Algebra: Lap 9: Graphing and Analyzing Trig Functions

Formula for finding "endpoints" / asymptote location for a tan graph:

-90<(bt+c)<90

principle axis / midline / mean line

-The horizontal line that the function oscillates about (prior to any shifting, the midline is the x axis, y=0) -midline: y=d

What is a periodic function?

-a function that repeats itself over and over against in a horizontal direction -ex: sin, cos, and tan functions

What is a trigonometric identity?

-a trigonometric equation that is true for all values of the variable in the domain -Pythagorean identities, reciprocal identities, quotient identities, periodicity identities, cofunction identities, negative angle identities

What does periodic mean?

-happening at regular intervals -Ex: the tidal variations in the depth of the water fluctuate in a cyclical pattern throughout the day

b

-horizontal stretch by a factor of 1/b if 0<b<1 -horizontal shrink by a factor of 1/b if b>1 -reflection over the y axis (or any other vertical line) if b is negative -in an equation, you must factor out the coefficient on the x in order to see what b and c truly are

formula for finding max and min points

-sin has max at 2 and min at 4 -cos has max at 1 and 5 and min at 3 -make sure to reverse these locations if there is a shift (i.e. -a flips the graph) -min and max points are always the length of the amplitude away from the midline

formula for midline intercepts

-sin has midline intercepts at positions 1, 3, 5 -cosine has midline intercepts at positions 2, 4

What is an even function?

-symmetrical about the y axis -ex: cosine and secant

amplitude

-the distance between the midline and either the max or min point -always positive because its distance -tan graphs have an undefined amplitude - amplitude = IaI

what is an odd function?

-the graph maps back onto itself when rotated around the origin -ex: sin, csc, and tan, cot

period

-the length of one cycle of a periodic function -the chunk of the graph before it starts to repeat again -for sin and cos: period = 360/b or the difference of the endpoints -for tan: period = 180/b or the difference of the endpoints

c

-the same thing as h for other functions -if c is positive, then the function is shifted left c units -if c is negative, then the function is shifted right c units

d

-the same thing as k for other functions -if d is positive, then the function is shifted up d units -if d is negative, then the function is shifted down d units -the midline will be at y=d

a

-vertical stretch by a factor of a if a>1 -vertical shrink by a factor of a if 0<a<1 -reflection over the x axis (or any other horizontal line) if a is negative -IaI = amplitude for sin and cos

What does it mean to verify a trig expression?

-work to simplify ONE side of the equation -use the trig identities -make that side match the other side in order to "prove" that the trig expression was indeed true -If one of the chunks isn't squared, multiply by the conjugate (the binomial, just with the opposite sign) in order to apply the (a+b)(a-b)=a²-b² formula

Formula for Endpoints for sine and cosine

0≤(bx+c)≤360

Steps for graphing a tangent function:

1. Find the asymptotes ("endpoints") 2. Find the period 3. Find the spacing 4. Draw the asymptotes and five tick marks 5. draw the midline 6. draw the midline intercept at position 3 7. draw the two other points which are the amplitude's length from the midline 8. draw a curve through all three points

Steps for graphing sine and cosine functions

1. find the endpoints 2. find the period 3. find the spacing 4. Draw five tick marks 5. label the tick marks starting at the first endpoint and spaced correctly 6. draw the midline 7. draw midline intercepts 8. Find amplitude 9. Graph max and min points 10. Connect points

what are the parts of a periodic function?

1. principle axis/ midline / mean line 2. period 3. amplitude 4. maximum point 5. minimum point 6. crest / hill 7. trough / valley

formula for finding period of a tan graph

180/b or subtract the endpoints

Formula for period for sine and cosine

360/b or subtract the endpoints

What does a basic tangent graph look like?

ALWAYS TRUE -tangent is NOT a wave like the sin and cos graphs -instead, it is a series of "cubic" looking curves that reach towards asymptotes in every period -asymptotes at each period -one midline intercept at position 3 -domain: x ≠ asymptotes -range: y is all real numbers -amplitude = undefined -max value = none -min value = none ONLY TRUE OF A *BASIC* TAN GRAPH -starts low and ends high -y intercept: (0,0) -x intercept: (0+180n, 0) -period = 180 -midline: y=0

What does a basic cosine graph look like?

ALWAYS TRUE: -starts and ends OFF the midline -midline intercepts at positions 2 and 4 -max at positions 1 and 5 -min at position 3 -domain: x is all real numbers ONLY TRUE OF *BASIC* COSINE GRAPH -hill, valley, hill -range: [-1, 1] -y intercepts: (0, 1) -x intercepts: (90º + 180ºn, 0) -period = 360º -amplitude = 1 -max value = 1 -min value = -1 -midline: y=0

What does a basic Sine graph look like?

ALWAYS TRUE: -starts and ends ON the midline -midline intercepts are at positions 1, 3, & 5 -max point at position 2, min point at position 4 -domain: x is all real numbers ONLY TRUE OF *BASIC* SIN GRAPH: -hill, then valley -range: [-1, 1] -x intercepts= (0º + 180ºn, 0) -y intercepts= (0,0) -period = 360º -amplitude = 1 -min value = 1 -max value = 1 -midline: y=0

formula for finding spacing of a tan graph

period /4

formula for spacing for sine and cosine

period/4

What are the periodicity identities?

sin(x+360) = sinx csc(x+360)=cscx cos(x+360)=cosx sec(x+360)=secx tan(x+180)=tanx cot(x+180)=cotx -note: this basically just says that the trig functions for conterminal angles are equal -all have 360, but tan and cot use 180

What are the cofunction identities?

sine and cosine are cofunctions secant and cosecant are cofunctions tangent and cotangent are cofunctions

crest / wave

the "hills" of the graph that curve upwards

minimum / maximum point

the tip of a valley (min) or the tip of a crest (max)

What does it mean to simplify a trig expression into one type of trig function?

use the trig identities to reduce the expression so that it only contains one of the trig functions

formula for finding asymptote equation of a tan graph

x= (location of left most asymptote)+(period)n

What is the equation for a cosine function?

y = (a)cos(bx+c)+d

What is the equation for a sine function?

y = (a)sin(bx+c)+d

what is the equation for a tangent function?

y = (a)tan(bx+c)+d

What are the six trigonometric functions?

y = Sin(xº) y= cos(xº) y=tan(xº) y=csc(xº) y=sec(xº) y=cot(xº)

formula for midline

y=d

What are the negative angle identities?

even functions (cos,sec): f(-x)=f(x) odd functions (sin,csc,tan,cot): f(-x)=-f(x)

What is the general equation for all periodic functions?

F(x) = f(x+p), where p is the period of the function

What are the Pythagorean identities?

Note: these only work if its squared

trough / valley

The "lows" of the graph that curve downward

What is a sinusoidal function?

a graph or a function that can take the form of either a sine or cosine function depending on what period of the graph you look at (if it starts on the midline, then the sin will be easier to see; if it starts above the midline, then the cosine will be easier to see)

formula for amplitude

absolute value of a