Test 1 Study Guide
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