Unit 6: Quadratic Functions

what is the goal of writing a trinomial as a perfect sqaure?

(. )^2

what is center radius form and what does it show?

(x-h)^2+(y-k)^2=r^2 the center is (h,k) with opposite signs

How do you find the center and radius of a circle when given the general form?

1) first you have to convert it to center radius form convert by completing the square. a) put x's and y's next to each other just be rearranging them. move c, the number with no variable, over onto the right side with nothing else b)complete the square twice, once with x and once with y. Put parenthesis around the x's and y's , treat it as a binomial so find c by doing b/2=answer^2 c) once you found c in both, add or subtract both values of c onto the other side d) then condense by putting into ( )^2 form using b/2 answer this will get you the center radius form 2) from there, find the center by taking the opposite sign of k (h,k) find the radius by taking the square root of the =answer

what is a root like if the discriminant is greater than 1 and NOT a perfect square?

2 real non-equal irrational roots never ending decimals b.c not a perfect square

what is a root like if the discriminant is greater than 1 and a perfect square?

2 real nonequal rational roots

what is a root like if the discriminant is greater than 0? what are the two options here?

2 real nonequal roots if d is a perfect square then roots are rational if d is not a perfect square then roots are irrational

how can you remember the sign rules?

all pos=all pos all neg=1 pos 1 neg neg comes first=both neg neg comes second=neg second. neg second=same sign order as is written

what should you set equal to 0 when solving for x?

anything with an x in it inside or outside of the parenthesis, so you might have more than 2 solutions

what does the discriminant tell us?

discriminant tells what kind of roots/solutions the quadratic has

what do you do when asked to find x intercepts/roots/solutions algebraically?

first factor then set equal to 0 solve for x

how can you find the radius from center radius form?

whatever is the answer the you have in center radius form take the square root of that answer that square root answer will be your radius

what are the steps to completing the square? what should your answer look like?

1) in the quadratic equation ax^2+bx+c=0, move C to the right side of the equation by adding/subtracting it 2) take half of b and square that number. add that answer to both sides of the equation 3) factor the trinomial on the left side (it will be the same number as b/2's answer and same in both (. ) solve for x by consensing to get (x. )^2 and taking the square root your answer should be x=number plus or minus something in a radical sign

what are the 3 things that you will be asked for in a world problem about quadratic? what do you need to keep in mind about these 3 things?

x intercept y intercept turning point/vertex/maximum or minimum you will be given 2 or the information needed to find 2 your answer/what you are trying to find will be the third one

How do you write the equation of a vertical line?

x=

what is general form of a circle?

x² + y² + ax + by + c

how do you write the equation of a horizontal line?

y=

vertex form of a quadratic equation

y=a(x-h)^2+k (h,k) is the vertex point

intercept form of a quadratic equation

y=a(x-p)(x-q) p and q are the two x intercepts this is the factored form b.c the roots, since that is what you find when factoring where intersects with the x axis, so intercepts form the x intercepts

standard form of a quadratic equation

y=ax^2+bx+c a b and c are just numbers

how do you know when you see a quadratic function based on the form?

y=ax^2+bx+c the largest power has to be a square

what is the root like if the discriminant is equal to 0?

2 real equal roots (so 2 roots that have the same value) means the graph only goes through the x axis one time

how should you read inequality signs?

left to right

neg, neg

pos, neg

what does the discriminant refer to?

what is underneath the radical, doesn't include the radical

how do you know that you have a parabola by looking at a table of values?

when coordinate points repeat

when do you need to complete the square with circles?

when given general form and you have to get center radius form in order to find (h,k)

what are pieces of information that you can get from a quadratic function?

when you look at a graph, you should be able to list all of these things: domain and range increasing and decreasing intervals axis of symmetry vertex/turning point x intercepts/roots/solution y-intercepts where it is negative and positive intervals

how do you know where a y intercept is?

where x equals 0

how do you know where a root is?

where y equals 0

how should you write the axis of symmetry?

write y= or x= so just what you get for x or y

what is the quadratic formula?

a b and c from ax^2+bx+c

what is the shape of a quadratic function?

a parabola it looks like a U

whenever factoring what should you look for first? what if that doesn't work for a binomial? what about a trinomial?

always look for the GCF first then if don't have that can move onto AC, DOTS, quadratic formula binomial= 1)GCF 2) then DOTS trinomial= 1) GCF 2) then AC method if no GCF 3) if no AC then quadratic formula

why can something have roots and solutions but NOT x intercepts?

b.c when d is less than 0 the graph floats it still has solutions/roots, but they are imaginary numbers so they never cross the x axis so can't be called x intercepts b.c never cross x axis, since the roots/solutions are imaginary

what is the equation for the discriminant?

b^2 -4ac always get it to equal 0 before finding the value of the discriminant and stating the type of solutions

what are the two options in terms of shading?

both out (greater than) both in/between (less than)

what do you do if you see bx on the right side or just a number on the left side?

