Presentation on theme: "Warm-Up Draw the Unit Circle on a sheet of paper and label ALL the parts. DO NOT LOOK AT YOUR NOTES."— Presentation transcript:
Warm-Up Draw the Unit Circle on a sheet of paper and label ALL the parts. DO NOT LOOK AT YOUR NOTES
So far, in our study of trigonometric functions we have: defined all of them learned how to evaluate them used them on the unit circle So logically, the next step would be to study the graphs of the functions.
Unit 2: Graphs and Inverses of Trigonometric Functions LG 2-1 Graphing Trig Functions (Quiz 9/6 LG 2-2 Circular Functions (Quiz 9/10) LG 2-3 Evaluating Inverse Trig Functions (Quiz 9/12) LG 2-4 Graphing Inverse Trig Functions (Quiz 9/14)
What’s Your Temperature? Scientists are continually monitoring the average temperatures across the globe to determine if Earth is experiencing Climate Change (Global Warming!). One statistic scientists use to describe the climate of an area is average temperature. The average temperature of a region is the mean of its average high and low temperatures.
The graph shows the average high (red) and low (blue) temperature in Atlanta from January to December. 1. How would you describe the climate of Atlanta? 2. If you wanted to visit Atlanta, and prefer average highs in the 70’s, when would you go? 3. Estimate the lowest and highest average high temperature. When did these values occur? 4. What is the range of these temperatures? 5. Estimate the lowest and highest average low temperature. When did these values occur? 6. What is the range of these temperatures?
What’s Your Temperature? A function that repeats itself in regular intervals, or periods, is called periodic. a. If you were to continue the temperature graph, what would you consider its interval, or period, to be? b. Choose either the high or low average temperatures and sketch the graph for three intervals, or periods.
Periodic Functions A periodic function is a function whose values repeat at regular intervals. –Sine and Cosine are examples of periodic functions The part of the graph from any point to the point where the graph starts repeating itself is called a cycle. The period is the difference between the horizontal coordinates corresponding to one cycle. –Sine and Cosine functions complete a cycle every 360°. So the period of these functions is 360°.
Periodicity is Common in Nature Day/night cycle (rotation of earth) Ocean Tides Pendulums and other swinging movements Ocean waves Birth/marriage/death cycle Menstrual cycle Eating and sleeping cycle Musical rhythm Linguistic rhythm Dribbling, juggling Calendars Fashion cycles, for example, skirt lengths or necktie widths Economic and political cycles, for example, boom and bust economic periods, right-wing and left-wing political tendencies
Whenever you have to draw a graph of an unfamiliar function, you must do it by point-wise plotting, or calculate and plot enough points to detect a pattern. Then you connect the points with a smooth curve or line. Objective: Discover by point-wise plotting what the graphs of the six trig functions look like. Homework: Finish the characteristics table and do the discovery task LG 2-1 Graphing Trigonometric Functions
Graphing Calculator Introduction Put your calculator in degree mode. Graph y = sin Θ. Trace along the graph. What do you observe? Repeat for y = cos Θ but this time put your calculator on radian mode.
Graphing Sine and Cosine Functions The graph of the sine and cosine functions are made by evaluating each function at the special angles on the unit circle. The input of the function is the angle (in degrees OR radians) measure on the unit circle. The output is the value of the function for that angle. We can “unwrap” these values from the unit circle and put them on the coordinate plane.unwrap
Exploration: Parent Sinusoids Sinusoid – a graph of a sine or cosine function “sinus” coming from the same origin as “sine,” and “– oid” being a suffix meaning “like.”
The Graph of Sine x (angles) y (evaluate for sine) 0o0o 90 o 180 o 270 o 360 o
The Graph of Cosine x (angles) y (evaluate for cos) 0o0o 90 o 180 o 270 o 360 o
Important Words Sinusoidal axis - the horizontal line halfway between the local maximum and local minimum: y = 0 for parent function Convex – bulging side of the wave Concave – hollowed out side of the wave –Concave up –Concave down Point of inflection - point on a curve at which the sign of the curvature (the concavity) changes.
The Graph of Tangent x (angles) y (evaluate for tan) 0o0o 90 o 180 o 270 o 360 o
Co-Trig Functions (Reciprocal Functions) Each of the co-functions relate to the original graph. Plot the “important points” for the sine function on the cosecant graph and then sketch the sine curve LIGHTLY in pencil (borrow one if you need to!)
The Graph of Cosecant x (angles) y (evaluate for csc) 0o0o 90 o 180 o 270 o 360 o
The Graph of Secant x (angles) y (evaluate for sec) 0o0o 90 o 180 o 270 o 360 o
The Graph of Cotangent x (angles) y (evaluate for cot) 0o0o 90 o 180 o 270 o 360 o
Discontinuous Functions The graphs of tan, cot, sec, and csc functions are discontinuous where the function value would involve division by zero. What happens to the graph when a function is discontinuous?
Trig Function Characteristics As always, we need to talk about domain, range, max and min, etc. You will fill out the table for each of the 6 trig functions (and finish for HW).
Period The period of a trig function is how long it takes to complete one cycle. What is the period of sine and cosine? What about cosecant and secant? Tangent and cotangent? The period of the functions tangent and cotangent is only 180° instead of 360°, like the four trigonometric functions.
Domain and Range When we think about the domain and range, we have to make sure we are considering the entire function and not just the part on the unit circle.
Maximum and Minimum Local max & min Absolute max & min
Intercepts x-intercepts y-intercepts
Points of Inflection point on a curve at which the sign of the curvature (the concavity) changes. Will all graphs have points of inflection?
Intervals of Increase and Decrease Positive slope? Negative slope?
Classwork… Fill out the characteristics table for each of the 6 trig functions
PeriodDomainRangeMaximumMinimum Sine Cosine Tangent Cotangent Cosecant Secant
Increasing Interval Decreasing Interval X-intercept(s)Y-intercept(s)Point of Inflection sine cosine Tangent Cotangen t cosecant secant
FunctionPeriodDomainRange y = tan Θ180° Θ = all real #’s of degrees except Θ = 90° + 180°n, where n is an integer y = all real #’s y = cot Θ180° Θ = all real #’s of degrees except Θ = 180°n, where n is an integer y = all real #’s y = sec Θ360° Θ = all real #’s of degrees except Θ = 90° + 180°n, where n is an integer y = csc Θ360° Θ = all real #’s of degrees except Θ = 180°n, where n is an integer
What’s Your Temperature? Sine and Cosine functions can be used to model average temperatures for cities. Based on what you learned about these graphs, why do you think these functions are more appropriate than a cubic function? Or an exponential function?