Graphs of Sine and Cosine Functions PDF

Graphs of the Sine and Cosine Functions  Graphs of 𝒚 = 𝐬𝐢𝐧 𝒙 and 𝒚 = 𝐜𝐨𝐬 𝒙 Recall that the function 𝑦 = sin 𝑥 is always defined for any real number 𝑥. This is also true for 𝑦 = cos 𝑥. Let us take a look at the graphs of sine and cosine functions....

Graphs of the Sine and Cosine Functions  Graphs of 𝒚 = 𝐬𝐢𝐧 𝒙 and 𝒚 = 𝐜𝐨𝐬 𝒙 Recall that the function 𝑦 = sin 𝑥 is always defined for any real number 𝑥. This is also true for 𝑦 = cos 𝑥. Let us take a look at the graphs of sine and cosine functions. 2 Graphs of 𝒚 = 𝐬𝐢𝐧 𝒙 and 𝒚 = 𝐜𝐨𝐬 𝒙 Graph of 𝑦 = sin 𝑥 3 Graphs of 𝒚 = 𝐬𝐢𝐧 𝒙 and 𝒚 = 𝐜𝐨𝐬 𝒙 Graph of 𝑦 = cos 𝑥 4 Graphs of 𝒚 = 𝐬𝐢𝐧 𝒙 and 𝒚 = 𝐜𝐨𝐬 𝒙 Properties One cycle occurs Domain: ℝ Range: −1 ≤ 𝑦 ≤ 1 every 2𝜋. 5  Graphs of 𝒚 = 𝒂 𝐬𝐢𝐧 𝒙 and 𝒚 = 𝒂 𝐜𝐨𝐬 𝒙 Graph of 𝒚 = 𝒂 𝐬𝐢𝐧 𝒙 Consider the graphs of 𝒚 = 𝟐 𝐬𝐢𝐧 𝒙 and 𝒚 = 𝟑 𝐬𝐢𝐧 𝒙, and let us compare these functions to 𝒚 = 𝐬𝐢𝐧 𝒙. What do you notice about the graphs? 6  Graphs of 𝒚 = 𝒂 𝐬𝐢𝐧 𝒙 and 𝒚 = 𝒂 𝐜𝐨𝐬 𝒙 Graph of 𝒚 = 𝒂 𝐬𝐢𝐧 𝒙 In this case, 𝑎 is negative. What do you notice about the graph of 𝒚 = − 𝐬𝐢𝐧 𝒙 as compared to the graph of 𝒚 = 𝐬𝐢𝐧 𝒙? 7 What do you think will happen to the graph of a sine function if 𝒂 increases or decreases? 8  Graphs of 𝒚 = 𝒂 𝐬𝐢𝐧 𝒙 and 𝒚 = 𝒂 𝐜𝐨𝐬 𝒙 Graph of 𝒚 = 𝒂 𝐜𝐨𝐬 𝒙 Consider the graphs of 𝒚 = 𝟐 𝐜𝐨𝐬 𝒙 and 𝒚 = 𝟑 𝐜𝐨𝐬 𝒙 and let us compare these functions to 𝒚 = 𝐜𝐨𝐬 𝒙. What do you notice about the graphs? 9  Graphs of 𝒚 = 𝒂 𝐬𝐢𝐧 𝒙 and 𝒚 = 𝒂 𝐜𝐨𝐬 𝒙 Graph of 𝒚 = 𝒂 𝐜𝐨𝐬 𝒙 In this case, 𝑎 is negative. What do you notice about the graph of 𝒚 = − 𝐜𝐨𝐬 𝒙 as compared to the graph of 𝒚 = 𝐜𝐨𝐬 𝒙? 10  Graphs of 𝒚 = 𝒂 𝐬𝐢𝐧 𝒙 and 𝒚 = 𝒂 𝐜𝐨𝐬 𝒙 Amplitude of the Sine and Cosine Functions The value of |𝒂| in the graphs of 𝑦 = 𝑎 sin 𝑥 and 𝑦 = 𝑎 cos 𝑥 is called the amplitude of the graphs. It is the vertical distance of the highest and lowest points of the graph from the 𝑥-axis. Note that 𝑎 ≠ 0. 11  Graphs of 𝒚 = 𝒂 𝐬𝐢𝐧 𝒙 and 𝒚 = 𝒂 𝐜𝐨𝐬 𝒙 Amplitude of the Sine and Cosine Functions If 𝒂 < 𝟏, the graphs will shrink vertically. If 𝒂 > 𝟏, the graphs will stretch vertically. 12  Graphs of 𝒚 = 𝒂 𝐬𝐢𝐧 𝒙 and 𝒚 = 𝒂 𝐜𝐨𝐬 𝒙 Amplitude of the Sine and Cosine Functions The graphs of 𝑦 = −𝑎 sin 𝑥 and 𝑦 = −𝑎 cos 𝑥 are vertical reflections of the graphs of 𝑦 = 𝑎 sin 𝑥 and 𝑦 = 𝑎 cos 𝑥, respectively. 