Presentation on theme: "10.1 Graph Sine, Cosine, and Tangent Functions"— Presentation transcript:
110.1 Graph Sine, Cosine, and Tangent Functions What graphs are periodic?What graphs have maximum and minimum values?What values do you need to graph the sin𝑥 or the cos𝑥?
2VocabularyAmplitude of each functions graph: half the difference of the maximum 𝑀 and the minimum 𝑚, or ½ (𝑀−𝑚) = ½[1−(−1)] = 1.Periodic: the graph has a repeating pattern.Cycle: the shortest repeating portion of the graph.Period: the horizontal length of each cycle.
3Parent Graph of 𝑦 = sin𝑥 D: all real numbers R: −1≤𝑦≤1.This graph has a horizontal length and a period of 2.The x-intercepts for 𝑦=sin𝑥 occur when 𝑥=0, ±𝜋, ±2𝜋, ±3𝜋,…
4Parent Graph of 𝑦 = cos𝑥 D: all real numbers R: −1≤𝑦≤1.This graph has a horizontal length and a period of 2.The x-intercepts for 𝑦=cos𝑥 occur when 𝑥= ± 𝜋 2 , ± 3𝜋 2 , ± 5𝜋 2 ,± 7𝜋 2 ,…
7𝑦=𝑎 sin 𝑏𝑥 𝑦=𝑎 cos𝑏𝑥Each graph shows five key x-values on the interval 0≤𝑥≤ 2𝜋 𝑏 when 𝑎>0 and 𝑏>0.Maximum x values occur at the blue points.Minimum x values occur at the green points.X-intercepts occur at the red points.
8Graph (a) y = 4 sin xSOLUTION2bπ=12π.a. The amplitude is a = 4 and the period isIntercepts:(0, 0);12( π, 0)(π, 0); (2π, 0)=Maximum:( π, 4)142π( , 4)=Minimum:( π, – 4)3423π( , – 4)=
9Graph (b) y = cos 4x.SOLUTIONb. The amplitude is a = 1 and the period is2bπ=4= .Intercepts:( , 0)14π2= ( , 0);83= ( , 0)3πMaximums:(0, 1);2π( , 1)Minimum:( , –1)12π= ( , –1)4
10Graph the function.1. y = 2 cos xSOLUTIONThe amplitude is a = 2 and the period is2bπ=1= 2πIntercepts:( π, 0)14( π, 0)= ( , 0);π23= ( , 0)3π( , 2)Maximums:(0, 2);2πMinimum:( π, –2)12= ( , –2)π4
11Graph the function.2. y = 5 sin xSOLUTIONThe amplitude is a = 5 and the period is2bπ=1= 2πIntercepts:(0, 0)(π, 0)= ( π, 0) =12= ( , 0)2πMaximums:( π, 5) =14( , 5)π2
12Changes in a and bNotice how changes in a and b affect the graphs of 𝑦=𝑎 sin𝑏𝑥 and 𝑦=𝑎 cos𝑏𝑥.When the value of a increases, the amplitude increases.When the value of b increases, the period decreases.
13Graph the function.3. f (x) = sin πxSOLUTIONThe amplitude is a = 1 and the period is2bπ== 2Intercepts:(0, 0)(1, 0); (2, 0)= ( , 0) =12Maximums:( , 1 =( , 1)124Minimum:( , –1)34= ( , –1)2
14Graph the function.4. g(x) = cos 4πxSOLUTIONThe amplitude is a = 1 and the period is2bπ=4π1Intercepts:( , 0) = ( , 0); ( , 0) = ( , 0)14283Maximums:12( , 1)( 0, 1);Minimum:= ( , –1)14( , –1)2
15What graphs are periodic? The graphs of 𝑦=sin𝑥 and 𝑦=cos𝑥 are periodic.What graphs have maximum and minimum values?The graphs of 𝑦=sin𝑥 and 𝑦=cos𝑥 have maximum and minimum values.What values do you need to graph the sin𝑥 or the cos𝑥?You can use the maximum and minimum values and the x-intercepts to graph 𝑦=sin𝑥 and 𝑦=cos𝑥 .
1610.1 Assignment, day 1Page 617, 3 – 15No graphing calculators are to used for this assignment. Graphing solutions need to include:intercepts informationmaximum informationminimum informationgraphsLearn where this information comes from!
1710.1 Graph Sine, Cosine, and Tangent Functions, day 2 What graphs have asymptotes?What is frequency?What values do you need to graph tangents?
