AP Pre-Calc Gold Sheet Review - Unit 1-4

concave up

• “like a cup”

• rate of change is increasing

concave down

• “like a frown”

• rate of change is decreasing

Average rate of change from x=a to x=b (AROC)

• Slope of a secant line (connects two points)

• f(b)-f(a)/b-a

• old school slope

instantaneous rate of change at x=a

• slope of a tangent line (only at one point)

  • estimate by finding AROC for points close to x=a

slope intercept form for linear functions

• y=mx+b

  • m = slope

  • b = y-intercept

point slope form for linear functions

• y-y1=m(x-x1)

  • m = slope

  • (x1 , y1) = point

standard form for linear functions

• Ax+By=C

vertex form for quadratics

• y=a(x-h)²+k

  • vertex = (h,k)

  • Axis of symmetry (AOS) = x=h

intercept form for quadratics

• y=a(x-p)(x-q)

  • p & q = zeros

standard form for quadratics

• y=ax²+bx+c

  • c = y-intercept

  • vertex = (-b/2a)

fundamental therom of algebra

• a polynomial of degree n has a total of n zeros (real and non-real)

quadratic formula

• x=-b+-square root b²-4ac/2(a)

  • find zeros

even function

if f(x)=f(-x)

odd function

if f(-x)=-f(x)

end behavior

limf(x) = limf(x) =

x →-infinity x→infinity

finding vertical asymptotes of a rational function

• factor the denominator and find what makes the factors equal zero

• x=#

finding a hole for a rational function

• factoring will help…if you have the same factor on top and bottom, then there is a hole

horizontal asymptotes

• compare the degree of the top and bottom

  • y=1

  • limf(x)=#

finding zeros/x-int. of a rational function

• factor the numerator and find what values make each factor = to zero

the binomial therom (a+b)^n

• (n0) a^n b^0 + (n1) a^n-1 b^1+…(nn) a^0 b^n

slant asymptote

• use synthetic division of long division to find a linear equation that represents the end behavior

• the numerator must be 1 degree greater than the denominator

synthetic division

• must be in the form (x+-a) to use this type of division

arithmetic sequences

• a sub n = a1+d(n-1)

  • d = common differnce

  • a1 = first term

• a sub n = a0+d(n)

  • a0 = zero term

• a sub n = ak+d(n-k)

  • k = any term

geometric sequences

• a sub n = a1(r)^n-1

  • r = common ration

• a sub n = a0(r)^n

• a sub n = ak(r)^n-k

exponential functions

• f(x)=a(b)^x

  • a = y-int

  • b = constant proportion/multiplier

• f(x)=y1(b)^x-x1

  • (x1 , y1) = any input-output pair

compound interest

• A = p0 (1+r/n)^nt

  • p0 = initial amount

  • r = rate as a decimal

  • n = number of times compounded per year

interest compounded continuously

• A = Pe^rt

  • P = initial amount

  • r = rate as a decimal

  • t = time

half life

• y=y0(1/2)^t/n

  • y0 = initial amount

  • t = time of half life