13 A function 𝑓 is said to be an odd function if for any number 𝑥, 𝑓 −𝑥 = −𝑓 𝑥. 14 A function 𝑓 is said to be an even function if for any number 𝑥, 𝑓 −𝑥 = 𝑓 𝑥. 15 Sine function is an odd function. Thus, sin −𝑥 = − sin 𝑥. Cosine function is an even function. Thus, cos −𝑥 = cos 𝑥. 16  Graphs of 𝒚 = 𝐬𝐢𝐧 𝒃𝒙 and 𝒚 = 𝐜𝐨𝐬 𝒃𝒙 Graph of 𝒚 = 𝐬𝐢𝐧 𝒃𝒙 Consider the graphs of 𝒙 𝒚= 𝐬𝐢𝐧 and 𝒚 = 𝐬𝐢𝐧 𝟐𝒙 , 𝟐 and let us compare these functions to the graph of 𝒚 = 𝐬𝐢𝐧 𝒙. What do you notice about the graphs? 17  Graphs of 𝒚 = 𝐬𝐢𝐧 𝒃𝒙 and 𝒚 = 𝐜𝐨𝐬 𝒃𝒙 Graph of 𝒚 = 𝐜𝐨𝐬 𝒃𝒙 Consider the graphs of 𝒙 𝒚= 𝐜𝐨𝐬 and 𝒚 = 𝐜𝐨𝐬 𝟐𝒙 𝟐 and let us compare these functions to the graph of 𝒚 = 𝐜𝐨𝐬 𝒙. What do you notice about the graphs? 18  Graphs of 𝒚 = 𝐬𝐢𝐧 𝒃𝒙 and 𝒚 = 𝐜𝐨𝐬 𝒃𝒙 Period of the Sine and Cosine Functions The value of 𝑏 in the graphs of 𝑦 = sin 𝑏𝑥 and 𝑦 = cos 𝑏𝑥, determines the period of the graphs. Note that 𝑏 ≠ 0. The period is the horizontal distance required for the sine and cosine functions to complete one full cycle. 19  Graphs of 𝒚 = 𝐬𝐢𝐧 𝒃𝒙 and 𝒚 = 𝐜𝐨𝐬 𝒃𝒙 Period of the Sine and Cosine Functions 𝟐𝝅 The period of the functions is. If 𝒃 > 𝟏, the graphs will 𝒃 shrink horizontally. If 𝟎 < 𝒃 < 𝟏, the graphs will stretch horizontally. 20  Graphs of 𝒚 = 𝐬𝐢𝐧 𝒃𝒙 and 𝒚 = 𝐜𝐨𝐬 𝒃𝒙 Period of the Sine and Cosine Functions The graph of 𝒚 = 𝐬𝐢𝐧 −𝒃𝒙 is the same as the graph of 𝒚 = − 𝐬𝐢𝐧 𝒃𝒙, while the graph of 𝒚 = 𝐜𝐨𝐬 −𝒃𝒙 is the same as the graph of 𝒚 = 𝐜𝐨𝐬 𝒃𝒙. 21  Graphs of 𝒚 = 𝐬𝐢𝐧(𝒙 − 𝒄) and 𝒚 = 𝐜𝐨𝐬(𝒙 − 𝒄) Graph of 𝒚 = 𝐬𝐢𝐧(𝒙 − 𝒄) 𝝅 𝝅 Consider the graphs of 𝒚 = 𝐬𝐢𝐧 𝒙 + and 𝒚 = 𝐬𝐢𝐧 𝒙 − , 𝟐 𝟐 and let us compare these functions to 𝒚 = 𝐬𝐢𝐧 𝒙. What do you notice about the graphs of the functions? 22  Graphs of 𝒚 = 𝐬𝐢𝐧(𝒙 − 𝒄) and 𝒚 = 𝐜𝐨𝐬(𝒙 − 𝒄) Graph of 𝒚 = 𝐜𝐨𝐬(𝒙 − 𝒄) 𝝅 𝝅 Consider the graphs of 𝒚 = 𝐜𝐨𝐬 𝒙 + and 𝒚 = 𝐜𝐨𝐬 𝒙 − , 𝟐 𝟐 and let us compare these functions to 𝒚 = 𝐜𝐨𝐬 𝒙. What do you notice about the graphs of the functions? 23  Graphs of 𝒚 = 𝐬𝐢𝐧(𝒙 − 𝒄) and 𝒚 = 𝐜𝐨𝐬(𝒙 − 𝒄) Phase Shift of the Sine and Cosine Functions The value of 𝑐 in the graphs of 𝑦 = sin(𝑥 − 𝑐) and 𝑦 = cos(𝑥 − 𝑐) determines the phase shift of the graphs. Phase shift refers to how far the graph is shifted to the left or the right from the base function. 24  Graphs of 𝒚 = 𝐬𝐢𝐧(𝒙 − 𝒄) and 𝒚 = 𝐜𝐨𝐬(𝒙 − 𝒄) Phase Shift of the Sine and Cosine Functions If 𝒄 < 𝟎, the graphs are shifted 𝑐 units to the left. If 𝒄 > 𝟎, the graphs are shifted 𝑐 units to the right. 