Presentation on theme: "Graphing Sine and Cosine"— Presentation transcript:
1Graphing Sine and Cosine Pre CalculusGraphing Sine and Cosine
2What you will learn How to graph sine and cosine functions. How to translate sine and cosine functions (shift, left, right, vertical stretch, horizontal stretch)How to use key points to “sketch” a graph.
3Plan Discuss how to use the Unit Circle to help with graphing Graphing Sine and Cosine and their translations
5Review of Even and Odd Functions Cosine and secant functions are evencos (-t) = cos t sec (-t) = sec tSine, cosecant, tangent and cotangent are oddsin (-t) = -sin t csc (-t) = -csc ttan (-t) = -tan t cot (-t) = -cot t
7Key Things to Discuss Shape of the functions Using the Unit Circle to help identify key pointsPeriodic NatureTranslations that are the same as other functions we have studiedTranslations that are different than others we have studiedUsing the calculator and correct interpretation of the calculator
8Shape of Sine and Cosine The unit circle: we imagined the real number line wrapped around the circle.Each real number corresponded to a point (x, y) which we found to be the (cosine, sine) of the angle represented by the real number.To graph the sine and cosine we can go back to the unit circle to find the ordered pairs for our graph.
9Let’s convert some of these numbers to decimal form – start with cosine
10Cosine Key Features to define the shape Input: xAngleOutput:Cos x1maxπ/6.87π/4.71π/3.5π/2int2π/3-.53π/4-.715π/6-.87π-1min7π/65π/44π/33π/25π/37π/411π/62π
12Cosine For the function: The angle is the input or independent variable and the cosine ratio is the output or dependent variable.From a unit circle perspective, the input is the angle and the output is the “x” coordinate of the ordered pair.Remember “coterminal angles” every 2π the values will repeat – this is called a periodic functionWe will use the interval [0, 2π] as the reference period.
13Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts.1-1cos xxThen, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.yxy = cos xCosine Function
19Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts.-11sin xxThen, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.yxy = sin xSine Function
20Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties:1. The domain is the set of real numbers.2. The range is the set of y values such that3. The maximum value is 1 and the minimum value is –1.4. The graph is a smooth curve.5. Each function cycles through all the values of the range over an x-interval of6. The cycle repeats itself indefinitely in both directions of the x-axis.Properties of Sine and Cosine Functions
21Transformations – A look back Let’s go back to the quadratic equation in graphing form: y = a(x – h)2 + kIf a < 0: reflection across the x axis|a| > 1: stretch; and |a| < 1: shrink(h, k) was the vertex (locator point)h gave us the horizontal shiftk gave us the vertical shift
22Transforming the Cosine (or Sine) y = a (cos (bx – c)) + d Let’s look at a |a| is called the amplitude, like our other functions it is like a stretch If a < 0 it also causes a reflection across the x-axis Graph y = cos x and y = 3 cos x y = cos x and y = -3 cos x
23Key PointsAmplitude – increase the output by a factor of the amplitudeRemember the amplitude is always positive so you have to apply any reflectionsy = 3 cos x1-1cos xx3-3cos xx
24Example: Sketch the graph of y = 3 cos x on the interval [–, 4]. Partition the reference interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.maxx-intmin3-3y = 3 cos x2xyx(0, 3)( , 3)( , 0)( , 0)( , –3)Example: y = 3 cos x
25If |a| > 1, the amplitude stretches the graph vertically. The 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.yxy = 2 sin xy = sin xy = sin xy = – 4 sin xreflection of y = 4 sin xy = 4 sin xAmplitude
26Transforming the Cosine (or Sine) y = a (cos (bx – c)) + d Let’s look at d Just like in our other functions, d is the vertical shift, if d is positive, it goes up, if it is negative, it goes down. Graph y = cos x and y = cos x + 1 y = cos x and y = cos x – 2
27Vertical Shift1-1cos xxy = cos x + 1 Begin with y = cos x Add d to the output to adjust the graph Then shift up one unityx21cos xx
28Transforming the Cosine (or Sine) y = a (cos (bx – c)) + d Let’s look at c Just like in our other functions, c is the horizontal shift, if c is positive, it goes right, if it is negative, it goes left. If there is no b present, it is the same as other functions Graph: y = cos x and y = cos (x + π/2) y = cos x and y = cos (x - π/2)
29Horizontal shift y = cos (x + π/2) Begin with y = cos x 1-1cos xxy = cos (x + π/2)Begin with y = cos xNow you must translate the input… the angleThen shift left π/2 unitsyx1-1cos xx
30Transforming Cosine (or Sine) y = a (cos (bx – c)) + dLet’s look at bGraph:y =cos x and y =cos 2x (b = 2)y =cos x and y =cos ½ x (b = ½ )What happened?
31Transforming the Period b has an effect on the period (normal is 2π) If b > 1, the period is shorter, in other words, a complete cycle occurs in a shorter interval If b < 1, the period is longer or a cycle completes over an interval greater than 2πTo determine the new period = 2π/b
32If b > 1, the graph of the function is shrunk horizontally. The period of a function is the x interval needed for the function to complete one cycle.For b 0, the period of y = a sin bx isFor b 0, the period of y = a cos bx is alsoIf b > 1, the graph of the function is shrunk horizontally.yxperiod:period: 2If 0 < b < 1, the graph of the function is stretched horizontally.yxperiod: 4period: 2Period of a Function
33Summarizing …Standard form of the equations: y = a (cos (bx – c)) + d y = a (sin (bx – c)) + d“a” - |a| is called the amplitude, like our other functions it is like a stretch it affects “y” or the outputIf a < 0 it also causes a reflection across the x- axis“d” – vertical shift, it affects “y” or the output“c” – horizontal shift, it affects “x” or “θ” or the input“b” – period change (“squishes” or “stretches out” the graphThe combination of “b” and “c” has another effect that we will discuss next time.
34Next classWe will get into the detail of transforming the period when both b and c are presentWe will graph using a “key point” methodWe will talk about how the calculator can help and how you need to be careful with setting windows.
35Calculator Issues Window settings Using your reference period to set your windowSetting your scale