Presentation on theme: "13.4 – The Sine Function. The Sine Function Use the graph. Find the value(s) of each of the following. 1.the period 2.the domain 3.the amplitude 4.the."— Presentation transcript:
13.4 – The Sine Function
The Sine Function Use the graph. Find the value(s) of each of the following. 1.the period 2.the domain 3.the amplitude 4.the range
The Sine Function Solutions 1.the period: 2 units 2.the domain: all real numbers 3. the amplitude: = 1 unit 4. the range: –1 y 1, where y is a real number 2222 < – < –
Graphing the Sine Function A Video Introduction
Highlights of the Sine Function The sine function, matches the measure of an angle in standard position with the y-coordinate of a point on the unit circle. Within one cycle of the function the graph will “zero” by touching the x axis three times ( ); reach a minimum value of -1 at and a maximum value of 1 at.
The Sine Function Use the graph of the sine function. a. What is the value of y= sin for = 180°? b. For what other value(s) of from 0° to 360° does the graph of sin have the same value as for = 180°? The value of the function at = 180° is 0. When y = 0, = 0° and 360°.
The General Equation Suppose:
The Sine Function Estimate each value from the graph. Check your estimate with a calculator. a. sin 3 sin 3 = 0.1411200081 Use a calculator to check your estimate. The sine function reaches its median value of 0 at 3.14. The value of the function at 3 is slightly more than 0, or about 0.1.
The Sine Function (continued) b. sin 2 sin = 1Use a calculator to check your estimate. 2 The sine function reaches its maximum value of 1 at, so sin = 1. 22
Graphing the Sine Function Sketch the graph of Steps: 1.Determine the amplitude. In this case a = 2. 2.Determine the period using the formula. This will be the outer boundary of your graph. Period = 3. Use five points equally spaced through one cycle to sketch a cosine curve. The five–point pattern is zero-max–zero–min–zero. Plot the points. 8
Graphing the Sine Function Sketch the graph of Steps: 4. Make a smooth curve through the points to complete your graph. 8
The Sine Function Use the graph of y = sin 6. a. How many cycles occur in this graph? How is the number of cycles related to the coefficient of in the equation? b. Find the period of y = sin 6. The graph shows 6 cycles. 2 ÷ 6 =Divide the domain of the graph by the number of cycles. 3 The number of cycles is equal to the coefficient of. The period of y = sin 6 is. 3
The Sine Function This graph shows the graph of y = a sin for values of a = and a = 3. 3434 a. Find the amplitude of each sine curve. How does the value of a affect the amplitude? In each case, the amplitude of the curve is | a |. b. How would a negative value of a affect each graph? When a is negative, the graph is a reflection in the x-axis. The amplitude of y = 3 sin is 3. The amplitude of y = sin is 1, and the amplitude of y = sin is. 3434 3434
Step 1: Choose scales for the y-axis and the x-axis that are about equal ( = 1 unit). On the x-axis, mark one period (4 units). Step 2: Mark equal spaces through one cycle by dividing the period into fourths. Step 3: Since the amplitude is 3, the maximum 3 and the minimum is –3. Plot the five points and sketch the curve. The Sine Function a. Sketch one cycle of a sine curve with amplitude 3 and period 4.
The Sine Function (continued) b. Use the form y = a sin b. Write an equation with a > 0 for the sine curve in part a. The amplitude is 3, and a > 0, so a = 3. The period is 4, and 4 =, so b =. 2 b2 b 2 An equation for the function is y = 3 sin x. 2
Divide the period into fourths. Using the values of the amplitude and period, plot the zero-max-zero-min-zero pattern. The Sine Function Sketch one cycle of y = sin 3. 5353 | a | =, so the amplitude is. 5353 5353 b = 3, so there are 3 cycles from 0 to 2. =, so the period is. 2 b2 b 2 3 2 3 Sketch the curve.
The Sine Function Find the period of the following sine curve. Then write an equation for the curve. According to the graph, one cycle takes 3 units to complete, so the period is 3. To write the equation, first find b. period = Use the relationship between the period and b. 2 b2 b 3 = Substitute. 2 b2 b b = Multiply each side by. 2 3 b3b3 2.094Simplify. Use the form y =a sin b. An equation for the graph is y = 5 sin. 2 3