Ch 4 Graphs of the Circular Functions

4.1 Graphs of the Sine and Cosine Functions

A periodic function is a function f such that f x  f x  np, for every real number x in the domain of f, every integer n, and some positive real number p. The least possible positive value of p is the period of the function. sin x  sinx  n2 

. The graph is continuous over its entire domain, (–¥, ¥). . Its x-intercepts are of the form np, where n is an integer. . Its period is 2p. . The graph is symmetric with respect to the origin, so the function is an odd function. sin(–x) = –sin x.

A function f is an odd function if for all x in the domain of f, f  x   f x cos x  cosx  n2 

. The graph is continuous over its entire domain, (–¥, ¥).  . Its x-intercepts are of the form2n 1 , 2 where n is an integer. . Its period is 2p. . The graph is symmetric with respect to the y-axis, so the function is an even function. cos(–x) = cos x.

A function f is an even function if for all x in the domain of f, f  x  f x Graphing y  asin x Vertical stretch or shrink

The amplitude of a periodic function is half the difference between the maximum and minimum values.

Graphing y  sin bx Horizontal stretch or shrink

For b > 0, the graph of y  sin bx will resemble that of y  sin x, but with 2 period . b Also, the graph of y  cosbx will resemble 2 that of y  cos x, but with period . b

Graph y  2sin x

Graph y  sin 2x Guidelines for Sketching Graphs of Sine and Cosine Functions

To graph y = a sin bx or y = a cos bx, with b  0, follow these steps.

2 1 Find the period, . Start with 0 on the b 2 x-axis, and lay off a distance of . b 2 Divide the interval into four equal parts. 3 Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and x-intercepts. 4 Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude a . 5 Draw the graph over additional periods as needed. Graph y  2sin 3x Reflection across x axis 2 Graph y  cos x 3

Graph y  3cosx Determine an equation of form y  a cosbx or y  asin bx, b > 0, for The average temperature in °F at Mould Bay, Nunavut, can be approximated by   f x  34sin x  4.3 , x is the month#. 6  a. 0,25by 45,45 b. Average temperature in May?

c. Average annual temperature? 4.2 Translations of the Graphs of the Sine and Cosine Functions

y  f x  d  Translations

The graph of the function y  f x  d  is translated horizontally compared to the graph of y  f x. The translation is d units to the right if d  0 and is |d| units to the left if d  0. With circular functions, a horizontal translation is called a phase shift.

   Graph y  sin x    3     Graph y  3cos x    4 

Graph y  2cos3x   over 2 periods Graph y  3 2cos3x over 2 periods

Further Guidelines for Sketching Graphs of Sine and Cosine Functions Method 1 1. Find an interval whose length is one 2 period by solving the three-part inequality b 0  bx  d   2 2. Divide the interval into four equal parts. 3. Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and points that intersect the line y  c. 4. Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude a . 5 Graph over additional periods, if needed. Method 2 1. Graph y  asin bxor y  a cosbx. The amplitude of the function is a , and the 2 period is . b 2. Use translations to graph the desired function. The vertical translation is c units up if c  0and |c| units down if c  0. The horizontal translation (phase shift) is d units to the right if d  0and |d| units to the left if d  0.

Graph y  1 2sin4x   over 2 periods The maximum average monthly temperature in New Orleans is 83°F and the minimum is 53°F. Using only the maximum and minimum temperatures, determine a function of the form f x  asinbx  d  c where a, b, c, and d are constants, that models the average monthly temperature in New Orleans. 4.3 Graphs of the Tangent and Cotangent Functions

A vertical asymptote is a vertical line that the graph approaches but does not intersect. As the x-values get closer and closer to the line, the function values increase or decrease without bound.

(1314) The line x  k is a vertical asymptote of f x if f x   or f x   as x approaches k from the left or the right.

Graph of tan x tan x   tan x    Domain: x x  2n 1 , n is an integer  2 Graph of cot x cot x  cot x Domain: x x  n, n is an integer Guidelines for Sketching Graphs of Tangent and Cotangent Functions  1. Determine the period, . To locate two b adjacent vertical asymptotes, solve the following equations for x:   y  a tan bx :bx   and bx  2 2 y  a cot bx : bx  0 and bx   2. Sketch the two vertical asymptotes found in Step 1. 3. Divide the interval formed by the vertical asymptotes into four equal parts. 4. Evaluate the function for the first-quarter point, midpoint, and third-quarter point, using the x-values found in Step 3. 5. Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph if needed. Graph y  tan 2x

1 Graph y  3tan x 2 1 Graph y  cot 2x 2

Graph y  2  tan x    Graph y  2  cot x    4 

Determine an equation for each graph 4.4 Graphs of the Secant and Cosecant Functions secant function f x  sec x sec x  sec x    Domain: x x  2n 1 , n is an integer  2 Range:  ,11,

1 sec x  cos x cosecant function f x  csc x csc x  csc x

Domain: x x  n, n is an integer Range:  ,11,

1 csc x  sin x Guidelines for Sketching Graphs of Cosecant and Secant Functions

1. Graph the corresponding reciprocal function as a guide, using a dashed curve. To Graph Use as a Guide y = a csc bx y = a sin bx y = a sec bx y = a cos bx

2. Sketch the vertical asymptotes. They will have equations of the form x = k, where k is an x-intercept of the graph of the guide function. 3. Sketch the graph of the desired function by drawing the typical U-shaped branches between the adjacent asymptotes. 1 Graph y  2sec x 2

3    Graph y  csc x   2  2  Determine an equation for each graph Addition of Ordinates New functions can be formed by adding or subtracting other functions. y  cos x  sin x

Find functional values (ordinates) for certain x, then for cos x and sin x. Then add to see cos x  sin x, and join the resulting points with a sinusoidal curve. 4.5 Harmonic Motion In part A of the figure, a spring with a weight attached to its free end is in equilibrium (or rest) position. If the weight is pulled down a units and released (part B), the spring’s elasticity causes the weight to rise a units ( a  0) above the equilibrium position, as seen in part C, and then to oscillate about the equilibrium position. Simple Harmonic Motion The position of a point oscillating about an equilibrium position at time t is modeled by either st  a cost or st  asint where a and ω are constants, with  0.

The amplitude of the motion is a , the period 2  is and the frequency is oscillations  2 per time unit.

Suppose an object is attached to a spring, and pulled down 5cm from equilibrium. The time for one oscillation is 4 sec.

Give an equation that models position wrt t.

Determine position at t 1.5sec

Find frequency. Suppose an object oscillates according to the model st  8sin 3t . Analyze. Damped Oscillatory Motion

Most oscillatory motions are damped by the force of friction which causes the amplitude of the motion to decrease gradually until the weight comes to rest. This motion can be modeled by st  et sin t  et  et sin t  et A typical example of damped oscillatory motion is provided by the function sx  ex cos 2x. We use x rather than t to match the variable for graphing calculators.

 ex  ex sin x  ex