14-1 Graphing the sine, cosine, and tangent functions 14-2 Transformations of the sine, cosine, and tangent functions

What you’ll learn in section (14-1) 1.The appearance of the graph of: Sin x, Cos x, and Tan x 2. What the amplitude of each graph is. 3. What the period of each graph is. 4. How to find the x-intercepts of each graph.

Vocabulary Function: a rule that matches each input to exactly one output. Range: the set of all output values of a function (the ‘y’ values). What is a function? f(x) Domain: the set of all allowable input values of a function (the ‘x’ values).

Function Notation y = f(x) “y is a function of x” ‘y’ equals ‘f’ of ‘x’ A function is a rule that matches input values to out put values. f(x) = 2x + 1 (Input) x(rule) 2x + 1 2x + 1(output) y 2 2(2) f(2) = 5 3 2(3) f(3) = 7

The sine function (as opposed to the sine ratio) r = 1 x y Sine of the angle is the y-value of the point on the circle. is the y-value of the point on the circle

Sin (90) The Sine Function (Input) θ(rule) sin(θ) sin(θ)(output) f(θ) f(θ) 0 Sin (0) 0 30 Sin (30) The input value is the angle. The output value is the ratio (opp/hyp). The input value is plotted on the x-axis. 45 Sin (45) 60 Sin (60) Sin (120) 135 Sin (135) 150 Sin (150) 180 Sin (180) 0 + y + x

Graph of the Sine Function radians Output value: the y-value of the point on the circle corresponding to the angle. Input value: the angle

Graph of the Sine Function radians Think of a dot traveling around the circle to the right. At the exact same time, a corresponding dot is traveling along the x-axis to the left.

Graph of the Sine Function radians When r = 1, angle in radians equals arc length. equals arc length.

Graph of the Sine Function: radians Think of sin θ as the distance above (or below) the x-axis determined by ‘ θ ’ (the distance around the circle). determined by ‘ θ ’ (the distance around the circle).

Sinusoid 1. Domain = ? 2. Range = ? Your turn: 1

Sinusoides in the Real World Day light (Minutes/day) (Philidelphia, PA) Date 3. Approximately how many minutes of daylight were there on the shortest day of the year? Your turn: 4. And on the longest day about how many minutes of daylight were there?

The Cosine function (as opposed to the cosine ratio) r = 1 x y cosine of the angle is the x-value of the point on the circle. is the x-value of the point on the circle. The output of the cosine function of an angle (input value) is the x-value of the point on the circle.

Graph of the Cosine Function radians Output value: the x-value of the point on the circle of the point on the circle corresponding to the angle. corresponding to the angle. Input value: the angle

Graph of the Cosine Function radians Think of a dot traveling around the circle to the right. At the exact same time, a corresponding dot is traveling along the x-axis to the left.

Graph of the Cosine Function radians When r=1, angle in radians equals arc length. equals arc length.

Graph of the Cosine Function radians Think of cos θ as the distance to the right (or left) of the y-axis as determined by ‘ θ ’ (the distance around the circle). as determined by ‘ θ ’ (the distance around the circle).

I2 34 When the angle is between 0 and 90 (quadrant 1) What is the sign (+/-) of: + + +

I When the angle is between 90 and 180 (quadrant 2) What is the sign of in quadrant 2? in quadrant 2? + 6. What is the sign of when when 7. What is the sign of ofwhen Your turn:

θ (radians)sin θcos θtan θ Construct the Tangent Function y = Sin θ y= Cos θ

UNIT CIRCLE -60º -30º 60º 30º 0º0º0º0º 90º 1 -90º Vertical asymptote

UNIT CIRCLE 0º0º0º0º 90º 1 -90º Pattern repeats

Your turn: 8. What are the “zeroes” of the tangent function? (θ = ? For tan Ɵ = 0) (θ = ? For tan Ɵ = 0) 10. What are the vertical asymptotes of the tangent function? 9. How many “zeroes” does the tangent function have?

