Unit 7: Trigonometric Functions Graphing the Trigonometric Function
2 Graphing Trigonometric Functions Amplitude: the maximum or minimum vertical distance between the graph and the x-axis. Amplitude is always positive
Amplitude 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. y x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x y = 2 sin x y = sin x
4 Period: the number of degrees or radians we must graph before it begins again. Graphing Trigonometric Functions
y x Period of a Function period: 2 period: 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 is. For b 0, the period of y = a cos bx is also. If, b > 1 the graph of the function is shrunk horizontally. If 0 < b < 1, the graph of the function is stretched horizontally. y x period: 2 period: 4
The sine function θsin θ 00 π/21 π0 3π/2−1 2π2π0 45° 90° 135° 180° 270° 225° 0° 315° 90° 180° 270° 0 360° I II III IV sin θ θ Imagine a particle on the unit circle, starting at (1,0) and rotating counterclockwise around the origin. Every position of the particle corresponds with an angle, θ, where y = sin θ. As the particle moves through the four quadrants, we get four pieces of the sin graph: I. From 0° to 90° the y-coordinate increases from 0 to 1 II. From 90° to 180° the y-coordinate decreases from 1 to 0 III. From 180° to 270° the y-coordinate decreases from 0 to −1 IV. From 270° to 360° the y-coordinate increases from −1 to 0 Interactive Sine Unwrap
Sine is a periodic function: p = 2π One period 2π2π 0 3π3π 2π2π π −2π−2π−π −3π−3π sin θ θ sin θ: Domain (angle measures): all real numbers, (−∞, ∞) Range (ratio of sides): −1 to 1, inclusive [−1, 1]
The cosine function θcos θ 01 π/20 π−1 3π/20 2π2π1 45° 90° 135° 180° 270° 225° 0° 315° 90° 180° 270° 0 360° IV cos θ θ III I II Imagine a particle on the unit circle, starting at (1,0) and rotating counterclockwise around the origin. Every position of the particle corresponds with an angle, θ, where x = cos θ. As the particle moves through the four quadrants, we get four pieces of the cos graph: I. From 0° to 90° the x-coordinate decreases from 1 to 0 II. From 90° to 180° the x-coordinate decreases from 0 to −1 III. From 180° to 270° the x-coordinate increases from −1 to 0 IV. From 270° to 360° the x-coordinate increases from 0 to 1
Cosine is a periodic function: p = 2π One period 2π2π π 3π3π −2π2π2π −π −3π 0 θ cos θ cos θ: Domain (angle measures): all real numbers, (−∞, ∞) Range (ratio of sides): −1 to 1, inclusive [−1, 1]
10 Properties of Sine and Cosine graphs 1. The domain is the set of real numbers 2. The range is set of “y” values such that -1≤ y ≤1 3. The maximum value is 1 and the minimum value is The graph is a smooth curve 5. Each function cycles through all the values of the range over an x interval or 2π 6. The cycle repeats itself identically in both direction of the x-axis
Sine Graph Given : A sin Bx Amplitude = IAI period = 2π/B Example: y=5sin2X › Amp=5 › Period=2π/2 = π π π/2 π/43π/2
y x Example: y = 3 cos x Example: Sketch the graph of y = 3 cos x on the interval [– , 4 ]. Partition the interval [0, 2 ] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. maxx-intminx-intmax y = 3 cos x 22 0x (0, 3) (, 0) (, 3) (, –3)