Rising Algebra Honors Complete the following packet and return it on the first da of school. You will be tested on this material the first week of class. 50% of the test grade will be completion of this packet; ou must show our work to receive full credit. Unless otherwise stated, work should be done without a calculator. Algebra 1 Skills Needed to be Successful in Algebra A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif an algebraic epression. Simplif polnomial epressions using addition and subtraction. Multipl a monomial and polnomial. B. Solving Equations Objectives: The student will be able to: Solve multi-step equations. Solve a literal equation for a specific variable, and use formulas to solve problems. C. Rules of Eponents Objectives: The student will be able to: Simplif epressions using the laws of eponents. Evaluate powers that have zero or negative eponents. D. Binomial Multiplication Objectives: The student will be able to: Multipl two binomials. E. Factoring Objectives: The student will be able to: Identif the greatest common factor of the terms of a polnomial epression. Epress a polnomial as a product of a monomial and a polnomial. Find all factors of the quadratic epression a + b + c b factoring and graphing. F. Radicals Objectives: The student will be able to: Simplif radical epressions. G. Graphing Lines Objectives: The student will be able to: Identif and calculate the slope of a line. Graph linear equations using a variet of methods. Determine the equation of a line. H. Regression and Use of the Graphing Calculator Objectives: The student will be able to: Draw a scatter plot, find the line of best fit, and use it to make predictions. Graph and interpret real-world situations using linear models. 4
A. Simplifing Polnomial Epressions I. Combining Like Terms - You can add or subtract terms that are considered "like", or terms that have the same variable(s) with the same eponent(s). E. 1: 5-7 + 10 + 5-7 + 10 + 15-4 E. : -8h + 10h - 1h - 15h -8h + 10h - 1h - 15h -0h - 5h II. Appling the Distributive Propert - Ever term inside the parentheses is multiplied b the term outside of the parentheses. E. 1: (9 " 4) # 9 " # 4 7 "1 E. : 4 (5 + 6) 4 " 5 + 4 " 6 0 5 + 4 III. Combining Like Terms AND the Distributive Propert (Problems with a Mi!) - Sometimes problems will require ou to distribute AND combine like terms!! E. 1: (4 " ) +1 # 4 " # +1 1 " 6 +1 5 " 6 E. : (1 " 5) " 9("7 +10) #1 " # 5" 9("7) " 9(10) 6 "15+ 6" 90 " 54 + 48 5
PRACTICE SET 1 Simplif. 1. 8! 9 + 16 + 1. 14 +! 15 +. 5n! (! 4n) 4.! (11b! ) 5. 10 q (16 + 11) 6.! ( 5! 6) 7. (18z! 4w) + (10z! 6w) 8. ( 8c + ) + 1(4c! 10)! 9. 9(6! )! (9 ) 10.! (! ) + 6(5 + 7) 6
I. Solving Two-Step Equations B. Solving Equations A couple of hints: 1. To solve an equation, UNDO the order of operations and work in the reverse order.. REMEMBER! Addition is undone b subtraction, and vice versa. Multiplication is undone b division, and vice versa. E. 1: 4 " = 0 + + 4 = 4 4 = 8 E. : 87 = "11 + 1 " 1 " 1 66 = "11 "11 "11 " 6 = II. Solving Multi-step Equations With Variables on Both Sides of the Equal Sign - When solving equations with variables on both sides of the equal sign, be sure to get all terms with variables on one side and all the terms without variables on the other side. E. : 8 + 4 = 4 + 8 " 4 " 4 8 = 4 + 4 " 4 " 4 4 = 4 4 4 = 6 III. Solving Equations that need to be simplified first - In some equations, ou will need to combine like terms and/or use the distributive propert to simplif each side of the equation, and then begin to solve it. E. 4 : 5(4 " 7) = 8 + 45+ 0 " 5 =10 + 45 "10 "10 10 " 5 = 45 + 5 + 5 10 = 80 10 10 = 8 7