c and b are based off of how they look, NOT their positions so c is always just a number and b is always number x so if they are on the wrong side, move them over by adding/subtracting so that they are on the correct side

what do you do when asked which equation in (. )^2=number form is equal to a trinomial given?

complete the square, the answer that you get before taking the square root will be the equation that is correct

what is the root like if the discriminant is less than 0, meaning when you have a negative root?

complex/imaginary/distinct roots the graph floats b.c the roots are imaginary so they never touch the x axis. this means that there are roots/solutions but not x-intercepts

what can you do if the a is negative and you are solving a quadratic inequality?

divide by -1 so that all of the signs change also means that the inequality sign flips over

how do you find the solutions if you have a binomial, meaning that you don't have (. ) ( ) as the factored form?

draw a line between the x's set anything with x equal to 0 so set the thing with x outside of the parenthesis and the thing with x inside of the parenthesis equal to 0 ONLY DO THIS WITH A PROBLEM THAT STARTS WITH A BINOMIAL. WITH A TRINOMIAL JUST LEAVE THE OUTSIDE NUMBER OUT AND DO THE TWO (. ). SINCE YOU ONLY HAVE ONE (. ) WITH A BINOMIAL YOU HAVE TO USE THE OUTSIDE NUMBER

How do you factor using GCF?

find the biggest number that divides evenly into all terms, including all coefficients and variables do coefficient first, then pull out the smallest variable in the exponent as part of the GCF you will add the exponents to check, and mulitply the coefficients

how do you answer when asked to find the discriminate and state the type of solutions?

find the discriminate by plugging in the values then see if it is greater than, equal to, or less than 0 that answer will determine the nature of the solutions

how do you algebraically find the vertex of a parabola?

find the x value using x=-b/2a (the b and a are just the values from the equation) plug in the x value from step 1 into the quadratic equation. f(x) . this will get you the y value

what do you do if just asked to factor and not clear if should use GCF or DOTS?

first do GCF if there is no GCF, then to DOTS if there is a GCF, then see if in the ( ) from that there is a perfect square. if there is then use DOTS and keep the coefficient outside of it one time if no perfect squares, then keep answer ALWAYS GCF FIRST, THEN DOTS

how do you solve quadratic inequalities with one variable?

first factor the quadratic ignoring the inequality sign. get the x values using ac usually and setting ( ) equal to 0 then plot the x values on the number line test each interval to find the solution set/which way to shade, usually want to use 0 to see if it is included or not

what do you do when factoring and you have x^3 not just squared?

first find the GCF then do the AC method for the parenthesis, with your final answer being a binomial with the original coeffcient on the outside

what do you do if you are given a very long equation and told to find the solutions?

first simplify so that 0 is on one side and everything else is on the other then factor set each part/x in the factor equal to 0 and solve

when do you shade outside of the values? what about inside of them?

greater than=outside values less than=inside values use this to check your answer

when you are finding the vertex using algebra but you also have a graph, what is a good idea to do?

guess/approximate where the vertex will be based on your graph then once you do your algebra check it to make sure you didn't make a mistake in your calculation

how do you make a. perfect square trinomial? How do you find c? why is this needed?

half b then square it to get c this is one of the first parts of completing the square

how can you find vertex form from completing the sqaure?

have y=(condensed version in parenthesis)^2 with the other number negated (so if was neg=pos, pos=neg) vertex is the opposite sign of the inside comma the outside number as is will also tell you how many down or up (x) or right/left (y)

when given a problem that equals y and told to solve for x, what should you do?

have y=0 and then factor and set equal to 0 don't be confused by the y

what does completing the square help with?

helps go from standard to vertex form shifting=parent function moves, wnat to know how far down/up or over it went

what does the graph do if a<0, meaning that the number in front of x^2 is negative and less than 1?

if number in front of x^2 is less than 1 and negative, then the graph open down if negative, less than 1=looks like a sad face

what does the zero product law mean?

if the product of multiple factors is equal to zero then at least one of the factors must be zero (b.c 0 times ____ is 0)

what is the inside number in the ( ) always when completing the square?

inside number is always the b/2 answer but before it got squared so underline this when it is found b.c will need to use it again

from a graph how can you tell the nature of the discriminant?

intersects twice=positive d equal roots, intersects once=d is 0 floating parabola=d is a negative number

what do you do when asked between what two consecutive integers does the larger solution to a quadratic equation=o lie?

it is asking between what two integers does the largest x value lie, or the largest root/solution so look for roots and see what integers it is between. that is your answer

when you have a word problem that talks about the ground, what is that ground?

it is the x intercept b.c y is 0

what do you do if you have a quadratic and both x's are teh same?

just do it onceb.c the x's are equal so shading more simple

what is the only difference with inequalities with quadratics?

just have to factor first to find x instead of finding it algebraically

what do you do if yo have one of the center coordinates as 0 and have to write the equation of a circle?

just have x^2 or y^2 for that coordinate, no point in writing anything

what do you do when factoring and finding the solutions/roots/x intercepts and there is a number in front of the (. )?

