H.Algebra Summer Review Packet 1
Correlation of Algebra Summer Packet with Algebra 1 Objectives A. Simplifing Polnomial Epressions Objectives: The student will be able to: Use the commutative, associative, identit, and distributive properties. Appl the appropriate arithmetic operations needed to simplif an algebraic epression. Simplif polnomials 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: Simpl 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: Add, subtract, multipl, divide and simplif square roots. Find the real solutions of a quadratic equation of the form a + b + c = 0 using factoring and the quadratic formula. 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 an equation of a line. Rewrite an equation in standard form to slope intercept form and vice versa.
H. Regression and Use of the Graphing Calculator Objectives: The student will be able to: Use linear models to interpret real-world situations. 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.
I. Combining Like Terms A. Simplifing Polnomial Epressions - You can add or subtract terms that are considered "alike", or terms that have the same variable with the same eponent EX 1: 5-7 + 10 + 5-7 + 10 + 15-4 EX : -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 : 4 0 (5! 5 5 + 4 + 6) + 4! 6 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
PRACTICE 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)
I. Solving Two-Step Equations B. Solving Equations A couple of hints: 1. To solve an equation, UNDO the order of operations.. Save the operation that is directl related to the variable as the last operation ou will undo.. 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 Equals 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 8 = 4 + 4! 4 4 = 4 4 = 6! 4 4! 4 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
PRACTICE: 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 18 = 6 = for E : 5a! 10b = 0 + 10b = + 10b 5a = 0 + 10b 5a 0 10b = + 5 5 5 a = 4 + b
PRACTICE 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
C. Eponent Rules Multiplication: Recall ( m )( n ) ( m+ n) = E: 4 5 4 1 5 5 6 ( )(4 ) = (! 4)(! )(! ) = 1 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 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 ( )( )
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)
NOTE: Special Case Recall: 4 = 4 4 = E. ( + 5) ( + 5) = ( + 5)(+5) Now ou can use the FOIL method to get a simplified epression. PRACTICE Multipl. Write our answer in simplest form. 1. ( + 10)( - 9). (7 + )( - 1). ( - 10)( - ) 4. ( - 8)( + 81) 5. ( - 1)(4 + ) 6. (10 - )(5-9) 7. (- - 4)( + 4) 8. ( + 10) 9. (5 - ) 10. ( - )
E. Factoring I. Determining the greatest common factor (GCF). 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 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) 1 0 15 10 5 6 Note: Not all epression 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) 4! 100 E. 4(! 5) 4 (! 5)( + 5) Since and 5 are perfect squares separated b a subtraction sign, ou can appl the difference of two squares formula.
PRACTICE 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
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!! 4 This is not simplified completel because 1 is divisible b 4 (another perfect square) PRACTICE Simplif each radical. 1. 11. 90. 175 4. 88 5. 486 6. 16 7. 6 500 8. 147 9. 8 475 10. 15 9
G. Graphing Lines I. Finding the slope of the line that contains each pair of points. Given two points with coordinates ( ) and ( ) 1, 1 the line containing the points is! m = 1.! 1, E. (, 5) and (4, 1) E. (-, ) and (, ) 1! 5! 4! 1 m = = =! m = = 4!! (! ) 5 1 The slope is -. The slope is 5, the formula for the slope, m, of PRACTICE 1. (-1, 4) and (1, -). (, 5) and (-, 1). (1, -) and (-1, -) 4. (, -4) and (6, -4) 5. (, 1) and (-, -) 6. (5, -) and (5, 7) 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 = m + b. E. =! 1 E. =! + 4 Slope: Slope:! 4 -intercept: -1 -intercept:
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 1 1. = + 5. =! Slope: Slope: -intercept: -intercept
. =! + 4 5 4. =! Slope: Slope: -intercept: -intercept 5. =! + 6. = Slope: Slope: -intercept: -intercept
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
PRACTICE 1. + =. 5 + = 10. = 4 4. 4! = 9
5.! + 6 = 1 6. =!
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. 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 scatterplot 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 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 scatterplot 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.
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 or 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.
Algebra Summer Review Packet Student Answer Ke A. Simplifing Polnomial Epressions 1. 4 +.! 15 + 7 +. 9 n! 4.! b + 6 5. 160 q + 110q 6.! 5 + 6 7. 74 z! 4w 8. 56 c! 117 9.! 7 + 54! 9 10.! + 1 + 4 B. Solving Equations a. Equations 1. = 7. = 6. = 9. 5 4. = 1 5. =. 5 6. = 16 7. = 19 8. =!. 8 9. = 9. 5 10. = 0 b. Literal Equations 1. V = W! Y. w = 9 r. 5. 9! d f =! + d =! 4. P P + 160 g = + 9 = 180 180 6. 10! t 10 = =! d d 9 + u + 5h = 4 t d
C. Eponent Rules 1. 8 c. 1 m. 0 k 4. 1 5. p 11 q 7 6. 9 9z 7. 1! t 8. f 9. 8 60h k 5 10. a b c 4 11. 8 4 81m n 1. 1 4 1. 0 a b c 14. 4 15. 4 7 4 D. Binomial Multiplication 1. +! 90.! 5! 84.! 1 + 0 4. + 7! 648 5. 8 +! 6. 50! 100 + 18 7.! 6! 0 + 16 8. + 0 + 100 9. 5! 10 + 10. 4! 1 + 9
E. Factoring 1. ( + ). 4ab ( a! 4b + c). (! 5)( + 5) 4. ( n + 5)( n + ) 5. ( g! 4)( g! 5) 6. ( d + 7)( d! 4) 7. ( z! 10)( z + ) 8. ( m + 9) 9. 4 (! )( + ) 10. 5 ( k + 9)( k! ) F. Radicals 1. 11. 10. 5 7 4. 1 5. 9 6 6. 8 7. 60 5 8. 1 9. 40 19 10. 5 5 G. Graphing Lines 1. Slope:. Slope: 1 -intercept: 5 -intercept -
. =! + 4 4. =! 5 Slope:! Slope: - 5 -intercept: 4 -intercept 0 5. =! + 6. = Slope: -1 Slope: 1 -intercept: -intercept 0
1... 4. 5. 6.
H. Regression and Use of the Graphing Calculator 1. a. Variable Quantit Lower Bound Upper Bound Interval Math Test Scores 40 100 5 Science Test Scores 5 95 5 b. SEE CALCULATOR c. =.976 +.6 d. Use the table feature of the calculator (in ASK mode) and input 8 in the X- column. The answer is 8.1, so the test score is approimatel 8. e. Graph = 5 along with the line of best fit graph. Find the point of intersection of the two lines. The answer will be a score of 5.046 or approimatel 5.. a. Variable Quantit Lower Bound Upper Bound Interval Year 1964 199 Time (Sec) 1. 0.1 b. SEE CALCULATOR c. = -0.048 + 117.7608. This was determined b entering the ears in L 1 and the time in L. Then a LinReg calculation was performed. d. Use the table (in ASK mode) and input 01. The answer is 0.51 seconds. This answer is reasonable since it is not significantl outside the range of the data.