PRACTICE SET Solve each equation. You must show all work. 1. 5! =. 140 = 4 + 6. 8 (! 4) = 196 4. 45! 70 + 15 = 60 5. 1 = 4(1! 9) 6. 198 = 154 + 7! 68 7.! 11 =! 5(! 8) + 6 8.! 7! 10 = 18 + 9. 1 + 8! 15 =! (! 8) 10.! ( 1! 6) = 1 + 6 IV. Solving Literal Equations - A literal equation is an equation that contains more than one variable. - You can solve a literal equation for one of the variables b getting that variable b itself (isolating the specified variable). E.1: =18, Solve for. = 18 = 6 E. : 5a "10b = 0, Solve for a. +10b =+10b 5a = 0 +10b 5a 5 = 0 5 + 10b 5 a = 4 + b 8
PRACTICE SET Solve each equation for the specified variable. 1. Y + V = W, for V. 9wr = 81, for w. d f = 9, for f 4. d + t = 10, for 5. P = (g 9)180, for g 6. 4 + 5h = 10 + u, for 9
C. Rules of Eponents Multiplication: Recall ( m )( n ) ( m+ n) = E: ( 4 )(4 5 )=(" 4)( 4 " 1 )( " 5 )=1 5 7 Division: Recall m ( m n)! n 5 5 4m j ' 4 $ ' m $ ' j $ = E: = 14m j % " =! 1 m j % m " % j "! &! #& #& # Powers: Recall ( m ) n ( m! n) = E: 4 1 4 9 1 (! a bc ) = (! ) ( a ) ( b ) ( c ) =! 8a b c 0 Power of Zero: Recall = 1,! 0 E: 0 4 4 4 5 = (5)(1)( ) = 5 PRACTICE SET 4 Simplif each epression. 15 5 m 1. ( c )( c)( c ). m. (k 4 ) 5 4. 0 4 7 5 d 5. ( q )( p q ) p 6. 45 z 5 z 10 7. (! t 7 ) 8. g 0 5 f 9. (4h k )(15k h ) 10. 4 6 1a b 6ab c 11. ( n m ) 4 1. ) 0 ( 1 1. (! 5a b)(ab c)(! b) 14. 4 0 ( ) 15. 4 ( )( ) 10
I. Reviewing the Distributive Propert D. Binomial Multiplication The distributive propert is used when ou want to multipl a single term b an epression. E 1: 8(5 8 " 5 40! 9) + 8 " (! 9)! 7 II. Multipling Binomials the FOIL method When multipling two binomials (an epression with two terms), we use the FOIL method. The FOIL method uses the distributive propert twice! FOIL is the order in which ou will multipl our terms. First Outer Inner Last E. 1: ( + 6)( + 10) FIRST OUTER First " ------> ( + 6)( + 10) Outer Inner 10 -----> 10 6 ------> 6 INNER LAST Last 6 10 -----> 60 + 10 + 6 + 60 + 16 + 60 (After combining like terms) 11
Recall: 4 = 4 4 = E. ( + 5) ( + 5) = ( + 5)(+5) Now ou can use the FOIL method to get a simplified epression. PRACTICE SET 5 Multipl. Write our answer in simplest form. 1. ( + 10)( 9). ( + 7)( 1). ( 10)( ) 4. ( 8)( + 81) 5. ( 1)(4 + ) 6. (- + 10)(-9 + 5) 7. (- 4)( + 4) 8. ( + 10) 9. (- + 5) 10. ( ) 1
E. Factoring I. Using the Greatest Common Factor (GCF) to Factor. Alwas determine whether there is a greatest common factor (GCF) first. E. 1 4! + 90 In this eample the GCF is. So when we factor, we have (! 11 + 0). Now we need to look at the polnomial remaining in the parentheses. Can this trinomial be factored into two binomials? In order to determine this make a list of all of the factors of 0. 0 0 1 0 15 10 5 6-1 -0 - -15 - -10-5 -6 Since -5 + -6 = -11 and (-5)(-6) = 0 we should choose -5 and -6 in order to factor the epression. The epression factors into (! 5)(! 6) Note: Not all epressions will have a GCF. If a trinomial epression does not have a GCF, proceed b tring to factor the trinomial into two binomials. II. Appling the difference of squares: a! b = ( a! b)( a + b) E. 4 "100 ( ) 4 " 5 ( )( + 5) 4 " 5 Since and 5 are perfect squares separated b a subtraction sign, ou can appl the difference of two squares formula. 1
PRACTICE SET 6 Factor each epression. 1. + 6. 4 a b! 16ab + 8ab c.! 5 4. n + 8n + 15 5. g! 9g + 0 6. d + d! 8 7. z! 7z! 0 8. m + 18m + 81 9. 4! 6 10. 5k + 0k! 15 14
