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

# 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