are the same?The function is an ______function.
Every sinusoidal graph has a period which is divided into four sections. We can describe the boundary points of the sections as follows:
The graphs of and are all congruent, but start on different boundary points.
Function / / / /Starting point
Graph the following vertically reflected sinusoidal functions:
Amplitude______Ymax____
Vertical
Translation____ Yavg____
Ymin____
Period____
Horizontal Translation______/
b)
Amplitude______Ymax____
Vertical
Translation____ Yavg____
Ymin____
Period____
Horizontal Translation______/
MATH 12 PRINCIPLES E6 NOTES
TRIG APPLICATIONS
1. GRAPHING PRE-SKILLS
1) Graph the following trig functions.
a)A: VT: P: HT:
Ymax: Ymin:
/ b)
A: VT: P: HT:
Ymax: Ymin:
2) a) Find a cosine equation with a positive horizontal translation to describe the following graph:
/ A:V:
P:
H:
Equation:
______
b) For the function graphed above, find y (to 2 decimal places) when t = 2.8
c) Use the graph above to estimate the two smallest positive values for t to solve:
plot the horizontal line y=4.0, then find where this line intersects the graph of the cosine function.
d) Use a graphing calculator to find the two smallest positive values for t (to 2 decimal places) to solve the equation .
step 1: Press Y= , then enter
step 2: Press WINDOW, then enter the dimensions:
Xmin = ____ Ymin = _____
Xmax= ____ Ymax = _____
Xscl = ____ Yscl = _____
(Xscl and Yscl set the spacing between the axes markings)
Graph the resulting functions, and label the axes with the values for Xmin, Xmax, Ymin, and Ymax.
/ step 3: Find the intersections of and:
Press 2nd then CALC then 5
The calculator asks:
______? Press ENTER
The calculator asks:______? Press ENTER
The calculator asks:______? Using the blue left and right arrow keys, move the cursor near the intersection point closest to the x-axis, then
press ENTER. The calculator determines the point of intersection: ( ) –round coordinates to 2 decimal places
To find the second intersection point, repeat step 3, but move the cursor near the second intersection point. The second intersection point is ( )
Step 4: Write the relationship between the graph used and the solution obtained:
______
______
Step 5: Write out the solution: t = _____ or ______
e) Find the general solution to the equation
The two solutions listed above are the values for t which satisfy the equation in the first period of the graph. To find any other solution, just add a multiple of the period to either of the solutions in the first period:
Practice: For the relation:
a) find y ( to 2 decimal places) when t =5.0 / b) Use the graphing calculator to find the general solution to the equation:= 9.5
II. TRIG APPLICATIONS
1) Ferris Wheels
Example 1
Kyle and Katie decide to relax with a romantic ride on the Ferris Wheel at Playland after writing their Trig test. The Ferris Wheel has a radius 10 m, and the bottom of the Ferris Wheel is raised 2.0 m above the ground. The Ferris Wheel starts rotating just after they sit down and takes 60 seconds to complete one full rotation.
a) Construct a graph showing Kyle and Katie’s height above the ground, y meters, as a function of the time, t seconds, after the Ferris Wheel starts rotating.
Time, t (seconds) / 0 / 15 / 30 / 45 / 60 / 75 / 90Height, y
(meters)
Height
(y meters)
b) Determine a cosine equation with a positive horizontal translation to describe
K and K’s height (y meters) above the ground as a function of the time (t seconds) after the ride starts.
A:____
V:____
P:____
H:____
c) Determine a cosine equation with no horizontal translation to describe
K and K’s height (y meters) above the ground as a function of the time (t seconds) after the ride starts.
c) Determine how the features of the equation are related to the properties of the Ferris Wheel and of the graph.
Feature of the equation / Property of the Ferris Wheel / Feature of the graphAmplitude
Vertical Translation
Period
Horizontal Translation
(cosine equation)
Horizontal Translation
(negative cosine equation)
Horizontal Translation
(sine equation)
Horizontal Translation
(negative sine equation)
d) Determine K and K’s height above the ground 37 seconds after the ride starts.
e) Use a graphing calculator to determine the times (to the nearest hundredth of a second) during the first rotation when K and K’s height will be 15 m.
Equation to solve:______
Xmin = ____ Ymin = _____
Xmax= ____ Ymax = _____
Xscl = ____ Yscl = _____
Explanation of how the graph leads to the solution:
______
______
Solutions:______
Example 2: A Ferris Wheel Coney Island Pier has a radius of 8m, has its bottom 2m under the pier and rotates with a period of 45 seconds. When the ride starts, Karlo is at the same height as the middle of the Ferris Wheel, and is rotating towards the top of the wheel.
a) Find an equation with no horizontal translation to describe Karlo’s height above the pier t seconds after the ride starts.
A______
V______
P______
b) Find Karlo’s height above the pier 17 seconds after the ride starts.
c) Find the times during the first rotation, to the nearest hundredth of a second, when Karlo’s height will be 12 m.
Equation to solve:______
Xmin = ____ Ymin = _____