F. Radicals To simplif a radical, we need to find the greatest perfect square factor of the number under the radical sign (the radicand) and then take the square root of that number. E. 1: 7 6 " 6 E. : 4 90 4 " 9 " 10 4 " " 10 1 10 E. : 48 16 4 OR E. : 48 4 1 1 4 " " This is not simplified completel because 1 is divisible b 4 (another perfect square) 4 PRACTICE SET 7 Simplif each radical. 1. 11. 90. 175 4. 88 5. 486 6. 16 7. 6 500 8. 147 9. 8 475 10. 15 9 15
G. Graphing Lines I. Finding the Slope of the Line that Contains each Pair of Points. Given two points with coordinates ( 1, 1) and (, ) the line containing the points is! m = 1.! E. (, 5) and (4, 1) E. (-, ) and (, ) 1! 5! 4! 1 m = = =! m = = 4!! (! ) 5 1 The slope is -. The slope is 5 1, the formula for the slope, m, of PRACTICE SET 8 1. (-1, 4) and (1, -). (, 5) and (-, 1). (1, -) and (-1, -) 4. (, -4) and (6, -4) 5. (, 1) and (-, -) 6. (5, -) and (5, 7) 16
II. Using the Slope Intercept Form of the Equation of a Line. The slope-intercept form for the equation of a line with slope m and -intercept b is E. =! 1 E. =! + 4 Slope: -intercept: -1 Slope:! -intercept: 4 = m + b. Place a point on the -ais at -1. Place a point on the -ais at. Slope is or /1, so travel up on Slope is -/4 so travel down on the the -ais and over 1 to the right. -ais and over 4 to the right. Or travel up on the -ais and over 4 to the left. PRACTICE SET 9 1 1. = + 5. =! Slope: -intercept: Slope: -intercept: 17
. =! + 4 5 4. =! Slope: Slope: -intercept: -intercept 5. =! + 6. = Slope: Slope: -intercept: -intercept 18
III. Using Standard Form to Graph a Line. An equation in standard form can be graphed using several different methods. Two methods are eplained below. a. Re-write the equation in = m + b form, identif the -intercept and slope, then graph as in Part II above. b. Solve for the - and - intercepts. To find the -intercept, let = 0 and solve for. To find the -intercept, let = 0 and solve for. Then plot these points on the appropriate aes and connect them with a line. E.! = 10 a. Solve for. OR b. Find the intercepts:! =! + 10 let = 0 : let = 0:! + 10 =!! (0) = 10 (0)! = 10 10 =! = 10! = 10 = 5 10 =! So -intercept is (5, 0) & 10 # So -intercept is $ 0,'! % " On the -ais place a point at 5. 10 On the -ais place a point at! Connect the points with the line. =! 1 19
PRACTICE SET 10 1. + =. 5 + = 10. = 4 4. 4! = 9 0
5.! + 6 = 1 6. =! 1
H. Regression and Use of the Graphing Calculator Note: For guidance in using our calculator to graph a scatterplot and finding the equation of the linear regression (line of best fit), please see the calculator direction sheet included in the back of the review packet. PRACTICE SET 11 1. The following table shows the math and science test scores for a group of ninth graders. Math Test Scores Science Test Scores 60 40 80 40 65 55 100 90 85 70 5 90 50 65 40 95 85 90 Let's find out if there is a relationship between a student's math test score and his or her science test score. a. Fill in the table below. Remember, the variable quantities are the two variables ou are comparing, the lower bound is the minimum, the upper bound is the maimum, and the interval is the scale for each ais. Variable Quantit Lower Bound Upper Bound Interval b. Create the scatter plot of the data on our calculator. c. Write the equation of the line of best fit. d. Based on the line of best fit, if a student scored an 8 on his math test, what would ou epect his science test score to be? Eplain how ou determined our answer. Use words, smbols, or both. e. Based on the line of best fit, if a student scored a 5 on his science test, what would ou epect his math test score to be? Eplain how ou determined our answer. Use words, smbols, or both.
