Algebra Honors Unit Review of Algebra Day Combining Like Terms and Distributive Property Objectives: SWBAT evaluate and simplify expressions involving real numbers. SWBAT evaluate exponents SWBAT combine like terms SWBAT plug values into expressions SWBAT apply Distributive and Double Distributive Properties Exponent Vocabulary Base The number that will be multiplied Coefficient the number in front of an exponent (assume it s a if there is none). Exponent the number of times the base will be multiplied Examples Simplify the following expressions.. 4. ( ) 4. ( 5) 8 ( ) ( ) ( ) ( ) 6 ( 5)( 5)( 5) 5 4. 4 5. x 6. (y) 4 4 6 x ( y)( y)( y) 8y Adding and Subtracting Expressions Vocabulary Terms Like Terms A part or piece of an expression or equation separated by a + or Terms with the exact variable makeup Constant Simplified Number with no variable attached Combining all like terms or solving
Distributive Property Double Distributive Property (FOIL) a c d a( b c) ab ac b ac ad bc bd Examples Simplify the following expressions. 7. (7z +z) + (6z +z) 8. (9v 8w) (0w 5v) 7z z 6z z 7z 6z z z z 5 z 9v 8w 0w 5v 9v 8w 0w 5v 9v 5v 8w 0w 4v8w 9. x7x 4 0. x x x x 5 x (7x 4) 7 x 4 x x(x x x 5) 4 6 6 4 0 x x x x. (x 9)(x + 4). (x + 4)(x + 6) 6x 8x 7x 6 6x 9x6 4 x 8x 4x 4 4 x x 4
Imposters of Double Distributive For the following examples, explain why they are not a double distributive, and then simplify.. (x 5y) (4x + y) 4. (4m ) There is a +/- sign in-between the ( ) There is only one set of ( ) So it is not double distributive, it is so that means this is just Combining like terms distributive property x 4x 5y y x 8y (4m ) 8m 4 8m 7 5. (x )(5x x + ) 6. m (x )(5x x ) 0x x 6x 5x x 9 0x 7x 9x 9 mmm m m m 4 m m m m m 4m 4 m m 4m 4m m m m m m 4 8 4 8 m m m 6 8
Day Solve Linear Equations and Inequalities Objectives: SWBAT solve linear equations and inequalities. SWBAT use a formula. SWBAT isolate any variable in a formula or literal equation. GOLDEN RULE OF SOLVING LINEAR EQUATIONS Six Steps to help solve equations.. PEMDAS. Combine like terms that are on the same side of the = sign. If there are variables on each side of the = sign, find the smallest variable and move it to the other side. 4. Solve the two step equation by isolating the variable term 5. Multiply/Divide the coefficient of the variable so that there is a single positive variable = a number 6. Check your work Examples Solve the following equations. x. 8 7 x 8 7 8 8 x 5 7 x 7 5 7 7 x 5. 5 5 w. 5 m 6 w 5 5 5 5 8 w 5 5 8 5 w 5 0 w 5 m 6 5 m 6 6 6 6 m 6 m m m
4. n 6 n 4 5. 4x x 7 4x x7 n 6 n4 n n 6 n 4 4 4 8 n 6. x 4 6x 6 7. x 4 x 4x 0 7x 7 7 7 7 7x 7 7x 7 x 6 4 7 x (x) 4 6x 6 6x 4 6x 6 6x 6 6x 6 6 x 6x 6 6 All Real Numbers ( x 6) 4 7 x x 4 7 x x6 7 x x x 6 7 No Solution Formulas and Rewriting Equations 5 Example: The equation C F is used to change temperature from Fahrenheit to 9 Celsius. Find the equivalent Celsius temperature for each of the following Fahrenheit temperatures. 8. F 9. 70 C 5 C 9 5 C 80 9 Examples C 00 Solve each equation for the indicated variable A b b h, for b A b b h A b b h 0. a + b = c, for a. a b c b b a c b a c b a c b 5 70 F 9 9 5 9 70 F 5 9 5 6 F h A b b h b b A b b h F 58
Day Slope Intercept Form and Writing Equations of Lines Objectives: SWBAT graph lines SWBAT write the equation of lines from their graphs SWBAT write the equation of lines from a point and a slope Slope rise y y m run x x Slope intercept form y mx b m slope b y intercept y mx b begin with move with " b" " m" Examples State the slope and the y-intercept for each of the following equations.. y = x. x y. y = 5 4. x = m b m b 0 m 0 or no slope b 5 m undefined b doesn ' t have one Rewrite each equation into slope-intercept form. 5. x + y = 5 6. x y = 0 xy 5 x x y x5 5 y x x y 0 x x y x y x
Graph each equation. 7. f ( x) x 8. x y = 4 m or b Solve for " y" first y x m b Put a dot on - on y axis, Put a dot on - on y axis, then move up right then move up right 9. y 0. x 5 m 0 or no slope b m undefined b doesn ' t have one Horizontal Lines: y = any number Vertical Lines: x = any number
