Unit 7: Trigonometric Functions Graphing the Trigonometric Function

Graphing Trigonometric Functions Amplitude: the maximum or minimum vertical distance between the graph and the x-axis. Amplitude is always positive 2

If |a| > 1, the amplitude stretches the graph vertically. 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 = sin x y = sin x y = 2 sin x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x Amplitude

Graphing Trigonometric Functions Period: the number of degrees or radians we must graph before it begins again. 4

If b > 1, the graph of the function is shrunk horizontally. 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 0 < b < 1, the graph of the function is stretched horizontally. y x period: period: 2 If b > 1, the graph of the function is shrunk horizontally. y x period: 4 period: 2 Period of a Function

The sine function 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 90° 180° 270° 360° I II III IV sin θ θ 45° 90° 135° 180° 270° 225° 0° 315° θ sin θ π/2 1 π 3π/2 −1 2π Interactive Sine Unwrap

Sine is a periodic function: p = 2π One period 2π 3π π −2π −π −3π sin θ θ sin θ: Domain (angle measures): all real numbers, (−∞, ∞) Range (ratio of sides): −1 to 1, inclusive [−1, 1] sin θ is an odd function; it is symmetric wrt the origin. sin(−θ) = −sin(θ)

Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. -1 1 sin x x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = sin x Sine Function

The cosine function 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 45° 90° 135° 180° 270° 225° 0° 315° 90° 180° 270° 360° IV cos θ θ III I II θ cos θ 1 π/2 π −1 3π/2 2π

Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1 -1 cos x x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = cos x Cosine Function

Cosine is a periodic function: p = 2π One period 2π π 3π −2π −π −3π θ cos θ cos θ: Domain (angle measures): all real numbers, (−∞, ∞) Range (ratio of sides): −1 to 1, inclusive [−1, 1] cos θ is an even function; it is symmetric wrt the y-axis. cos(−θ) = cos(θ)

Properties of Sine and Cosine graphs The domain is the set of real numbers The rage is set of “y” values such that -1≤ y ≤1 The maximum value is 1 and the minimum value is -1 The graph is a smooth curve Each function cycles through all the values of the range over an x interval or 2π The cycle repeats itself identically in both direction of the x-axis 12

Sine Graph Example: y=5sin2X Amp=5 Period=2π/2 = π Given : A sin Bx π Amplitude = IAI period = 2π/B Sine Graph Example: y=5sin2X Amp=5 Period=2π/2 = π π/2 π π/4 3π/4

Cosine Graph Example: y=2cos 1/2 X 4 π Given : A sin Bx 4π Amplitude = IAI period = 2π/B Cosine Graph Example: y=2cos 1/2 X Amp= 2 Period= 2π/(1/2) 4 π 2π 4π π 3π

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. max x-int min 3 -3 y = 3 cos x 2  x y x (0, 3) ( , 3) ( , 0) ( , 0) ( , –3) Example: y = 3 cos x

Use basic trigonometric identities to graph y = f (–x) Example : Sketch the graph of y = sin (–x). The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. y x y = sin (–x) Use the identity sin (–x) = – sin x y = sin x Example : Sketch the graph of y = cos (–x). The graph of y = cos (–x) is identical to the graph of y = cos x. y x Use the identity cos (–x) = – cos x y = cos (–x) y = cos (–x) Graph y = f(-x)

Use the identity sin (– x) = – sin x: Example: Sketch the graph of y = 2 sin (–3x). Rewrite the function in the form y = a sin bx with b > 0 Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x period: 2 3 = amplitude: |a| = |–2| = 2 Calculate the five key points. 2 –2 y = –2 sin 3x x y x ( , 2) (0, 0) ( , 0) ( , 0) ( , -2) Example: y = 2 sin(-3x)

More Transformations We have seen two types of transformations on trig graphs: vertical stretches and horizontal stretches. There are three more: vertical translations (slides), horizontal translations, and reflections (flips).

More Transformations Here is the full general form for the sine function: Just as with parabolas and other functions, h and k are translations: h slides the graph horizontally (opposite of sign) k slides the graph vertically Also, if a is negative, the graph is flipped vertically.

More Transformations To graph a sine or cosine graph: Graph the original graph with the correct amplitude and period (like section 14.1). Translate h units horizontally and k units vertically. Reflect vertically at its new position if a is negative (or reflect first, then translate).

Examples Describe how each graph would be transformed:

Examples State the amplitude and period, then graph: x -2π 2π

Examples State the amplitude and period, then graph: x -2π 2π

Examples State the amplitude and period, then graph: x -2π 2π

Examples Write an equation of the graph described: The graph of y = cos x translated up 3 units, right π units, and reflected vertically.