EOC Review. Algebra I

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1 EOC Review Algebra I

2 Order of Operations PEMDAS Parentheses, Eponents, Multiplication/Division, Add/Subtract from left to right. A. Simplif each epression using appropriate Order of Operations (0 6) 5. (0 7) 6. 5 [5( + )]. ( + 7) Solving Equations The five steps to solving an equation are: Get rid of parentheses Simplif the left side and the right side of the equation as much as possible, i.e. combine an and all like terms Get the variable term on just one side Get the variable term b itself Solve for the variable B. Solve for the variable in each problem. 7. 5( ) r (r + 8) 6. (8 + ) - 7 (- 8) 8. (6 + ) 0 ( 7) 0. (c + ) (c + ) + c. ( 5) Solving Proportions Remember: Use Cross Productions to write and equation Solve the equation Eamples. 6. t + 77 (t + ) 6(77) t t 9 t 7 w w + w (w + ) w w + 6 w 6 w C. Solve the following: a. 9a 8. 5z + z 5. 8

3 Answer the following: 6. A recipe that ields buttermilk biscuits calls for cups of flour. How much flour is needed to make 0 biscuits? 7. It took 7. minutes to upload 8 digital pictures from our computer to a website. At this rate, how long will it take to upload 0 pictures? Solving Inequalities Eamples: < 8 Subtract from each side 8 > Divide each side b > -.5 Simplif Divide b - and change < to > D. Solve and graph the following inequalities. 8. f < f ( ) ( ) 0. c < 0 Graphs and Equations of Lines Slope Intercept Form m + b, where m slope and b -intercept Graphing Equations in Slope-Intercept Form. Write the equation in slope-intercept form for. Find the -intercept and use it to plot the point where the line crosses the -ais.. Find the slope and use it to plot at least two more points on the line.. Draw a line through the points. Writing the Equation: Given the Slope and a - intercept Eample: Write an equation of the line that passes through (0, ) and has a slope of -5. (These can also be given on a graph) Step : Substitute 5 for m. Step : Substitute for b (since it is the intercept) b

4 Point Slope Form m( ) where m slope and (, ) is the point. Graphing Equations in Slope-Intercept Form. Plot the point (, ).. Find the slope and use it to plot a second point on the line.. Draw a line through the two points. Writing the Equation: Given a point and a slope Eample: Write an equation of the line that passes through the point (, 5) and has a slope of. (These can also taken from a graph) Substitute for, 5 for, and for 5 ( -) Given Two Points Step : Find the slope of the line using the two points and the formula m Step : Choose either point and follow the steps above depending on the form ou are asked to use. Standard Form a + b c where a is a positive, and a and b are whole numbers. Graphing in Standard Form: Find the and intercepts and graph the line that contains them. Writing the Equation: Write the equation using slope-intercept or point-slope form, then rearrange to standard form. Eample: Write the equation of the line that passes through the point (, 5) and has a slope of ½. Step : Write in Point-Slope Form 5 ½ ( ) Step : Distribute 5 ½ Step : Subtract ½ and add 5 - ½ + Step : Multipl b - to make a a positive, whole number - E. Find the slope of the line containing each pair of points.. (5, 0) and (6, 8). (, -) and (6, -). (-, -) and (-9, -7) F. Find the slope of each line G. Find the equation of the line with the given slope through the given point. Write the answer in slope-intercept form. 8. m ; (, ) 9. m -; (, 7) 0. m ;(,-) H. Write an equation of the line that passes through the given point and is parallel to the given line.. (-, ); +. (, 7); (-0, 0); I. Write an equation of the line that passes through the given point and is perpendicular to the given line.. (, -); (8, -); + 6. (5, ); 5

5 J. Write the equation of the line in point-slope form. 7. The line containing (-, -) and (5, ) 8. The horizontal line passing through (, 5) K. Write the equation of the line in slope-intercept form. 9. The line containing (, ) and (, 8) 0. The line containing (, ) and (-6, 9). The line with slope 5 and containing (-, 7) Graph the following equations. Graph three points and label the line with its equation.. ( ). -( 5) 6. ( + ) L. Point-Slope Form. 5 ( ) ( ) 7. ( + 6) M. Slope-Intercept Form

6 M. Slope-Intercept Form N. Standard Form Sstems of Linear Equations Substitution Method: Use when an equation is solved for one variables ( or ) Solve: Solution: Substitute 5 for. 5 6(5 ) Then substitute for : 5 () - Answer: (, -) Solve each sstem b substitution

7 Elimination Method: Use addition when the coefficients of a variable are opposites. Use subtraction when the coefficients are the same. + 9 Eample - Solve: Solution: + 9 (+) - - ADD - 6 Then substitute - for : + (-) 9-7 Answer: (7,-) Use multiplication when ou have neither same or opposite coefficients Eample Solve: Solution: (5-9) ( + ) (+) Then substitute - for : (-) ADD Answer: (-, ) Solve each sstem b elimination Graphing Method: Graph or more equations on the same coordinate plane Scenario Intersecting lines ( solution point of intersection) Scenario Parallel Lines (no solution) Scenario Coinciding Lines (Infinitel Man Solutions {IMS})