Write the equation of each line graphed below. a c d b e Line a: yx Line d: y 7 Line b: y x Line e: y x 6 Line c: x 9 Point Slope form of a linear equation Traditional Formula (H, K) y y m x x m slope x x y y x, y a point y a x h k a slope x x y y hk, a point Use point slope form to write an equation for each problem. Then change it to slope intercept form. Pick either formula to use they both will work. It passes through (4, ) with a slope of 4 y 4 x 4 y 4x6 y 4x8 y 4 x 4 y 4x6 y 4x 8
. Contains (4,) and the slope is undefined Since the slope is undefined means you will have a vertical line. x =. Contains the origin and ( 6, ) You must find the slope between those two points y y 0 m x x 6 0 6 You can now use either point so use (0,0) because it is the easiest y 0 x 0 y 0 x 0 y x y x 0 0 y x00 y x 4. Contains an x-intercept of and y-intercept of You must find the slope between those two points (,0) and (0,) y y 0 m x x 0 You can now use either point so use (0,) because it is the easiest y x 0 y x y x y x 0 y x 0 y x
Day 4 Solving Systems of Equations by Graphing and Substitution Objectives: SWBAT solve systems of equations by graphing. Objectives: SWBAT solve systems of equations by Substitution What is a system of linear equations? Two or more equations with the same variable make-up What is a solution to a system of equations? A Point Finding Solutions to Systems of Equations by Graphing ) Graph Both Lines ) Find where they intersect ) y = x + and y = x y x m b y x m b Intersection point Check, y x y x
Introduction to Substitution Take the value of one variable and plug it into the second equation. Find the solution to the system. x 6 y x. x 6 y 8 y x is already solved for, so plug it into the other equation. yx y 6 y y 9 y is already solved for, so plug it into the other equation. x6y8 x 6 8 x 8 x 0 x 0 6,9 0,
Solving Systems of Equations by Substitution ) Look at the system of equations. Decide which problem will be the easiest to isolate a variable. You have 4 ways to proceed WORKER SMARTER NOT HARDER ) Decide which variable you want to solve for, and isolate that variable. ) Plug or Substitute the equation into the variable into the other equation, solve for the single variable. 4) Substitute the value of the first variable, and plug it back into the first equation. Solve for the Second equation. This will produce the part of your answer. 5) Check it! Put both values into both equations to see if it checks out. 6) ***** If the two equations are the same, the answer is infinite solutions. 7)****** If the two equations are parallel, then there is no solution. ***** If your Math is good, it does not matter where you start substituting! The answer will be the same******* Examples Solve the following systems of equations using substitution.. x = y 4 and 8x y = 6 ) Let s look at the two equations, which equation would be easier to isolate a particular variable? (EQN since it is already set equal to x ) Take the y-4, and plug into the equation 8x -y =6. It should now read, 8(y+4) - y =6 ) Solve for y. Then take that y and plug the value of y into x=y-4 4) Solve for x. 5) Check it. 8( y 4) y6 x (8) 4 0 (8) 4 4y y6 y 6 y 68 y 8 x 4 4 x 0 0 4 4 8(0) 0 (8) 6 60 4 6 6 6
4. x y = 4 and x y = 4 xy4 xy4 ( y4) y4 6y 40 y4 5y 40 4 5y 60 y 5 x ( 5) 4 x 04 4 x 0 (0) ( 5) 4 90 5 4 4 4 0 ( 5) 4 0 04 4 4 4 5. x + 6y = 9 and x 5y = 55 x 6y 9 x 6y 9 x y ( y) 5y 55 4y 6 5y 55 9y 6 55 9y 8 y 9 x 6( 9) 9 (5) 6( 9) 9 x 54 9 5 54 9 x 5 9 9 x 5 (5) 5( 9) 55 0 45 55 55 55
