New Functions from Old: Stretches and Shifts (1/28/09) Stretches: If you multiply y (the output) by a positive constant c, it stretches the graph vertically (if c > 1) or compresses it (if c < 1). If c is negative, it also turns the graph upside down! Multiplying x (the input) by c > 1 compresses the graph horizontally, etc. Shifts: Replacing y by y + k shifts the graph upward if k is positive and downward if k is negative. Replacing x by x – k shifts to the right if k >0, etc.

Clicker Question 1 The graph of f (x) = (x+2) can be gotten from the graph of f (x) = x 2 by shifting it A. left 2 and down 3. B. right 2 and up 3. C. left 2 and up 3. D. left 3 and up 2. E. right 3 and down 2.

New Functions from Old Obviously, one way to get a new function from old ones is to add, subtract, multiply, or divide them. For example, a polynomial is obtained by adding or subtracting certain “monomials”. For example, a rational function is obtained from dividing two polynomials.

New Functions from Old: Composite Functions Composite functions: If you apply one function, and then apply another function to the output of the first function, this is called a composite. Example: f ( x) = (3x – 4) 4 results from first applying the linear function 3x – 4 and then applying the “raise to the 4 th power” function. The composite of f followed by g is commonly denoted g  f (note the order!). More examples…

Clicker Question 2 The function f (t) = sin 2 (3t +2) is a composite of: A. a linear function followed by a trig function followed by a power function. B. linear followed by power followed by trig. C. trig followed by power followed by linear. D. trig followed by exponential followed by linear. E. linear followed by trig followed by exponential.

Trigonometric Functions Sin, cos, tan, and so on, are called “trigonometric” because their origins were in the study of right triangles. However, what they really should be called are Circular Functions or Periodic Functions, since their definitions are in terms of circles, and because they repeat themselves.

Radians Start with a unit circle (i.e., a circle whose radius is 1 unit) whose center is at the origin. An angle measured counterclockwise from the x-axis has measure t radians if its arc on the circle is t units long. Since the circumference is 2 , there are 2  radians in a whole circle, so 2  radians = 360°

Sin and Cos If we have an angle of t radians, the sin(t ) is defined to be the y-coordinate on the unit circle. The cos(t ) is the x-coordinate. Hence both sin and cos repeat themselves every 2  radians (i.e., they are periodic functions). Also, sin 2 (t ) + cos 2 (t ) = 1

Clicker Question 3 What is sin(  /4)? A. 1 B.  2 C.  2 / 2 D. ½ E.  3 / 2

Amplitude and Period The amplitude of any periodic function of time is half the distance between its highest and lowest points. The period is the shortest time before the function begins to repeat. Hence the function y = A sin(Bt ) has amplitude A and period 2  /B.

Tangent The tangent function is defined as tan(t ) = sin(t )/cos(t ). Note that this is simply the slope of the line which the angle t makes. Hence tan(t ) is periodic with period  rather than 2 .

Clicker Question 4 What is the tan(  /4)? A. 0 B. ½ C. 1 D.  2 E. undefined

Secant and Cosecant Though less commonly used, we need to know that: The secant (denoted sec) is the reciprocal of the cosine, and The cosecant (denoted csc) is the reciprocal of the sine. Hence, again, both these functions are period of period 2 .

Assignment for Friday Read Section 1.3. In Sec 1.3, do Exercises 1, 3, 9-17 odd, 23, 29, 31, 33, and 41. Read Appendix D as needed (many of the formulas will not be of use to us). In Appendix D, do Exercises 1, 3, 9, 13, 23, 29, 37, 65, 67, and 77.