Your turn: 11. Why does the tangent function have vertical asymptotes?

Your turn: 12. f(x) = tangent θ 13. f(x) = sin θ 14. f(x) = cos θ Match the graph with its function: a. b. c.

The “Transformation Equation” Vertical stretch shift shift Horizontal shift Vertical shift Reflection across x-axis reflection

Without graphing predict how the graph of the following functions are going to be different from the parent function. functions are going to be different from the parent function. Vertically stretched by a factor of 2, shifted right 4 and up 5 Vertically stretched by a factor of 0.25, shifted left 1 and down

Compare: a: Vertical stretch factor Vocabulary: For the sine and cosine functions, we call the coefficient ‘a’ the amplitude of the function. 3 1

Vocabulary: The distance along the x-axis (the distance around the circumference of the circle in radians or degrees) that the function takes to make a complete cycle is called the period.

Vocabulary: the coefficient of the x variable (the input angle in either degrees or radians) is the reciprocal of the horizontal stretch factor. Horizontal stretch factor shrinks or expands the period.

What is the period of g(x) ? Compare: 1/b: horizontal stretch shrink factor = ?. What is the period of f(x) ?

What is the period of g(x) ? Your turn: What is the horizontal stretch shrink factor ?. What is the period of f(x) ? What is the amplitude of f(x) ?

Vertical and now horizontal stretch factors b: horizontal stretch shrink factor. shrink factor. a: Vertical stretch, shrink factor shrink factor Reflected across x-axis. Vertically stretched by a factor of 2. Horizontally shrunk by a factor of

Your turn: horizontal stretch shrink factor. shrink factor. 21. Describe how f(x) is a transformation of the parent function g(x). Reflected across x-axis. Vertically stretched by a factor of 5. Horizontally shrunk by a factor of

More transformations b: horizontal stretch shrink factor. shrink factor. a. Vertical stretch, shrink factor shrink factor Not Reflected across x-axis. Vertically stretched by a factor of 3. Horizontally stretched by a factor of Shifted right by π radians

Your turn: 22. Describe how the parent function has been transformed. Reflected across x-axis. Vertically stretched by a factor of 3. Horizontally shrunk by a factor of Shifted left by 2π/3 radians

Sinusoid 1.Amplitude: ( ½ of peak to peak distance) = ( ½ of peak to peak distance) = 2. Period: length of horizontal axis encompassing one complete cycle = one complete cycle = 3. Frequency = Frequency = 1/period If a is negative: reflection across x-axis

Sinusoid 1.Amplitude: ( ½ of peak to peak distance) = ( ½ of peak to peak distance) = 2. Period: length of horizontal axis encompassing one complete cycle = one complete cycle = 3. Frequency = Frequency = 1/period

Your turn: 23. What is the amplitude? ( ½ of peak to peak distance) = ( ½ of peak to peak distance) = 24. What is the period of the function? (length of horizontal axis encompassing one complete cycle) one complete cycle) 25. What is the frequency ? Frequency = 1/period = 1 cycle for every 3 radians

Sine vs. Cosine Function Cos x = sin x shifted to the left by

Your turn: describe the transformations of f(x) 26. Vertical stretch/shrink 27. Horizontal stretch/shrink 28. Horizontal translation (phase shift) 29. Vertical translation

Vertical stretch/shrink: factor of 3 If a < 0: reflection across x-axis: none Horizontal stretch/shrink by factor of: If b < 0: reflection across y-axis: none Horizontal translation: left by Vertical translation: up by 5 units

Sinusoid Vertical stretch/shrink If a is negative: reflection across x-axis Horizontal stretch/shrink by factor of: If b is negative: reflection across y-axis Horizontal translation (phase shift)

Your turn: describe the transformations of f(x) 30. Amplitude = ? 31. Period = ? 32. Frequency = ?