25  Graphs of 𝒚 = 𝐬𝐢𝐧 𝒙 + 𝒅 and 𝒚 = 𝐜𝐨𝐬 𝒙 + 𝒅 Graph of 𝒚 = 𝐬𝐢𝐧 𝒙 + 𝒅 Consider the graphs of 𝒚 = 𝐬𝐢𝐧 𝒙 − 𝟏 and 𝒚 = 𝐬𝐢𝐧 𝒙 + 𝟏, and let us compare these functions to 𝒚 = 𝐬𝐢𝐧 𝒙. What do you notice about the graphs of the functions? 26  Graphs of 𝒚 = 𝐬𝐢𝐧 𝒙 + 𝒅 and 𝒚 = 𝐜𝐨𝐬 𝒙 + 𝒅 Graph of 𝒚 = 𝐜𝐨𝐬 𝒙 + 𝒅 Consider the graphs of 𝒚 = 𝐜𝐨𝐬 𝒙 − 𝟏 and 𝒚 = 𝐜𝐨𝐬 𝒙 + 𝟏, and let us compare these functions to 𝒚 = 𝐜𝐨𝐬 𝒙. What do you notice about the graphs of the functions? 27  Graphs of 𝒚 = 𝐬𝐢𝐧 𝒙 + 𝒅 and 𝒚 = 𝐜𝐨𝐬 𝒙 + 𝒅 Vertical Shift of the Sine and Cosine Functions The value of 𝑑 in the graphs of 𝑦 = sin 𝑥 + 𝑑 and 𝑦 = cos 𝑥 + 𝑑 determines the vertical shift of the graphs. Vertical shift is how far the graph is shifted downward or upward from the base function. 28  Graphs of 𝒚 = 𝐬𝐢𝐧 𝒙 + 𝒅 and 𝒚 = 𝐜𝐨𝐬 𝒙 + 𝒅 Vertical Shift of the Sine and Cosine Functions If 𝒅 < 𝟎, the graphs are shifted 𝑑 units downward. If 𝒅 > 𝟎, the graphs are shifted 𝑑 units upward. 29  Graphs of 𝒚 = 𝐬𝐢𝐧 𝒙 + 𝒅 and 𝒚 = 𝐜𝐨𝐬 𝒙 + 𝒅 Vertical Shift of the Sine and Cosine Functions Since the graph of the function will be vertically shifted, the range of the function will be − 𝒂 + 𝒅 ≤ 𝒚 ≤ 𝒂 + 𝒅. 30 Steps in Graphing Sine and Cosine Functions 1 Construct a table of values. 2 Plot the points in a coordinate plane. 3 Extend the curve to the left and right. 31  Steps in Graphing Sine and Cosine Functions Construct a table of values. 1. The endpoints of the 𝑥-coordinates in the table are 𝑐 2𝜋 and 𝑐 +. |𝑏| 2𝜋 2. The 𝑥-coordinates have interval of. 4|𝑏| 3. The 𝑦 values that you will get at the provided values of 𝑥 is either −|𝑎| + 𝑑, 𝑑, or |𝑎| + 𝑑. 32  Graphing Sine and Cosine Functions Plot the points in a coordinate plane. Plot the points in a coordinate plane and connect them using a smooth curve. 33  Graphing Sine and Cosine Functions Extend the curve to the left and right. Extend the curve in the previous step to the left and to the right by repeating the cycle. 34  Graphing Sine and Cosine Functions Example: Graph 𝑦 = sin 𝑥. 35  Graphing Sine and Cosine Functions Example: Graph 𝑦 = sin 𝑥. 36 Find the equation of the sine function with the same shape as 𝒚 = 𝟐 𝐬𝐢𝐧 𝒙 that is shifted 𝟐𝝅 units to the right and 5 units downward. 