8 Solve each sstem b graphing Remember: < or > Graph with a dotted line or Graph with a solid line < or Shade below the line (shade left of a vertical line > or Shade above the line (shade right of a vertical line Solutions are where the shaded regions overlap or on a solid boundar line Sstems of Inequalities < < + > Graph each sstem of inequalities. 7. < < + 7. > >

9 Domain and Range Domain: Set of values of the independent variable () for which a function is defined (Can also be seen as input or cause) Range: Set of values of a function (dependent variable, output, f(), or effect) Eamples: Find the domain and range for each of the following.. (, ), (5, 6), (, -), (-, 5), (7, ) Ans: Domain: {-,,, 5, 7} Range: { -,, 5, 6} **Written Least to Greatest, No Repeats! Ans: direction. Domain: All Real Numbers (ARN) Range: 0 **Remember that arrows mean continues on infinitel in that Ans: Domain: - < Range: - < **Remember() Open dots means not included (< or >), () Closed dots mean includes ( or ). Ans: Domain: {-, -, 0,, } Range: {0,, } P. Give the domain and range for the following: Domain: Range: 78. Domain: Range: 79. Domain: Range: 80. Domain: Range: Domain: Range: Domain: Range:

10 Eponents a 0 Eample: 5 0 a m a n a m+n Eample: + 6 a a m n a m n Eample: b b 7 7 b b (a m ) n a m(n) Eample: ( ) () m a m a Eample: Q. Simplif each epression a a 85. ( ) 86. ( 5a b )( a b) ( ab )( ab ) ( a )( 5a ) Multipling Polnomials 90. ( a ) 9. ( 5a ) 9. 5 c c c 9. ( )( ) Monomial Polnomial c (8c c c + 5) c c 9c + 5c 7 5 Distributeb multipling c b ever term inside the () Binomial Binomial Binomial Polnomial Use Punnett Squres E: ( )( ) Combine Like Terms

11 R. Find each product. 9. ( + )( + 9) 96. (6 + 5)( ) b (b 5 b + b ) 95. ( + )( + ) 97. ( )( + ) 99. (6 + 5) Factoring Polnomials Eamples: ) a b ( a b)( a b) + EX: a 6 ( a + )( a );5a 6 6 ( 5a + 6 )( 5a 6 ) ) a + ab + b ( a + b) EX: k + 0k + 5 ( k + 5)( k + 5) ( k + 5) k & 5 are perfect squares & 0k (k*5) ) a ab + b ( a b) EX: + 9 ( )( ) ( ) ) a b c & 9 are perfect squares & (*) + + EX: 6 8 ( )( ) ( )( 5) + a a 5 ( a 5)( a ) ( )( ) a b c a b c a b c since + 6 and * 8 + since and - * since and 5 * since and * - - S. Factor each of the following polnomials a a Solve using Square Roots Get numbers on one side of equation Problem: Divide b 5 Solving Quadratic Equations + 6 Problem ( ) 5 75 Square root both sides ( ) Square root of ( 6) ( 6) + + subtract 6 from both sides + 6 ± + 6 ± 6 6 Square root both sides ± 5 Answer: ± 6

12 T. Solve each quadratic equation using square roots ( ) 9 Solve using Factoring Problem a + a 5 a + 5 a Factor the problem ( )( ) Make each factor equal to zero and solve for a and a Answer a -5 a Solve each quadratic equation using factoring t 9t 0. p 6p m + m Solve Using Quadratic Formula ± b b ac a Put equation in proper format (a + b + c 0) Find a, b, and c Plug into the formula Do the math a little at a time. If the discriminate (b -ac) is positive there are real solutions, if 0, there is real solution, if negative, then there is NO real solution. Eamples. 5 0 a, b, c 5 ± ( ) ( ) ( ) ( 5 ) ± , ± ± , a, b 7, c ± ( ) 7 ± ± 7 7 (7 ) ( )( ) ,. 9,.8 9

13 Solve using the Quadratic Equation Solve b Graphing Rearrange to a + b + c Find a, b, and c b a Find ais of smmetr Plug in the ais of smmetr -value into the equation and find which together make the verte (, ) Make a table using two -values to the left of the verte, and two -values to the right of the verte. Graph all five points and connect with a smooth curved line. Solutions to Quadratics are called -intercepts, zeros, roots, and solutions. If the graph does not touch the -ais, there is no solution. Eample. a, b -, c - ( ) () () () - verte: (, -) Solutions: - and Solve b Graphing

14 Parent Functions Parent Function: The simplest version of an function Linear Functions: Absolute Value Functions: Quadratic Functions: Cubic Functions:

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