Algebra and Trig. I 4.5 Graphs of Sine and Cosine Functions The graph of The graph of. The trigonometric functions can be graphed in a rectangular coordinate system by plotting points whose coordinates satisfy the function. Thus, we graph by listing some points on the graph because the period of the sine function is 2π, we will graph the function on the interval [0,2π]. The rest of the graph is made up of repetitions of this period Period When the values of a function regularly repeat themselves, we say that the function is periodic. The values of the sine function repeat themselves every 2π Amplitude The maximum value of y on the graph of is, the amplitude. If then the curve is stretched, if then the curve is shrunk. The amplitude (half the distance between the maximum and minimum values of the function) will be, since distance is always positive. x 0 π 2π y=sinx 0 0 0 In plotting these points we will use approximations like instead of we will use 0.87 and instead of approximating π we will mark off the x-axis in terms of π. 1 P a g e
The zeros of y = sin x are at the multiples of π. And it is there that the graph crosses the x-axis. But what is the maximum value of the graph, and what is its minimum value? Sin(x) has a maximum value of 1 at, and has a minimum value of -1 at, and at all angles that are coterminal to them. Here is the x-axis marked off in terms of π, with the y-axis as the approximations of sin(x). Now just connect the curve Amplitude Period 2π 1 cycle The graph of the sine function allows us to visualize some of the properties of the sine function: 1. The range of is (i.e amplitude) 2. The domain is 3. The period is 2π. The graph s pattern repeats in every interval of length 2π. 4. The function is an odd function: this can be seen by noticing that the graph is symmetric with respect to the origin. 2 P a g e
When wanting to graph variations of it is helpful in noticing some key points of the graph of. Draw the picture of the sine function below: The key points to notice are the X-intercepts Maximum point Minimum Point One complete cycle of the sine curve includes 3 x-intercepts, one maximum and 1 minimum point. The x-intercepts are at the beginning, middle and end of the cycle. The x-intercepts are at (0,0) (π, 0) (2π, 0) The curve reaches its maximum point of the way through the cycle. The curve reaches its minimum point of the way through the cycle. Thus the key points in graphing sine functions are obtained by dividing the period into 4 equal parts (quarters). 3 P a g e
The x-coordinates of the five key points are as follows: Graphing Variations of 1. Identify the amplitude and the period 2. Find the values of x for the five key points the three x- intercepts, the maximum point and the minimum point. Start with the value of x where the cycle begins and add quarter periods that is to find successive values of x 3. Find the values of y for the five key points by evaluating the function at each value of x from step 2. 4. Connect the five key points with a smooth curve and graph one complete cycle of the given function 5. Extend the graph in step 4 to the left or right as desired. 4 P a g e
Example Determine the amplitude of for. Then graph 1. = Period = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 5 P a g e
Example Determine the amplitude of for. Then graph 1. = Period = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 6 P a g e
What to do if the function is of the form, where B is the coefficient of x and. The B indicates the number of periods in an interval of length. The graph of y=sin(bx) Since the graph of has period, then the constant B in indicates the number of periods in an interval of length. (In.) For example, if -- that means there are 2 periods in an interval of length. If B= 3 y = sin 3x -- there are 3 periods in that interval: While if B = ½ y = sin ½x -- there is only half a period in that interval: 7 P a g e
The constant B thus signifies how frequently the function oscillates; so many radians per unit of x. Amplitudes and Periods: The graph amplitude = has Amplitude period = Period: The graph of is the graph of horizontally shrunk by a factor of if and horizontally stretched by a factor of if 8 P a g e
Example Determine the amplitude and period of. Then graph for 1. = Overall Period = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 9 P a g e
The graph of The graph of is obtained by horizontally shifting the graph so that the starting point of the cycle is shifted from. If then shift right. If then shift left. The number is called the phase shift. Amplitudes and Periods and Staring Points: The graph has Starting Point Amplitude Period: amplitude = period = staring point: The phase shift is the amount that the curve is moved in a horizontal direction from its normal position. The displacement will be to the left if the phase shift is negative, and to the right if the phase shift is positive. 10 P a g e
Example Determine the amplitude, period and phase shift of. Then graph one period of the function. 1. = Overall Period = Phase Shift = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 11 P a g e
The graph of. The graph of. The trigonometric functions can be graphed in a rectangular coordinate system by plotting points whose coordinates satisfy the function. Thus, we graph by listing some points on the graph because the period of the cosine function is 2π, we will graph the function on the interval [0,2π]. The rest of the graph is made up of repetitions of this period x 0 π 2π y=cosx 1-1 1 You can also use degree measurements: The zeros of that the graph crosses the x-axis. are at the multiples of. And it is there 12 P a g e
Here is the x-axis marked off in terms of, with the y-axis as as the approximations of cos(x). Notice that the graph of cox(x) is the graph of sin(x) shifted left by -2π - π 2π Amplitude Period 2π 1 cycle The graph of the cosine function allows us to visualize some of the properties of the cosine function: 1. The range of is (i.e amplitude) 2. The domain is 3. The period is 2π. The graph s pattern repeats in every interval of length 2π. 4. The function is an even function: this can be seen by noticing that the graph is symmetric with respect to the y-axis. Amplitudes and Periods: The graph amplitude = period = has Amplitude Period: 13 P a g e
Example Determine the amplitude and period of. Then graph the function for. 1. = Overall Period = Phase Shift = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 14 P a g e
The graph of The graph of is obtained by horizontally shifting the graph so that the starting point of the cycle is shifted from. If then shift right. If then shift left. The number is called the phase shift. Amplitudes and Periods: The graph has Starting Point Amplitude x-axis amplitude = period = Period: starting point: The phase shift is the amount that the curve is moved in a horizontal direction from its normal position. The displacement will be to the left if the phase shift is negative, and to the right if the phase shift is positive. 15 P a g e
Example Determine the amplitude, period and phase shift of. Then graph one period of the function. 1. = Overall Period = Phase Shift = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 16 P a g e
Vertical Shifts and Sinusoidal Graphs We now look at sinusoidal graphs of The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation The constant D in the above formula causes vertical shifts in the graphs of. If D is positive, the shift is D units upward. If D is negative, the shift is D units downward. These vertical shifts result in sinusoidal graphs oscillating about the horizontal line rather than about the x-axis, thus the maximum y is and the minimum is Period: Amplitudes and Periods: The graph amplitude = has Starting Point Amplitude d units period = x-axis starting point: 17 P a g e
Example Graph one period of. 1. = Overall Period = Phase Shift = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 18 P a g e
Velocity of Air Flow (liters per second) Modeling Periodic Behavior The below graph shows one complete normal breathing cycle. The cycle consists of inhaling and exhaling. It takes place every five seconds. Velocity of air flow is positive when we inhale and negative when we exhale. It is measured in liters per second. If y represents velocity of air flow after x seconds, find a function of the form that models air flow in a normal breathing cycle. Inhaling Time (sec) Exhaling Period 5 seconds 19 P a g e