. Use the chart below of winning times for the women's 00-meter run in the Olmpics below to answer the following questions. Year Time (Seconds) 1964.00 1968.50 197.40 1976.7 1980.0 1984 1.81 1988 1.4 199 1.81 a. Fill in the table below. Remember, the variable quantities are the two variables ou are comparing, the lower bound is the minimum, the upper bound is the maimum, and the interval is the scale for each ais. Variable Quantit Lower Bound Upper Bound Interval b. Create a scatter plot of the data on our calculator. c. Write the equation of the regression line (line of best fit) below. Eplain how ou determined our equation. d. The Summer Olmpics will be held in London, England, in 01. According to the line of best fit equation, what would be the winning time for the women's 00- meter run during the 01 Olmpics? Does this answer make sense? Wh or wh not?
graph a function Press the Y= ke, Enter the function directl using the X, T,!, n ke to input. Press the GRAPH ke to view the function. Use the WINDOW ke to change the dimensions TI-8 Plus/TI-84 Graphing Calculator Tips How to and scale of the graph. Pressing TRACE lets ou move the cursor along the function with the arrow kes to displa eact coordinates. find the -value of an -value Once ou have graphed the function, press CALC nd TRACE and select 1:value. Enter the - value. The corresponding -value is displaed and the cursor find the maimum value of a function Once ou have graphed the function, press CALC nd TRACE and select 4:maimum. You can set the left and right boundaries of the area to be eamined and guess the maimum value either b entering values find the zero of a function Once ou have graphed the function, press CALC nd TRACE and select :zero. You can set the left and right boundaries of the root to be eamined and guess the value either b entering values find the intersection of two functions Once ou have graphed the function, press CALC nd TRACE and select 5:intersect. Use the up and down arrows to move among functions and press ENTER to select two. Net, enter lists of data Press the STAT ke and select 1:Edit. Store ordered pairs b entering the coordinates in L1 and the coordinates in L. You can calculate new lists. To moves to that point on the function. directl or b moving the cursor along the function and pressing ENTER. The -value and -value of the point with the maimum -value are then displaed. directl or b moving the cursor along the function and pressing ENTER. The -value displaed is the root. enter a guess for the point of intersection or move the cursor to an estimated point and press ENTER. The -value and -value of the intersection are then displaed. create a list that is the sum of two previous lists, for eample, move the cursor onto the L heading. Then enter the formula L1+L at the L prompt. 4
plot data Once ou have entered our data into lists, press STAT PLOT nd Y= and select Plot1. Select On and choose the tpe of graph ou want, e.g. scatterplot (points not connected) or connected dot for graph a linear regression of data Once ou have graphed our data, press STAT and move right to select the CALC menu. Select 4:LinReg(a+b). Tpe in the parameters L1, L, Y1. To enter Y1, press VARS draw the inverse of a function Once ou have graphed our function, press DRAW nd PRGM and select 8:DrawInv. Then enter Y1 if our function is in Y1, or just enter the function itself. create a matri From the home screen, press nd -1 to select MATRX and move right to select the EDIT menu. Select 1:[A] and enter the number of rows and the number of columns. Then fill in the matri b entering a value in each element. solve a sstem of equations Once ou have entered the matri containing the coefficients of the variables and the constant terms for a particular sstem, press MATRX (nd -1, move to MATH, and select B:rref. generate lists of random integers From the home screen, press MATH and move left to select the PRB menu. Select 5:RandInt and enter the lower integer bound, the upper integer bound, and the number of trials, separated b two variables, histogram for one variable. Press ZOOM and select 9:ZoomStat to resize the window to fit our data. Points on a connected dot graph or histogram are plotted in the listed order. and move right to select the Y-VARS menu. Select 1:Function and then 1:Y1. Press ENTER to displa the linear regression equation and Y= to displa the function. You ma move among elements with the arrow kes. When finished, press QUIT nd MODE to return to the home screen. To insert the matri into calculations on the home screen, press nd -1 to select MATRX and select NAMES and select 1:[A]. Then enter the name of the matri and press ENTER. The solution to the sstem of equations is found in the last column of the matri. commas, in that order. Press STO and L1 to store the generated numbers in List 1. Repeat substituting L to store a second set of integers in List. 5