Trig/Math Analysis Important Formulas and Info

key concepts to memorize for trigonometry and precalculus !!

Set Notation

D={x|xɛR} or R={y|yɛR}

• DOMAIN is x-values, remember to use x as variable

• RANGE is y-values, remember to use y as variable

Interval Notation

(-infinity, 5] or [7, 6] or (-infinity, infinity)

• Parentheses () for open points

• Brackets [] for closed points

Inequality Notation

-infinity < y < infinity and 5<=y<=9

• <= or >= for closed points

• < or > for open points

Difference Quotient

f(x+h)-f(x)/h

• h cannot equal 0

• same as slope (secant line on a graph, AROC) EXAMPLE: Find the difference quotient of f(x)=2x-1

• 1.) Replace x in the original function with (x+h), so 2(x+h)-1

• 2.) Take substituted function and subtract from original given function, so [2(x+h)-1]-(2x-1)/h

• 3.) Distribute, so [(2x+2h)-1]-2x+1/h

• 4.) Cancel out, and divide remaining h-terms by h; 2x and -2x cancel, -1 and 1 cancel, leaving (2h)/h ——— (2h)/h=2

• FINAL ANSWER: 2 — if you want to find the function of the secant line, then plug in resulting slope to point-slope form [y-y¹=m(x-x¹)]
  

Point-Slope Form

y-y¹=m(x-x¹)

• where m = slope and a known point = (x¹, y¹)

When listing multiple intervals demonstrating the same behavior (e.g. increasing, decreasing, constant), use…

u (symbol between intervals, meaning in unison)

EXAMPLE: Increasing from [-10, -4) u (5, 8)

Equation to determine if a function is an EVEN function

f(-x)=f(x)

• A function CANNOT be both even and odd; if passes even test, then there is no need to test for odd

• If it fails the even test, go on to test for odd

• Test even first! EXAMPLE: Determine if f(x)=-2x/x²-1 is even, odd, or neither

• 1.) Substitute (-x) for each x in the function; -2(-x)/(-x)²-1

• 2.) Distribute and simplify; 2x/x²-1

• 3.) Compare resulting function to the original; 2x/x²-1 ≠ -2x/x²-1, so NOT even

• 4.) Test for odd, substituting -f(x) into the original function; -(-2x/x²-1) — REMEMBER: An outside negative to a fraction will go to EITHER the numerator or denominator, NOT both (both would turn it positive)

• 5.) Distribute and simplify; 2x/x²-1

• 6.) Compare resulting function to your already-simplified f(-x) function from the even test; 2x/x²-1 = 2x/x²-1, so ODD function
  

Equation to determine if a function is an ODD function

-f(x)=f(-x)

• A function CANNOT be both even and odd; if passes odd test, then it is an odd function only

• If fails odd test, then it is NEITHER odd nor even

• See even function example
  

Average rate of change (AROC)

f(b)-f(a)/b-a

• a ≠ b

• usually given two x-values; plug x values into original function for both b AND a, simplify both functions, subtract b from a, and then divide by b plus a to find AROC

• Plug in one of the given x-values to original function to find y-value ——> (x, y)

• Once you have your AROC value and at least one point (x, y), plug into point-slope form for the equation of the secant line [y-y¹=m(x-x¹)]
  

Determine if an equation represents a function by…

Testing to see if an x-value will give you two different y-values (if yes, then NOT a function)

• Given a function, isolate y on one side (no degrees/powers on it)

• Using the isolated y function, substitute a number for x (usually 0) and solve

• If one answer, then yes a function; if multiple y answers, then not a function

A domain includes all real numbers, unless:

• There is an even index radical (e.g. square root, fourth root, sixth root)

• Expression inside radical must be GREATER THAN OR EQUAL TO ZERO (>=) EXAMPLE: fourth root of x+7 ; set x+7>=0 ; isolate variable x>=-7 ; D={x|xɛR, x>=-7}

• The equation has a denominator

• Denominator then must NOT be EQUAL TO ZERO (≠)

• EXAMPLE: 2x/x²-9 ; x²-9≠0 ; x²≠9 ; x≠±3

Distance formula

D = square root of (x²-x¹)²+(y²-y¹)²

Midpoint formula

M = (x²+x¹/2, y²+y¹/2)

Standard form of a line

Ax+By=C

• No fractions (if fractions, eliminate them)

• STANDARD stays on 0's side

General form of a line

Ax+By+C=0

• No fractions (if fractions, eliminate them)

• GENERAL runs from 0

Slope intercept form of a line

y=mx+b

• Fractions OKAY !!

• General form —> Standard form —> Slope intercept

• Slope intercept —> Standard form —> General form