18Graph y = cos 2π x.12SOLUTIONThe amplitude is a =12and the period isbπ=2π1.Intercepts:( , 0)14= ( , 0);( , 0)3= ( , 0)Maximums:(0, ) ;12(1, )Minimum:( , – )12= ( , – )
19FrequencyFrequency: the reciprocal of the period which gives the number of cycles per unit of time.The periodic nature of trig functions is useful to model oscillating motions or repeating patterns (sound waves, motion of a pendulum,…)
20A sound consisting of a single frequency is called a pure tone A sound consisting of a single frequency is called a pure tone. An audiometer produces pure tones to test a person’s auditory functions. Suppose an audiometer produces a pure tone with a frequency f of 2000 hertz (cycles per second). The maximum pressure P produced from the pure tone is 2 millipascals. Write and graph a sine model that gives the pressure P as a function of the time t (in seconds).Audio Test
21The pressure P as a function of time t is given by P = 2 sin 4000πt. SOLUTIONSTEP 1Find the values of a and b in the model P = a sin bt. The maximum pressure is 2, so a = 2. You can use the frequency f to find b.frequency =period12000 =b2π= bThe pressure P as a function of time t is given by P = 2 sin 4000πt.STEP 2Graph the model. The amplitude is a = 2 and the period is20001f=Intercepts:(0 , 0);( , 0)122000= ( , 0) ;4000( , 0)Maximum:( , 2)142000= ( , 2)8000Minimum:( , –2)3420001= ( , –2)8000
22Graph the function.5. y = sin πx14SOLUTIONThe amplitude is a =14and the period is2bπ=2.Intercepts:= (1, 0) ;(2, 0)(0 , 0);122, 0()Maximums:142,()1 ,2=Minimum:342,1()–=3 ,2
23Graph the function.6. y = cos πx13SOLUTIONThe amplitude is a =and the period is2π=b2.13=12,();342, 0)Intercepts:Maximums:132,)(0,);Minimum:122,3()–=1,
24Graph the function.7. f (x) = 2 sin 3xSOLUTIONThe amplitude is a = 2and the period is2π3b=Intercepts:(0 , 0);122π3,=π());Maximums:142π3,2=π6()Minimum:342π–2()π=
25Graph the function.g(x) = 3 cos 4xSOLUTIONThe amplitude is a = 3and the period is2π=4bIntercepts:14π2,=()83);3πMaximums:(0 , 3);π2,3()Minimum:12π–3()4=
26Graph of 𝑦= tan 𝑥The domain is all real numbers except odd multiples of 𝜋 2 . At these x-values, the graph has vertical asymptotes.The range is all real numbers.The function 𝑦= tan 𝑥 does not have a maximum or minimum value. The graph of 𝑦= tan 𝑥 does not have an amplitude.The graph has a period of 𝜋.The x intercepts of the graph occur when 𝑥=0, ±𝜋, ±2𝜋, ±3𝜋,…
28Graphing Key Points Graph the x-intercept Graph the x-values where the asymptotes occur.Graph the x-values half-way between the x-intercept and the asymptotes.At each halfway point, the function’s value is either 𝑎 or −𝑎.
29Graph one period of the function y = 2 tan 3x. SOLUTIONbπ=3.The period isIntercepts:(0, 0)Asymptotes:x =2bπ=2 3, or x = ;6x =2bπ–=2 3, or x =6Halfway points:( , a)4bπ=4 3( , 2)12( , 2);( , – a)4bπ–=4 3( , – 2)12( , – 2)
30Graph one period of the function. f (x) = 2 tan 4xSOLUTION𝜋 𝑏The period isIntercepts:𝜋 2 𝑏Asymptotes:𝑦= a tan 𝑏𝑥± 𝜋 4𝑏Halfway points:
31Graph one period of the function. g(x) = 5 tan πxSOLUTIONThe period is𝜋 𝑏Intercepts:𝜋 2 𝑏Asymptotes:𝑦= a tan 𝑏𝑥± 𝜋 4𝑏Halfway points:
32What graphs have asymptotes? 𝑦= a tan 𝑏𝑥What is frequency?The reciprocal of the period which gives the number of cycles per unit of time.What values do you need to graph tangents?You can use the asymptotes, the x-intercepts, and the x-values halfway between the x-intercepts and the asymptotes to graph 𝑦= tan 𝑥 .
3310.1 Assignment, day 2 Page 617, 16-24 all No graphing calculators are to used for this assignment. Graphing solutions need to include:intercepts informationmaximum informationminimum informationasymptotesgraphsLearn where this information comes from!