37 Find the equation of the sine function with the same shape as 𝒚 = 𝟐 𝐬𝐢𝐧 𝒙 that is shifted 𝟐𝝅 units to the right and 5 units downward. 𝒚 = ±𝟐 𝐬𝐢𝐧(𝒙 − 𝟐𝝅) − 𝟓 38 Find the equation of the sine function with the same shape as 𝒚 = − 𝐬𝐢𝐧 𝟑𝒙 that is shifted 𝝅 units to the left and 8 units upward. 39 Determine the amplitude, period, domain, and range of the function 𝒚 = −𝟑 𝐜𝐨𝐬 𝟔 𝒙 − 𝝅 + 𝟏. 40 Determine the amplitude, period, domain, and range of the function 𝒚 = −𝟑 𝐜𝐨𝐬 𝟔 𝒙 − 𝝅 + 𝟏. Amplitude: 𝟑 𝝅 Period: 𝟑 Domain: all real numbers Range: − 𝟐 ≤ 𝒚 ≤ 𝟒 41 Determine the amplitude, period, domain, and range of the function 𝒚 = 𝟐 𝐬𝐢𝐧 𝟓 𝒙 − 𝟑𝝅 − 𝟕. 42 Graph the function 𝒚 = 𝐬𝐢𝐧 𝟒𝒙. 43 Graph the function 𝒚 = 𝐬𝐢𝐧 𝟒𝒙. 44 Graph the function 𝒚 = 𝐬𝐢𝐧 𝟑𝒙. 45  𝒙 Graph the function 𝒚 = 𝟐 𝐜𝐨𝐬. 𝟑 46  𝒙 Graph the function 𝒚 = 𝟐 𝐜𝐨𝐬. 𝟑 47  𝟐𝒙 Graph the function 𝒚 = −𝟐 𝐜𝐨𝐬. 𝟑 48  𝝅 Graph the function 𝒚 = −𝟐 𝐬𝐢𝐧 𝒙 +. 𝟒 49  𝝅 Graph the function 𝒚 = −𝟐 𝐬𝐢𝐧 𝒙 +. 𝟒 50  𝟏 𝝅 Graph the function 𝒚 = 𝟑 𝐬𝐢𝐧 𝒙−. 𝟑 𝟐 51  𝟏 𝟐 𝝅 Graph the function 𝒚 = 𝐜𝐨𝐬 𝒙− + 𝟏. 𝟐 𝟑 𝟑 52  𝟏 𝟐 𝝅 Graph the function 𝒚 = 𝐜𝐨𝐬 𝒙− + 𝟏. 𝟐 𝟑 𝟑 53  𝟑 𝝅 Graph the function 𝒚 = 𝟐 𝐜𝐨𝐬 𝒙+ − 𝟒. 𝟐 𝟔 54 Determine the amplitude, period, domain, and range of the following functions. 1. 𝑦 = sin 8𝑥 𝑥 2. 𝑦 = −3 cos 2 5𝑥 3. 𝑦 = −3 sin − 7 6 7 1 3𝜋 4. 𝑦 = − cos 𝑥 − −9 2 2 4 1 2 11𝜋 5. 𝑦 = sin 𝑥 + +8 3 3 6 55 Graph the following functions in a rectangular coordinate system. 1. 𝑦 = − sin 4𝑥 2. 𝑦 = 3 cos 𝑥 𝜋 3. 𝑦 = sin 𝑥 + 6 2𝜋 4. 𝑦 = cos 𝑥 − 3 1 𝜋 5. 𝑦 = cos 𝑥 − −1 5 2 56  The sine and cosine functions are called periodic functions since the value of the dependent variable 𝑦 repeats its value in regular intervals of 𝟐𝝅 , which is the period of the graph of the 𝒃 functions. 57  The graph of sine and cosine functions of the form 𝑦 = 𝑎 sin 𝑏(𝑥 − 𝑐) + 𝑑 and 𝑦 = 𝑎 cos 𝑏(𝑥 − 𝑐) + 𝑑 have the following properties: o The amplitude |𝒂| determines the vertical shrinking or stretching of the graph. 58  𝟐𝝅 o The period is given by , and the value of 𝑏 |𝒃| determines the horizontal shrinking or stretching of the graph. o The phase shift 𝒄 determines the horizontal shifting of the graph. 59 o The vertical shift 𝒅 determines the vertical shifting of the graph. o The domain of the function is the set of real numbers. 60 o The range of the function is −|𝒂| + 𝒅 ≤ 𝒚 ≤ |𝒂| + 𝒅. 61