Presentation on theme: "Warm Up Using your unit circle find each value: Sin 0°= Sin "— Presentation transcript:
1Warm Up Using your unit circle find each value: Sin 0°= Sin 𝜋 2 =
2Chapter 4 Graphs of the Circular Functions Section 4.1 Graphs of the Sine and Cosine FunctionsObjective:SWBAT graph the sine and cosine functions with variations in amplitude and periods.
3This periodic graph represents a normal heartbeat. Periodic FunctionsMany things in daily life repeat with a predictable pattern, such as weather, tides, and hours of daylight.This periodic graph represents a normal heartbeat.A function that repeats itself after a specific period of time is called a Periodic Function. Sine and Cosine functions are periodic functions.
4Sine and Cosine Functions We are going to deconstruct the Unit circle and graph the sine and cosine functions on graph…Remember: Cosine is in the “x” spot in an ordered pair andSine is in the “y” spot.
5VocabularySin wave - is a repetitive change or motion which, when plotted as a graph, has the same shape as the sine function.Amplitude - is the maximum distance it ever reaches from zero.Period - is the time it takes to perform one complete cycle. (2π for cosine and sin)
6Sine FunctionTo sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. (5 total)yxsin xx1-1
7Sine Function f(x) = sin x The graph is continuous over its entire domain, (–, ).Its x-intercepts are of the form n, where n is an integer.Its period is 2.The graph is symmetric with respect to the origin, so the function is an odd function. For all x in the domain, sin(–x) = –sin(x).
8AmplitudeThe amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function.amplitude = |a|If |a| > 1, the amplitude stretches the graph vertically.If 0 < |a| > 1, the amplitude shrinks the graph vertically.If a < 0, the graph is reflected in the x-axis.The graph of y = a sin x or y = a cos x, with a 0, will have the same shape as the graph of y = sin x or y = cos x, respectively, except the range will be [|a|, |a|]. The amplitude is |a|.Amplitude
9Example: Graph y = 3 sin x compare to y = sin x. AmplitudeExample: Graph y = 3 sin x compare to y = sin x.Make a table of values.333sin x11sin x3/2/2xThe range of y = 3sin x is [–3, 3].
10Amplitude y y = sin x x y = sin x y = 2 sin x y = – 4 sin x reflection of y = 4 sin xy = 4 sin xAmplitude
11Period Divide the interval into four equal parts to For b > 0, the graph of y = sin bx will resemble that ofy = sin x, but withperiodDivide the interval into four equal parts toobtain the values for which sin bx equal –1, 0, or 1.(These values give the minimum points, x-intercepts, and maximum points on the graph)
12Graph y = sin 2x and compare to the graph of y = sin x. GRAPHING y = sin bxGraph y = sin 2x and compare to the graph of y = sin x.The coefficient of x is 2, so b = 2, and the period isThe endpoints are 0 and and the three points between the endpoints areThe x-values are:
13Graph y = sin 2x and compare to the graph of y = sin x. GRAPHING y = sin bxGraph y = sin 2x and compare to the graph of y = sin x.Y = sin2xXY(x, y)𝜋 4𝜋 23𝜋 4𝜋
14Graph y = sin 2x and compare to the graph of y = sin x. GRAPHING y = sin bxGraph y = sin 2x and compare to the graph of y = sin x.