Finals+Review

 § __ Slope __ - Take two points (2,4), (6,8) Next take y1-y2 /x1-x2 = 8-4/ 6-2 (It doesn’t matter which y you use first but the x of that group has to start.) You get an answer of 4/4 =1(If the fraction doesn’t come out to a whole number, it still works.) When graphing, if the slope is + then its up one over one if the slope is – then its down one over one. § __ Y intercept review- __ after graphing the equation you look at the graph and see if any part of the parabola crossed over the y axis and if it did then you mark how many points crossed. § __ Transformation- __ The form for a standard graph is f (x)= a (x-h)+k h  the graph moving right and left horizontally (h is always opposite if its –5 then you +5 when graphing.) The kto the graph moving up and down vertically. Transformation is the numbers you put into the equation so that the graph moves up, down, right, left, # units, reflection over x. ex, f(x-9) the graph would move right 9. Ex2, -f (x)+3 the graph would be doing a reflection because of the – in front of the f (x) and up 3. § __ Function Notation- __ f(x)=2x+3 and g(x)=2(x-3) when you see this f(2) and g(4) that means that f ‘s x is equal to 2 so you put 2 in for x in the equation (2(2)+3 then you get f=7.) You do the same for g and get g=2. § __ Function vs. Relation- a __ relation is a set of points given to you, such as, {(2,4), (3,5), (4,6), (9. -2), (3,7)}. What you have to do now is figure out if the points given is a function. Look at all the x numbers and if any of them come up more than once it is not a function. With those points 3 is stated twice therefore it is not a function. When stating the domain of points given in a line like the example, the domain is simply the x values but you do not have to repeat numbers if they are stated more than once. The domain of those numbers is (2,3,4,9)(in numerical order). When finding the domain of a graph you do the vertical line test, if you draw a vertical line through the function and it touches more than 1 point it is not a function. § __ Solve by completing the square- __ x^2+2x+4= 0, then subtract the 4 x^2+2x +_=-4, then take half of _x and add that number squared to both sides x^2+2x+1=-4+1 (x+1)^2=-3, then square both sides x+1=-+ i root 3, then subtract 1 and get, x=1 +,- i root 3 § __ Quadratic Formula- __ –b +- root b^2-4ac all over 2a 10x+3=3x^2, make it equal to 0 3x^2-10x-3=0, find a=3, b=-10, c=-3, then plug into the quadratic formula 10 +- root 100-4(3)(-3)/ 2(3) 10+- root 100+24/6 10+- root 124/6 10+-2 root 31/6 5/3+-1root31/3 § __  Extracting a square root-  __ (x-5)^2=28, next you square both sides x-5= +- root 28, then add 5 x=5+- 2root 7 § __ Factoring- __ x^2-9x+14=0 factor it out and you get (x-7) (x-2)=0 then solve for x, x =7 and x = 2 __ Special patterns- __ is when you can not factor completely but you can still simplify. Ex. 6//x^2//+ 6//x//, both numbers have 6x in common so you can factor that out, and get 6x(x+1) § __ Find the vertex of a parabola- __ starting out you want to get it in this form f (x)= a (x-h)+k ex. x^2+6x+7, do completing the square x^2+6x+_=-7 half of 6 is 3 so add 9 to both sides (x+3)^2=2 then instead of solving for x just put everything on 1 side 0=(x+3)^2 - 2 now just find vertex and the vertex is always opposite h and k   v= (-3, -2) § __ Add- __ 7/6x +5/8x =? First you want to find an LCD (least common denominator) They both go into 24 so you multiply the 7/6x (4/4)+5/8x(3/3)= 28/24x+15/24x, then add, and get //43/24x// § __ Subtracting- __ 3/3-x –5/x+2 the LCD is (3-x) and (x+2) multiply through and get 3(3-x)(x+2)/(3-x) – 5(3-x)(x+2)/ (x+2) the bottoms combine, so it looks like this, 3(x+2) - 5(3-x)/ (3-x)(x+2) do the subtraction and get = //–2x+21/(3-x)(x+2).// § __ Multiply- __ y^2+4y+4 / y^2+6y+9 X y+3/y+2 first simplify everything you can So it becomes (y+2)(y+2)/(y+3)(y+3) X y+3/y+2 the y+3 and y+2 cancel out and your answer is //(y+2)/(y+3)// § __ Divide- __ 4x/3 / 12x/15 next we use a step called KEEP CHANGE FLIP. This means you **keep** the first equation the same, **change** the sign to multiplication, then **flip** the second equation, so it looks like this, 4x/3 X 15/12x, then simplify 4 and 12 and 3 and 15 and your answer is, //5/3// § __ Synthetic division- __ x^3 +3x^2 –4x-10 / (x-2) write coefficients in order, 1 3 -4 -10 then times by the opposite of constant (2) __  2 10 12  __  1 5 6 2 > to write the answer, look at what you are dividing and take one less exponent. Now your answer is 1x^2 + 5x + 6 with a R of 2 ** Exponetials: ** g=2(1/2x+3)-3) the 2 and ½ cancel out F=1/2(2x-6)+3 take ½ of 2x-6 g=x+3-3 F= x-3+3 //g=x// //F=x//
 * Functions: **
 * Quadratics: **
 * Rationals: **
 * Inverses-you can tell if functions are inverses if you solve them and they answer each other. Ex. f(x) =1/2x+3 and g(x)=2(x-3)
 * f(g(x)) and g(f(x))

§ __ Logarithm and Natural log- __ Logarithm is taking an equation like this and 4^3=64 Log4 64=3 Solving these there are three ways, A) 4^x=8 first __make the bases the same__ (2^2)^x=2^3 set the exponents equal 2x=3   Solve for x and //x =3/2//    Second was is B) take inverse of both sides 3^x=5 next make it log3^x=log5 move the xlog3=log5 and solve for x by dividing log5 by log3, //x= 1.47// § __ Solving systems of 2 and 3 equations- __ There are two different types of ways to solve equations of 2 and 3. First is **substitution** 2x+y=7, 3x+2y=-4 solve the first of equation for y and you get y=-2x+7. Then plug in y in the second equation and get 3x+2(-2x+7)=-4.Then simplify and get 3x+-4x+14=-4 then –x=-18 and //x =18// next plug in x and solve for y. 2(18)+y=7, //y =-29.// §  The second way is **Elimination.** Instead of plugging in what y equaled you could have times the top by –2 and cancel out y and then add the x’s and the answers, then solve for x. (2x+y=7)-2, 3x+2y=-4 then the y’s cancel and you get –x=-18, x//=18// then solve for y.   §  __ Word problems- __ Three pens and two notebooks cost $ 8.25. Two pens and three notebooks cost $8.00. Find the cost of two pens and two notebooks. P= pens and N= notebooks, then set up the equations 3p+2n=8.25, 2p+3n=8.00, next times the first by 2 and the second by –3 and get 6p+4n=16.5, -6p-9n=-24 solve for n-5n=-7.5 then //n=1.5// Then plug in and get p, 2p+3(1.5)=8 then 2p+4.5=8 then 2p =3.5 solve for p and get //p=1.75//
 * Systems: **

LINKS: ( you can search anything you have a question on at purple math) http://www.purplemath.com/modules/fcnnot.htm http://www.purplemath.com/modules/logs3.htm http://www.purplemath.com/modules/rtnldefs.htm http://www.purplemath.com/modules/rtnlmult2.htm http://www.purplemath.com/modules/specfact3.htm