FOR ALL STUDENTS TAKING ALGEBRA II Honors SUMMER REVIEW PACKET

Transcription

1 FOR ALL STUDENTS TAKING ALGEBRA II Honors SUMMER REVIEW PACKET

2 Dear Student and Parent/Guardian, The math department at Central Dauphin School District wants you to be successful in Algebra II. We also want you to be prepared for PSSA and Keystone Exams. This summer packet is designed to help you reach these goals by reviewing necessary skills. Be sure to follow the key information below when completing this packet: The packet is due when you return to school in August. Every problem must be completed. None left blank. The packet is worth 10 times a regular homework grade. Work must be shown to receive credit no work, no points. Final answers must be shown on the answer pages at the back of the packet. A possible Quiz covering the material from the packet may be given at the end of the first week of school. These topics also tie in with the first few units of Algebra II. When you return in August, you will have the opportunity to ask questions. Math Help will also be available during the first week. We hope that you have an enjoyable summer and return to school ready to be successful in Algebra II! Helpful Websites (a website where you can subscribe for individual math lessons)

3 Chapter 1 Equations and Inequalities Example: Evaluate x y if x 4 and y x y 4 Replace x with 4 and y with 4 5 Add and 16 5 Evaluate 4 11 Subtract 5 from 16 Evaluate each expression if x, y 4, and z 1. y z. 8 x z y. 5 x 6 6y 4. y z xz Example: Solve 7x 56 5x 11 5x 5x Subtract 5x from each side x Simplify Subtract 56 from each side x 67 Simplify x 67 = Divide each side by x.5 Simplify and check the result Solve: n w 14 7w a 8. 5y 4 y 4 Example: Solve x 9 x 9 Divide each side by to isolate the absolute value x 9 11 Simplify x 9 11 or x 9 11 Split using definition of absolute value and solve each equation. x or x 0 Subtract 9 from each side x 1 or x 10 Divide each side by. Check your solutions Solve: 9. x w 6 1

4 y 1. n 4 Example: Solve 7x 5 6x 4 and graph 6x 6x Subtract 6x from each side x 5 4 Simplify 5 5 Add 5 to each side x 9 Simplify * Recall when you divide or multiply by a negative number you switch the inequality symbol Solve and graph: 1. 4x 7 x m m x y or y 4 Example: Solve x 1 6 Solve x 4 10 x 1 6 or x 1 6 Split 10 x 4 10 x 18 or x 6 Solve 14 x 6 x 6 or x 14 x 6-4⅔ Solve and graph: 17. 4k y n y 9 7

5 Chapter Linear Relations and Functions Forms of Linear Equations Standard Point-Slope Slope-intercept Ax By C y y m x y mx b 1 x 1 Slope Formula m y x y1 x 1 Example: Graph x y 18 Method A: using a table Method B: using slope-intercept form Find the x-intercept Solve for y x 0 18 x y 18 x 18 y x 18 x 9 y x 6 Find the y-intercept 0 y 18 y 18 y 6 Graph: 1. 5x y 0. y x 6

6 Example: Write equations of lines given slope and a point. m, passes through 4,1 y mx b 1 4 b Substitute the point for x and y and the slope for m 1 6 b Simplify 5 b Simplify y x 5 Write the equation in slope-intercept form Write the equation of the line with the given slope passing through the given point. m ; 5,7 4. m ; 1, Example: Write equations of lines given two points. 6,1 ; 8, 4 y y m Use the slope formula to find slope x x y mx b b Use the slope and one point to find b 1 15 b 16 b 5 y x 16 Write the equation Write the equation of the line passing through the two given points., ;,7,5 ;, 5. 6.

7 Example: Write equations of parallel and perpendicular lines. 5 5 If m, then the parallel slope is the same m and the perpendicular slope is the opposite reciprocal m 5 Write the equation of the line that passes through, 1, perpendicular to the graph of x y 6. First find the slope of the line (solve for y) x y 6 x x Subtract x y x 6 Divide by y x Simplify Perpendicular slope is m y mx b 1 b Use the slope and the point to find b 1 b 4 b y x 4 Write the equation Write the equation of parallel or perpendicular lines, 1, parallel to graph of x y ,1, perpendicular to line whose slope is Challenge:, 5, perpendicular to graph of x 4

8 Chapter Systems of Equations and inequalities Methods to solve a system of equations 1. Graphing 4x y 10 x y 1 Solve each equation for y 4x y 10 y x 5 x y 1 y x 1 The solution is,1, where the two graphs intersect.. Substitution. Elimination 4x y 10 4x y 10 x y 1 x y 1 Solve one equation for one variable Choose a variable to be opposite coefficients x y 1 x y 1 Let s choose y. Multiply nd equation by Substitute into other equation 4x y 10 4x y y x y 1 x y y 10 Solve the equation 6 x 1 Add 4 y 1 y 10 x 4y 4 y 10 6y 4 10 Plug the answer into one of the original 6y 6 equations to find the other variable. y 1 y 1 Plug the answer into one of the original y 1 equations to find the other variable. y 1 x 1 1 Solution is,1 x Solution is,1

9 Solve the following system using all methods. (Do all work on answer key) x y 6 x y 9 1. Graphing. Substitution. Elimination Solving systems of inequalities The solution is the intersection of the shaded areas of the graphs. x y 4 x y Solve for y x y 4 y x 4 x y y x Graph Solve the system of inequalities 4. x y 6 4x y

10 Factoring Steps for factoring: 1. Factor the Greatest Common Factor (if it has one). Count the number of terms.. If TWO terms: a. Is it a Difference of Squares? A² - B² = (A B)(A + B) b. Is it a Sum/Difference of Cubes? a³ + b³ = (a + b)(a² - ab + b²) a³ - b³ = (a b)(a² + ab + b²) 4. If THREE terms: a. Is it a perfect Square Trinomial? A² +AB + B² = (A + B)² A² -AB + B² = (A - B)² b. Is it ax² + bx + c, where a = 1? Multiply to get c, Add to get b c. Is it ax² + bx + c, where a 1? Rewrite as 4 terms and Group 5. If FOUR terms: a. Factor by grouping Factor the following:

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12 Name Summer Packet Answer Sheet You MUST show your work and circle your answers. Chapter

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14 Name Summer Packet Answer Sheet You MUST show your work and circle your answers. Chapter Parallel 8. Perpendicular 9. CHALLENGE:

15 Name Summer Packet Answer Sheet You MUST show your work and circle your answers. Chapter 1. Graph. Substitution. Elimination

16 4. Graphing Inequalities

17 Name Summer Packet Answer Sheet You MUST show your work and circle your answers. Factoring

18 Name Summer Packet Answer Sheet You MUST show your work and circle your answers. Factoring