You can graph the equation, then have the calculator find the solutions/roots/zeros/x intercepts.

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1 To find zeros, if you have a quadratic, x 2, then you can use the quadratic formula. You can graph the equation, then have the calculator find the solutions/roots/zeros/x intercepts. Apr 22 10:39 AM Graphing piecewise functions 1. establish the left side and right side 2. graph the left side to the stopping point 3. graph the right side to the stopping point Apr 22 11:35 AM 1

2 Absolute value equations or inequalities cannot equal zero prior to removing the absolute value. 1. isolate the absolute value 2. if the constant is now negative, no solution 3. write two cases, one as given, one with the signs changed on the side without the absolute value (change directions of inequality) 4. solve Apr 22 11:36 AM Composition of functions f(g(x)), put g into the x's of f. Apr 22 11:39 AM 2

3 Finding inverses 1. must be one to one or have a restricted domain (pass the horizontal line test) 2. switch x and y 3. resolve for y 4. compose f(f 1 (x))=x Apr 22 11:40 AM Rate of change slope using two points include units on answers You are hiking, after 2 hours you have climbed 400 ft. After 5 hours you have climbed 650 ft. Apr 22 11:41 AM 3

4 Simplify either add or subtract, multiply Apr 22 11:46 AM Factor start with GCF How many terms? 2, 3, 4 2 terms sum or difference? squares or cubes? **sum of squares will have imaginary factors completely factor linear factors will include imaginary's Apr 22 11:47 AM 4

5 quotient Long Division: divisor dividend 1. Place what you need to multiply to the first term of the divisor on top 2. Distribute to the entire divisor 3. Place in correct spot under the dividend 4. Change signs (subtract away) and add 5. Bring down remaining terms and repeat 6. If there is a remainder, you need to write it over the divisor Oct 1 2:45 PM dividend (what you are dividing into) Only use coefficients of dividend always drop down divisor the first term coefficients go here 3 you will multiply the divisor to the number that has been dropped then place under the next coefficient divisor divisor add to the term above and place below 10 now multiply the divisor to the newest dropped number then place under the next coefficient add to the term above and place below multiply again and place under the last term divisor add to the term above and place below Aug 28 8:33 AM Bottom numbers will become the coefficients of the answer. 5

6 Solve Radicals 1. Isolate Square Root 2. Square both sides 3. Solve for x Apr 24 9:35 AM Simplify Rationals (Fractions) Multiplication/Division: 1. Factor the numerator 2. Factor the denominator 3. Once it is a multiplication problem, cancel out like terms from numerator to denominator that are exactly alike in ( ) 4. If dividing, flip the second expression and multiply Addition/Subtraction: 1. Make sure denominators are completely factored 2. Simplify individual fractions 3. Establish the LCD (LCM for the denominators) 4. Times each fraction by what it needs to achieve the LCD 5. Combine to one fraction and simplify the numerator Apr 24 9:43 AM 6

7 Rules for solving rational equations! 1. Factor the denominators 2. Establish the LCD and multiply to achieve LCD 3. Add or subtract fractions 4. Clear out denominators and solve the equation just using the numerators 5. Check excluded values for domain failures. Dec 10 7:24 AM Graphing Exponentials and Logs Exponents: standard form: a = f(x) = ab x h + k if a is negative, then b = if b > 1, then if 0 < b < 1, then h = k = asymptote: y = k ** if x is negative, then the base changes from a growth to a decay or a decay to a growth Logs: Jan 8 9:09 AM 7

8 Inverse Functions: 1. Make sure your function is Switch x and y 3. Solve for y 4. If in function notation, f(x), then use inverse function notation, f (x) Points: Switch the x and the y value and plot. Apr 24 9:47 AM You can circle around the log to write it in exponential form and from the exponential to the log form. Start at the base "b" and circle underneath to y and back to x. y = log b x x = b y log b x=y b y =x Solve for Logarithms 1. Set log equal to x 2. Change to exponential form 3. Guess and check Steps to solving for variables 1. Get the base (exponent/logarithm) by itself 2. Covert from exponential form to log form or from log form to exponential form 3. Solve for variable log expo expo log Jan 20 7:16 AM 8

9 Basic Properties of Logarithms For 0 < b and b 1, x > 0 and any real number y (base e, base 10 or any base) log b 1 = 0 because b 0 = 1 log b b = 1 because b 1 = b log b b y = y because b y = b y b log b x = x because log b x = log b x Apr 24 9:56 AM Solving Logs: 1. 1 log = a number, log 2 (3x + 2) = 8 a. write in exponential form b. solve 2. multiple logs = a number, log 2 (x) + log 2 (x 6) = 4 a. use log rules to combine the logs to one b. write in exponential form c. solve 3. 1 log = 1 log (or multiple logs) log(3x 5) = log(5x 19) a. combine each side as one log b. set the insides equal c. solve **** ALWAYS check your answers for extraneous solutions! Apr 24 9:59 AM 9

10 Special Right Triangles triangle H LL short leg is directly across from 30 degrees, long leg is across from 60 degrees, hypotenuse is across from the right angle Given: SL Short Leg: Hypotenuse: multiply SL by 2 (double it) Long Leg: multiply SL by square root of 3 ( ) Long Leg: Short Leg: divide LL by square root of 3 ( ) Hypotenuse: multiply what you found for the SL by 2 Hypotenuse: Short Leg: divide Hypotenuse by 2 Long Leg: multiply SL by square root of 3 ( ) Feb 2 8:24 AM Special Right Triangles triangle Both legs are the same! (opposite of 45 degrees) Given: L H Leg: L Hypotenuse: multiply Leg by the square root of 2 ( ) Hypotenuse: Leg: divide the hypotenuse by the square root of 2 ( ) Feb 2 8:37 AM 10

11 Solving for parts of all Right Triangles! 1. If given 2 sides of a triangle, use the Pythagoran Theorem to find the last side. 2. When looking for an angle, use the inverse trig functions to solve. 3. To find the 3rd angle, then subtract both given angles from 180. Angles are written with capital letters, sides are written in lower case letters. Feb 2 9:03 AM Law of Sines Testing for the number of triangles **must use this whenever you are using Law of Sines SIDE RELATIONSHIP Greater Equal Less > = < Angle > 90 One triangle No triangle No triangle Angle = 90 One triangle No triangle No triangle Angle < 90 One triangle one triangle Run test Test: Rule holds for B and b, C and c. This is written for A and a. If then no triangle If If then one triangle then two triangles or when solving for sides when solving for angles Apr 24 10:00 AM 11

12 Law of Cosines: 1. SSS triangle All three sides 2. SAS triangle Two sides with the angle in between To find a missing side a 2 = b 2 + c 2 2bccosA b 2 = a 2 + c 2 2accosB c 2 = a 2 + b 2 2abcosC You will square root to get the final side measurement To find a missing angle: cosa = (a 2 b 2 c 2 ) ( 2bc) cosb = (b 2 a 2 c 2 ) ( 2ac) cosc = (c 2 a 2 b 2 ) ( 2ab) 2nd cosine to get the final angle measurement Feb 8 11:38 AM A c b B a C Apr 24 12:11 PM 12