Unit 8: Graphing Trig Functions

1 cycle includes

a minimum and a maximum

reciprocal

a number that, when multiplied by another number, gives 1 (flip the fraction upside down)... reciprocal ex's: csc(x) = 1 / sin(x)... sec(x) = 1 / cos(x)... cot(x) = 1 / tan(x)

When a number is in front of the equation (e: y = 2sin(x) or y = 1/2sin(x)) a number greater than 1 makes the curves

"taller" and a number between 0 and 1 makes the curves "smaller." (If the coefficient of x is greater than 1, then the wave is shorter, but if it is between 0 & 1, then the wave is longer)

in general, given y = a•sin(bx + c), the phase shift is

-c/b or bx + c = 0... *if the coefficient of x is NOT 1, it must be in factored form to see the shift clearly* (when looking at the original equation, - = to the right and + = to the left. BUT, when looking at the answer, x = __, - = left and + = right)

5 key points in a SINE pattern

I M. I. m. I n a n. i. n t. x. t. n. t e. i. e. i. e r. m. r. m. r c. u. c. u. c e. m. e. m. e p. p. p. t. t. t (Intercept, Maximum, Intercept, minimum, Intercept) (*Intercepts are where the function intercepts the midline*)

sinusoidal model coefficients for y = A sin (Bx) + C and y = A cos (Bx) + C

IAI = the amplitude or distance the sinusoidal model rises and falls above its midline... C = The midline or average y-value of the sinusoidal model (max & min)... B = The frequency of the sinusoidal model - related to the period, P, by the equation BP = 2π or p = (2π/b)... P = The period of the sinusoidal model - the minimum distance along the x-axis for the cycle to repeat.

Transformations:

change in amplitude: max & min are affected (ex: y = 3sin(x))... change in period: x-values are affected (ex: y = sin(2x)... vertical shifts (up & down): changes the midline/resting value (y-value)... phase shift (left & right): changed start point (x-values)... *reflections (over x-axis):* when the amplitude in the given equation is *negative*, a reflection occurs in which the *mins & max's switch.*

We can graph the cosine function, y = cos (beta), on a coordinate plane using the coordinate point

(beta, cos beta)

We can graph the parent function y = sin (beta) on a coordinate plane using the coordinate point

(beta, sin beta).

y = A tan [B(x - C)] + D

A = Amplitude(but not quite... waves have amplitude, and this is not a wave). A is a vertical stretch. A will stretch the distance from the middle to the two points NOT plotted on the midline... B = Affects the period. Since the basic period is π, our formula to calculate the period is: *Period = (π/b)*... C = the Phase shift (left and right). Moves in opposite direction of the symbol. WE SHIFT THE "ZERO" POINTS, AND BUILD AROUND IT... D = The vertical shift (up and down). Moves in the same direction of the symbol. (The range is always "all real numbers". The domain is always "all real numbers + (asymptote) + k (times the period) where k is an integer")

A number multiplied by its reciprocal is always

1... (the reciprocal of 1 is 1 and the reciprocal of -1 is -1)

Cosine is a periodic function with a period of

2π... For every x in the domain of the cosine function cos(x) = cos(x + 2π)

The coefficient b (absolute value) represents the

frequency of a periodic function

the reciprocal of a positive number less than 1 is a number

greater than 1... (1/2 --> 2 > 1)

the reciprocal of a positive number greater than 1 is a number

less than 1... (4 --> 1/4 < 1)

Sine starts at the

midline, cosine starts at a maximum

The function y = sin(x) is called a

periodic function with a *period* of 2π.... Notice that each portion of the graph in an interval of 2π is one *cycle* of the sine function For every x in the domain of the sine function sine(x) = sin(x + 2π) (domain = all real numbers // range = -1 ≤ y ≤ 1)

Depending on which portion you look at, the cosine and sine graphs look similar, so what's the difference? REMEMBER:

sine and cosine are COFUNCTIONS

end point

start point + 1 period

To determine the midline value from a graph

take the average of the two y-values from a maximum and minimum point (y1 + y2)/2 = midline

increment

the amount or degree by which something changes - divide period by 4

The period of a trigonometric function is

the distance (x-value) it takes to complete one cycle (i.e. from max to max)

phase shift

the horizontal translation of a trigonometric function (ex: y = sin (x - π/2) --> the sine function is shifted right π/2 units)

The frequency of a trigonometric function is

the number of cycles it completes on a given interval (for sine and cosine that interval is 2π)

The reciprocal of 0 is

undefined (1/0)

Given curves in general form:

y = A sin (Bx) + C... the value of C is called the *midline* or resting value of the trig function. It's the height of the horizontal line that the sinusoidal curve rises/falls above and below by a distance of IaI (the amplitude)

Each cycle of a sine/cosine curve has

Five key points which will be used to help us graph them

How to write a sinusoidal equation given a graph:

1.) identify the min/max to find the amplitude... 2.) Identify one cycle and use the difference of the x-coordinates to find the period... 3.) Find the value of b using the formula: Period = 2π/IbI (b = 2π/IperiodI)... 4.) Phase Shifts: - for sine, identify a point on the left or right of the origin that intersects the midline (x-axis for graphs that didn't shift vertically)... - for cosine, identify a point on the left or right of the origin that is a max or a min (*use the opposite sign for phase shift*)... 5.) Calculate the midline by finding the average of the max/min. This will be your vertical shift. (y1 + y2/2)...

Period =

2π/IbI

y = Asin(B(x - C) + D

A = Amplitude (height of the wave) // B = Frequency (number of cycles over 2π) // C = Phase Shift (horizontal movement) // D = Midline (Vertical Movement)

5 key points in a COSINE pattern

M. I. m. I. M a n. i. n. a x. t. n. t. x i. e. i. e. i m. r. m. r. m u. c. u. c. u m. e. m. e. m p. p. p t. t (Maximum, Intercept, minimum, Intercept, Maximum) (*Intercepts are where the function intercepts the midline*)

There is an inverse relationship between period and frequency, therefore:

Period = 1/f & f = 1/period (for us, one unit is 2π)

When graphing trig functions, your calculator must be in

RADIAN mode (mode --> radian) (also change your window accordingly using π and your x min/max)

Amplitude

The amplitude of a periodic function is the height from the *rest value* to the maximum/minimum vale (the number in front). The amplitude is the absolute value of one-half the difference between the max/min y-values (a = I (0.5)(max - min) I ) (For y = a•sin(x) and y = a•cos(x), the amplitude is IaI )

asymptote

a line that a graph approaches but never crosses. Asymptotes = (Period/2) (then add to the start point)

To determine the range of a trigonometric function

add and subtract the amplitude from the midline value (ex: y = 7sin(x) - 4 ... range = [-4-7, -4+7] --> [-11, 3])

start point

bx + c = 0 (solve for x)

When we graph a tangent curve, we will

phase shift the middle point ("zero") first, then place an asymptote one-half period in each direction! Once the pattern is established by one period, we will repeat it again.