just leave that number there the ONLY thing that you set equal to 0 in a trinomial is what is in side of the two (. )

what is the axis is symmetry?

line drawn that when folded, it would fold on itself write like y= or write like x=

what do you do when given a binomial?

look for difference of squares, meaning subtraction and two square roots then look for GCF

when does a variable count as a perfect square? what can be done with this?

means any even exponent/one that can be divided by 2 will be a perfect square if the exponent is bigger then you will still have a perfect square after DOTS=factor again with just the negative one, the positive (. ) stays in place

how do you factor trinomials when a is greater than 1?

multiply a and c, keeping the sign in front of it when you multiply find factors of that number, add to get b. remember to look at the signs to help you once you have those factors, start writing out the original expression only substituting the two factors with x as b. use the one that is easier to factor in each place you should end up with two (. ) that are the same. that is one of your answers, and the outsides together is the other answer

when do you use DOTS?2

must have 2 perfect squares and a subtraction sign between them not that 1 is a perfect square

neg, pos

neg, neg

how do you know if a function has a maximum or minimum?

negative quadratic/sad parabola=maximum positive quadratic/happy parabola=minimum

when you plug the x value back into the equation, that should be in the parenthesis?

only the x value, nothing else so if a negative existed in the equation, keep it outside the ( ) only negative inside is the one that came with the x value

when factoring, when should you use the quadratic formula? what should you start with?

only use when you can't factor using the ac method still start equalling the equation to 0 and then plug into formula

what should you look at to see if open or closed circle?

original inequality sign if has and equal to then closed if not then open

what is the goal of completing the square with a trinomial/parabola? what about with a circle?

parabola=find x values circle=find center of the circle

when told ot use a calculator to sketch a graph, what should you do?

plug in the equation look at the given x or y values and find the missing piece of the coordinate point in the table

how do you figure out where to shade with a quadratic inequality?

plug in the x value if it makes the inequality true then it is a solution, shade including the value if not true then it isn't a solution, shade away from the value usually do this with 0 b.c that will tell you to shade between are both away

how can you figure out where a vertex is based on a table of values?

points above and below it have the same y values vertex sandwiched between it

pos, neg

pos, neg

pos, pos

pos, pos

what does the graph do if a>0, meaning that the number in front of x^2 is positive and greater than 1?

positive and greater than 1=graph opens up if positive, greater than 1=looks like a smile if positive

how can you use the x and y values in completing the square to predict what might be the answer?

see what half of both b's are, that answer with opposite sign will likely be one of the center coordinate points

what should you start with when factoring trinomials?

see what the signs are, and write those in you can get credit just from those this is helpful b.c otherwise you might be led down the wrong path with the wrong terms

how do you find a solution form a factored binomial?

set each (. ) equal to 0 and solve draw a line down between any ( ) and separate them that way

how do you write set builder notation?

set of all x's that so {x| and then what is in problem} way 1 when shading going in different directions: xnumber U xnumber closed way 2 when shading in middle: number x xnumber so shading between=x between, shading in diff directions=U

what should you do if you are completing the square and the radical is a perfect square?

simplify it then add it one time and subtract it one time to get the two x values

How do you complete the square if a is more than1 1?

start normally, move c over then take the GCF of the binomial, not including any X s look at the (. ). the second number with x in it is b, so take half of that and then square it to get c write the c in the parenthesis, since your whole trinomial is in parenthesis then add the GCF outside of the parenthesis multiplied by the c inside of the parenthesis to the other side simplify and solve for x. divide by the GCF before taking the square root

when is a good idea to state the vertex when you are in the process of completing the square?

state the vertex before you take the square root so do it when you are in (. )^2=number form b.c it is easier to put it in then

How do you factor using DOTS?

take the square roots of both numbers multiply them by the conjugate, so the opposite sign

when you are given the equation of a circle and have to find the center, what do you do if there is just an x^2 or a y^2?

that is just 0, so x is 0 or why is 0 depending on which variable had no number with it

how can you find the center with center radius form?

the center is (h,k) but with the signs in front of h and k being the opposite of how they appear in the equation so if - in equation, positive in answer if positive in equation, negative in answer

How do you know what is the vertex when looking at a table of values?

the point that doesn't repeat in the table is the vertex should be framed by the same points

what do the factors become when using the zero product law?

the roots/solutions sometimes also the x intercepts

what is the vertex?

the turning point know where point is by looking at vertex form

what are the extreme values?

the values of x these and 0 go on the number line

what do you know abou the vertex when you get two of the same points when graphing?

the vertex will be a decimal between them when drawing the parabola, go a little above/below it to indicate that the vertex isn't in a straight line between them

when you have a starting point in a word problem what is that asking for?

the y intercept

what is the goal of completing the square?

to get x=

what is factor as a verb? what about a noun?

verb: to rewrite an equation/expression as an equivalent product noun: any individual component of a product

how do you write interval notation?

way 1 when shading going in different directions: write a U between two ( ) showing the starting and ending points, U shows where the shading skips way 2 when shading in same direction: just write one ( )