Piecewise functions

• f(x) = {

• Plug in values appropriately and solve (if it doesn’t work put undefined)

• Graph using an open circle for less/greater than and a closed circle for less/greater or equal than

• Find the domain and range based on the graph (use {3} and U for singular values)

• Discontinous = break in the graph, x = [number]

• Show work for hole, vertex, and 2 points of parabola

  • To find the graph of functions with holes, factor parts and try to cross out anything that cancels

Greatest integer functions

• Marked by [[x]]

• 

• Apply transformations as usual

• To find values, find the greatest integer less than or equal to x and apply transformations

Writing absolute value as piecewise

• Set the original greater than or equal to 0

• Solve for x, that is your first “for”

• Find the opposite (switch all signs) of the original and set it greater than 0

• Solve for x again, that is your second “for” and should be opposite

Synthetic division (basic operation)

• P(x) is product aka what they give you, (x-c) is the divisor, q(x) is the quotient/your answer and r is the remainder

• 

• Put the c in the top left (c = -5 or x+5)

• Put all the coefficients of the equation in order with their signs (if a term is skipped put a 0)

• Pull down the first number

• Multiply each bottom number by the top left and put it under the next term (3 x -5 = -15)

• Continue until done

• The last number is the remainder, the first few numbers are coefficients of the quotient/answer

• Plug it into the forms given

Remainder Theorem

• If a polynomial is divided by x-c, the remainder is P(c)

• Use synthetic division on the P they give you, as is (if you have P(4) then put 4 in the top left and divide the equation as normal)

• The answer is the remainder/the last number

• Big degrees = big numbers

Factor Theorem

• A polynomial has factor x-c if and only if P(c) = 0

• If r = 0, x-c is a factor of P(x)

• Answer with yes or no

• Use the x = as your left number and divide normally

• If the remainder is 0 then it is a factor

Intermediate Value Theorem (IVT)

• If P(a) and P(b) have opposite signs, then P(x) has a real zero in the interval [a, b]

• Do synthetic division for both numbers

• If the sign for the remainder changes (one is pos, one is neg), then the theorem applies (there is a solution in the interval)

  • Change in sign. By IVT,

• If the sign does not change, the theorem cannot apply

  • Same sign. IVT does not apply.

Creating polynomials

• If you have an i or a radical, make another one that is the opposite sign

  • 3i → -3i

• Once that is done, duplicate existing terms (numbers) until you reach the degree you need

• Then multiply out

• If you have to multiply things with 3 terms (x + 1 + i)(x + 1 - i) use a 3x3 box

  • Put terms along the side and top

  • Multiply into the boxes

  • Add

• Combine like terms

Rational Zeroes Theorem

• The rational zeros will be equal to in lowest terms, where p is factors of the constant (last term) and q is factors of the leading coefficient (first number)

• List all factors of the last number with a plus and minus sign in front of each on top

• List all factors of the first number with a plus and minus sign in front of each on the bottom

• Rewrite as numbers and fractions

Descartes’ Rule of Signs

• The number of positive zeros is the same as the number of sign changes in the original function, or an even number less

• If you get 4, the number is 4, 2, or 0

• The number of negative zeros is the same as the number of sign changes in P(-x)

  • Switch the signs of the odd degree terms

• Count sign changes in the original and the -x

• Make a table with positive, negative, and complex

• Start with the highest number of positive and negative zeros, putting complex next to it if needed to fulfill the number of zeros (equal to highest degree)

  • 2, 2, 0

• Then put the lower number of negative zeros in the next row, filling in the difference with complex numbers

  • 2, 0, 2

• Put the lower number of positive zeros and the higher number of negative zeros

  • 0, 2, 2

• Finally put the lowest number of each and fill in with complex

  • 0, 0, 2

Finding zeros of a function

• Check for 1 and -1 first

• Add up all the coefficients, if the sum is 0 then 1 is a zero

• Add up all coefficients of f(-x), if the sum is 0 then -1 is a zero

• Synthetic divide by 1 and/or -1 if they are zeros

• Use the calculator to peek at what a zero might be

• Synthetic divide (if the remainder is 0 you found one)

• After each synthetic division use the quotient for the next division to make it faster

• Once you get a quadratic factor the quadratic

  • If it cannot be factored use the pythagorean theorem to find radicals/imaginary solutions

• List solutions in set notation

  • x = {2, 3, 9} or x ∈ {2, 3, 9}

• Write the factored form based on the solutions if necessary

Graphing a polynomial

• Leading term test

• If the term is positive, the right side goes up

• If the term is negative, the right side goes down

• If the degree is odd, the function behaves like a line (opposite directions)

• If the degree is even, the function behaves like a parabola (same direction)

• Write rises or falls

• To find the relative maxima/minima, use the calculator (2nd trace)

• To find the x-intercepts, find the zeroes using synthetic division (show work)

• To find the y-intercept, use the constant (last term) and write f(0) (or whatever letter) as work

• Graph by plotting all points (add some if needed) and connect the dots

Zero-multiplicity-cross/touch tables

• Factor and find zeros like normal

• Put the zeroes in the first column

• Determine the multiplicity (how many times each zero appears)

• Decide if cross or turning point

  • Maximum # of turning points = highest degree - 1

  • Cross if odd multiplicity, tp if even