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Math 5
| Question | Answer |
|---|---|
| parent function def | the simplest, most basic form of a function in a family of related functions. It shows the general shape and behavior of that type of function without any transformations (like shifting, stretching, or reflecting). |
| vertex form for a parabola | y = a (x-h)^2 + k |
| parent function for a parabola in vertex form | y = x^2 OR f(x) = x^2 |
| how to decide if a parabola opens up and down | is the a is positive, then it opens up if the a is negitive, then it opens down |
| how to find the axsis of symetry of a parabola (vertex form) | x = h --> take the (x - h) from the paraenthis and solve for x = h, h is the x-value -take the opposite value of h write this as an equation (cause it's a vertical line) |
| find the vertex of a parabola in vertex form | (h,k) -h is the value from your axis of symetry -k is taken from the equations (same sign) |
| how to know if the vertex of a parabola is a maximum or minimum | if the parabola goes down, its a max if the parabola goes up, its a min |
| what is the domain of a parabola | (-∞, ∞) IT IS ALWAYS THIS -a parabola continues forever horizontally in every direction no matter what |
| what is the range of a parabola (vertex form) | if it goes up: [K, ∞) if it goes down: [-∞, K) |
| how to find the x-intercept for a parabola in vertex form | plug y=0 into the equation and solve |
| how to find the y-intercept for a parabola in vertex form | plugging in x = 0 and solving there will only be one but it might be an imaginary |
| how do you know how many x-intercepts you will have for a parabola (vertex form) | take the real square root of something = 2 x-intercepts vertex touches the x-axis = one x-intercept (is -k/a = 0) (the right hand side is 0) -square root of a negitive number = no real x-intercepts & 2 imaginary x-intercepts |
| how to graph a parabola | (1, 1a) (2, 4a) -make sure to reflect each oint across the axis olf symetry |
| standrard form for a parabola | y = ax^2 + bx + c |
| parent function for a parabola in standard form | y = x^2 f(x)= x^2 |
| how to find the axsis of symetry (standard form) | x = -b/2a |
| how to find the vertex in standard form | 1. find the axsis of symetry (this tells you what x is) 2. plug x into the equation and solve for y |
| what is the range of a parabola (standard form) | up: [y, ∞) down: (-∞, y] |
| find the x-intercept in standard form | plug in y= 0 into the equation -this gives you a quadratic 1. factor and solve 2. quadratic formula 3. completeing the square |
| quadratic formula | |
| completeing the square | |
| how to find the y-intercept in standard form | plug in x = 0 -this is always y = c - (0, c) |
| given a polynomial divided by a monomial, how do you simplifiy | indivisually divide each term in the dividend by the divisor |
| given a polynomial divided by a binomial, how do you simplify | long division: divide 1st term of polynomial by 1st term of binomial multiply # by ENTIRE binomial subtract new binomial (multiply binomial by -1 cause ur subtracting) bring down next term repeat remainders go in a fraction over the divisor |
| WARNINGS FOR DOING LONG DIVISION (a polynomial divided by a binomial) | -the polynomials must be in descending order (highest power to lowest power) -if it is missing an exponet (x^3 + x +1) than add in the missing term with a coefficent of 0 (0x^2) |
| synthetic dividion (part 1 - set up) | -only use when dividing by a binomial 1. take the coefficients of your dividend in descending order 2. draw between last coefficient and 2nd last (the right of this line is the remainder) 3. solve for x in the divisor (called k) |
| synthetic division (part 2 - solve) | 4. bring down 1st coefficient, multiply by k, add it to the next coefficient, multiply by k , add to next coefficent, repeat 5. use these numbers as coefficents and write a depressed polynomial (polynomial with one degree less than original) |
| how to know if a binomial is a factor of a polynomial using synthetic division | if there is no remainder, then the divisor is a factor of the dividend (the remainder is called f(k)) f(k) = remainder |
| how to know how many roots/solutions/degrees/# of zeros a polynomial has | the highest power is the degree |
| how to find possible rational roots | 1. (p) find constant from equation & list it's factors 2.(q) list the factors of leading coefficent (coefficent of 1st term when the expression is in standard form) 3. make ratios: + or - p/q (use every combination of p and q possibe, no repeats |
| descrates rule of signs | based on how many times the sign changes from positive to negitive or negitive to positive, you can find how many of that sign your answers will have |
| how to apply the descrates rule of signs | 1. find the degree 2. plug a +1 & -1 into the equation 3. count the # of times the signs change (+ to - or - to+) for each equation --> this is the # of solutions for that category |
| how to apply the descrates rule of signs (part 2) | 4. add the # of times, if doesnt add up to the degree, there're imaginary solutions (the impaginary, positive, and negitive solutions should add up to degree) 5. if the + or - cateogry is = or > than 2, then subtract 2 from category & adjust i category |
| when you write a remainder as a fraction... | make sure you add the fraction, do not subtract it -if the fraction is negative, then make the numeratornegative EX: # + -#/# NOT # - #/# |
| what is the degree of a polynomial | the number of degrees is also the number of zeros, solutions, roots, answers EX: x^3 = 0 is also x = 0 x = 0 x = 0 |
| given one zero of a polynomial function, find all the zeros | do synthetic division with ther given zero as the K solve the depressed equation write your answers (just list the numbers) INCLUDE THE GIVEN ZERO IN UR ANSWER |
| given a polynomial, list all possible zeros | list all possible rational roots |
| given a polynomial, find all rational zeros | find the PRs choose one PR use synthetic division with the chosen PR as K if the chosen PR results in a remainder of 0, then it is ONE OF the rational zeros either: try every PR to find all 0s OR use the found PR to solve depressed equation |
| given a polynomial, factor f(x) into factors | find all zeros (find PR, try out synthetic division until a remainder of zero) make these zeros into linear factors EX: zeros: x, g, -y and: (x - x) (x - g) (x + y) |
| given a polynomial function, find all the complex zeros of each poly function | pick a PR, do synthetic division to see if it is a factor, if it is, there'll be a remainder of 0, solve depressed equation, list ur answers IF YOU TAKE THE SQUARE ROOT, put the + or - symbol -include the k (the chosen PR that worked) in your answer |
| Multiplicities | multiples of the same zero --> how many times you get the same answer -the multiplicitiy is the power EX: X^3 = o is x = 0 , x = 0, x = 0 --> multiplicity of 3 |
| find a poly function with the zeros given | f(x) = x^2 - sum (x) + product -any time you have a radical or an imagiary zero, always include it's conjugate (treat like another zero) |
| find a poly function with the zeros given ( 2 zeros and a conjugate) | use the equation f(x) = x^2 - sum (x) + product for the conjugates only. Then, multiply that equation by the linear function of the other zero |
| find the poly function with the zeros given (you have 3 zeros and 1 conjugate) | use the equation f(x) = x^2 - sum (x) + product on the conjugates 1st, then on the remaining 2. Then multiply both equations |
| If a polynomial needs to be divided by a product of several linear factors, you can perform sequential synthetic division for each linear factor. | |
| alternate way to find the poly function when given the zeros | -write each zero as a linear factor (x-#) -rember to include the conjugate if it is imaginary or a radical -them multiply the conjugates first, then multiply the rest |
| reivew imaginary numbers | |
| when solving a polynomial using sythetic division with a PR as your k, your answer might not be a quadratic hoe to simplify this quadratic? | List possible rational roots again using new polynomial Use synthetic division on the new polynomial again with a new k Repeat until you get a quadratic. solve and write the roots from the first and second part in ur answer |
| To determine if a parabola has no x-intercepts from its equation |
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Blessed Rectangle 10