Day 5 Solving Systems of Equations by Elimination Solving Systems of Equations by Elimination Objectives: SWBAT solve systems of equations by Elimination ) Put both equations into the Standard_ form. Usually this will be Ax + By = C Form (X AND Y ON THE SAME SIDE). ) Get Matching coefficients in front of either the x or y variables. ) Add the two equations together. Make sure this cancel out one of our variables. Solve the remaining equation. You just found _First part of your answer. 4) Substitute the value for the solved variable back into either equation. Then solve the remaining Equation. This is the other Part of your answer. 5) Write your answer as a X=, and Y=. Or as an ordered pair. 6) If both variables cancel out, then if it is a true statement then it is all real numbers, and if it is a false statement then it is no solution. 7) Check your Work. Examples Solve the following systems of equations using elimination.. x y 7 and 5x y 7. 7x y and 7x y 0 xy7 5() y 7 7x y ( ) 5x y 7 5 y 7 ( ) 7x y0 8x 4 y 8 4y x y 4 y 7x 0 7x 7x x
. 6y + 7 = 5x and x 9 = 4y + 4 4. x 4y = 8 and 9x y = 54 6y7 5x 7 7 6y 5x7 5 x 5x 6y 5x 7 6y5x7 5 4y x 8y5x 0y5x5 y 4 y x 9 4y4 9 9 x4y 4 y 4y x4y 6 7 5x 7 5x 5 5x 5 5 5 x x 4y 8 9x y 54 9x y 54 9xy 54 0x0y 0 Infinately Many Solutions 5. 8x + 5 = y and + y = 6x 8x 5 y 8 x 8x 5 y8x y 6x y y 6x y 8x y 5 6x y 8x y 5 6x y 6xy 0 6x y 0x 0y No Solution
Day 6 Solving Systems of Equations with Variables Objectives: SWBAT to solve a system involving three equations and three unknowns. Ordered Triple - THE ELIMINATION METHOD FOR A THREE-VARIABLE SYSTEM ) Write two Elimination Equations, (top two and bottom two but you can pick any two) ) Eliminate a variable the first equation (you should be left with an equation with two variables). ) Eliminate the same variable in the second set of equations (you should be left with an equation with two variables). 4) Line up the equations with only two variables, and perform Elimination again. Then solve that equation (you will have one part of your ordered triple). 5) Back substitute that solution into one of the equations with two variables to get the second answer (you will get the second part of your ordered triple). 6) Take those two answers, and back substitute into one of the original equations for your third answer. 7) Write you answer as an ordered triple and Check your work. Examples on the next page.
Examples: Solve the System.. 4x y z. x y 5z 4 6x y 4z x y z 0 6x y z x 4y z 7 Write two Elimination Equations, (top two and bottom two) 4x y z x y 5z 4 x y 5z 4 6x y 4z x y z 0 6x y z 6x y z x 4y z 7 Eliminate a variable the first equation (you should be left with an equation with two variables). 4x y z x y z 0 x y 5z 4 6x y z 4x y z 4x 6y 0z 8 8y7z9 Eliminate the same variable in the second set of equations (you should be left with an equation with two variables). x y 5z 4 6x y 4z 6x y 4z 0 6x y z xz8 6x y z x 4y z 7 6x 9y 5z 4 6x y 4z 8yz4 x 4y z 4 x 4y z 7 x5z Line up the equations with only two variables, and perform Elimination again. Then solve that equation. xz8 8y7z9 x5z 8yz4 8y 7z 9 8yz4 8y 7z 9 8yz 4 4z 4z 4 z 5 xz8 x5z 60x5z90 9x5z 9 99x 99 99x 99 99 x
Back substitute that solution into one of the equations with two variables to get the second answer. 8y7z9 8y 7 9 8y 9 8y 8 8y 8 8 y x5z 5z 5z 5z 0 5y 0 5 y Take those two answers, and back substitute into one of the original equations for your third answer. 4x y z 4x 4x 9 4x 7 4x 8 4x 8 x 4 6x y z y 6 6 y y 4 y 6 y 6 y Write all three as an Ordered Triple This is your answer z y x,, x z y,,
. 4x y 5z 9 x y 8z x y z Write two Elimination Equations, (top two and bottom two) 4x y 5z 9 x y 8z x y 8z x y z Eliminate a variable the first equation (you should be left with an equation with two variables). 4x y 5z 9 x y 8z 4x y 5z 9 9x y 4z 6 Eliminate the same variable in the second set of equations (you should be left with an equation with two variables). 5x 9z 44 x y 8z x y z x y 8z x y z x 5z 8 Line up the equations with only two variables, and perform Elimination again. Then solve that equation. 5x 9z 44 x 5z 8 5x 9z 44 5 x 5z 8 5x 9z 44 5x 5z 40 4z 4 4z 4 4 z
Back substitute that solution into one of the equations with two variables to get the second answer. x 5z 8 x 5 8 x 5 8 x Take those two answers, and back substitute into one of the original equations for your third answer. x y z y 6 y y 9 y 4 Write all three as an Ordered Triple This is your answer z x y 4, 4, Special Cases 5. x 4y 0z 4 6. x y 5z 4 x 4y z 5 x y z x y z 9 x 4y 4z NO SOLUTION INFINATELY MANY SOLUTIONS
Day 7 Domain and Range Objectives: SWBAT state the domain and range from a set of points. SWBAT state the domain and range from a graph. Vocabulary ~ Domain~ The set of all inputs or x-coordinates of a function and/or graph Range~ The set of all outputs or y-coordinates of a function and/or graph Find the domain of the following sets of points Domain x coordinates inputs. (, 5), (5, 6), ( 0, 5), (, 0). ( 5, 7), (, ), (, ), (5, 5) Domain: 0,,, 5 Range: 6, 5, 0, 5 Domain: 5,,, 5 Range: 5,, 7 Find the domain and range of each function Domain: -9, -7, -5, -, - Range: -0, -6, -,, 6 domain: 9 x 9 Range: y 9
Domain: 4x 5 Range: 5 y 8 Domain: 8 x 0 Range: 8 y 8 Domain: 7 x 7 Range: 0 y 7 Domain: 4 x 5 Range: 6 y 6 Find the domain and range of each function Domain: x Range: y 0 Domain: x 8 Range: y
Domain: ARN Range: y Domain: x 0 Range: ARN Domain: ARN Range: ARN Domain: ARN Range: y 9
Open Interval Closed Interval Infinite Interval (Positive) Infinite Interval (Negative) Day 8 Notation Objective: SWBAT write domain and ranges in set, interval, and inequality notation Words A set of numbers greater than a and less than b A set of numbers greater than or equal to a and less than or equal to b A set of numbers where x is greater than a A set of numbers where x is less than or equal to b Inequality Notation Set Notation Interval Notation a < x < b {x a < x < b} (a, b) a x b {x a x b} [a, b] x > a x < b {x x > a} {x x a} {x x < b} {x x b} (a, + ) [a, + ) (, b) (, b] Graph Given the following graph, write the domain and range in inequality, set, and interval notation. Also, describe using words.. D Words Inequality Set Interval The domain is in-between -9 and 9 inclusive 9 x 9 {x 9 x 9} [ 9, 9] R The range is in-between and 9 inclusive y 9 {y y 9} [, 9]
D Words Inequality Set Interval The domain is greater than x > {x x > } (, ) R The range is less than - y < {y y < } (, ) D R Words Inequality Set Interval The domain is greater than or equal to - x {x x } [, ) The range is greater than or equal to y {y y > } [, ).
Words Inequality Set Interval D The domain is all real numbers < x < {x < x < } (, ) R The range is less than or equal to y {y y } (, ]
Day 9 Piecewise Functions Day Objectives: SWBAT graph functions with restrictive domains SWBAT evaluate piecewise functions Piecewise Function Functions or graphs that are formed by piecing together two or more functions onto one coordinate plane. Close Dot Indicates if an endpoint is included with that particular piece of the graph Open Dot Indicates if an endpoint is not included with that particular piece of the graph Restricted Domains When the domain is restricted to include a set of specific numbers Evaluating the piecewise functions given the following domains x, if x f ( x ) x, if x x, if x. when x = 0. when x = 8. for f( ) 4. for f(0) x, if x f (x) x, if x x, if x f 0 0 f 0 f 0 0 x, if x f (x) x, if x x, if x f 8 8 f 8 f 8 f (x) x, if x x, if x x, if x f (x) x, if x x, if x x, if x f f f f 0 00 f 0 0 f 0 00
Graph the following functions: You are not graphing the whole line, just the part that the restrictive domain tells you to.. f x = x, if x. f x = x 5, if x 5. f x= x 7, if x 6. f x = x 5 if x 6
Day 0 Piecewise Functions Day Graph the following piecewise function. Objectives: SWBAT graph piecewise functions 5 x if x. g( x ) x if x x, if x. f ( x ) x, if x x. f = x 4, x x, x x, x4 4. f x= x, x 8, x 4
Step functions A piecewise functions that involve all horizontal lines., 7 x 5 5. f x= 0, x 4, 4 x 8 6. Write a piecewise function for the graph shown below. if 0 x f x if x if x Given the following graphs, write a piecewise function. 7. 8. f x x if 0 x x 4 if x f x x if x 4 4 x if x
9. In 005, the cost C (in dollars) to send U.S. Postal Service Express Mail up to 5 pounds depended on the weight w (in ounces) according to the function below..65, if 0 x 8 7.85, if 8 x C w=.05, if x 48 4.0, if 48 x 64 7.0, if 64 x 80 A) Graph the function. B) What is the cost to send a parcel weighing pounds and 9 ounces? lbs 9 ozs 6 9 4 ozs.05