Summer Math Assignments for Students Entering Algebra II

Transcription

1 Summer Math Assignments for Students Entering Algebra II Purpose: The purpose of this packet is to review pre-requisite skills necessary for the student to be successful in Algebra II. You are expected to have a foundational understanding of each of the topics listed below before coming to class in the fall. For your convenience, each section will have notes provided before each practice set. There are also KhanAcademy.com links to videos embedded in the outline for each of the topics if you need further review or instruction. Instructions: Do all problems on your own paper showing all work in a neat and organized manner. You may receive help from any resource you would like as long as YOU understand how to do each problem. (KhanAcademy.com links are provided. Other internet resources include math.com, purplemath.com, mathisfun.com) This packet must be completed and turned in the first day of school in order to receive 2 points extra credit for the first quarter. Finally, it is suggested that you not wait until the last two weeks of summer to begin on this packet. If you spread it out, you will most likely retain the information much better. Once again this is due, completed with quality, on the first day of class. It is intended to help you be successful in the coming year

2 Algebra 1 Skills Needed to be Successful in Algebra 2 A. Simplifying Polynomial Expressions Objectives: The student will be able to: Apply the appropriate arithmetic operations and algebraic properties needed to simplify an algebraic expression. Simplify polynomial expressions using addition and subtraction. Multiply a monomial and polynomial. B. Solving Linear Equations Objectives: The student will be able to: Solve basic and multi-step equations. Solve a literal equation for a specific variable, and use formulas to solve problems. C. Rules of Exponents and Radicals Objectives: The student will be able to: Simplify expressions using the laws of exponents. Evaluate powers that have zero or negative exponents. Simplify radical expressions. D. Graphing Lines Objectives: The student will be able to: Identify and calculate the slope of a line. Graph linear equations using a variety of methods. Determine the equation of a line. E. Systems of Linear Equations Objectives: The student will be able to: Graph systems of 2 linear equations. Solve linear systems by graphing, substitution, and elimination. Write and solve linear system equations from a word problem F. Multiplying Binomials and Factoring Objectives: The student will be able to: Multiply two binomials with FOIL method. Identify the greatest common factor of the terms of a polynomial expression. Express a polynomial as a product of a monomial and a polynomial. Factor using difference of squares and perfect square trinomials. Find all factors of the quadratic expression ax 2 + bx + c by factoring. G. Quadratics Objectives: The student will be able to: Graph a quadratic equation in standard form. Identify the vertex, axis of symmetry, zeros or roots, x & y intercepts. Solve quadratics by graphing, square roots, and factoring. Solve quadratics using the quadratic formula. 2

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6 11. ½ x + 12 = (4x+1) -5 = 6(x-7) 6

7 5. Dave drove his car for 5 hours and 30 minutes for a distance of 220 miles. Find the average speed (rate) that Dave was driving. (Hint: d=r*t) 7

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9 II. Radicals Multiplying Radicals: Multiply the outside of one radical to the outside of the other, then multiply the inside of one radical to the inside of the other. Simplify the answer. Ex. 4: = 2* = = = 60 6 Adding Radicals: Treat like combining like terms- The radicals must be the same before you can add the coefficients together (the radicals do not change when you are adding) Ex. 5: =

10 PRACTICE SET 5 8. Multiply and simplify: Add or subtract and simplify:

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12 PRACTICE SET 6 Graph the lines using the given information

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14 Horizontal lines have the equation y = #, the graph goes straight left to right, have zero slope, a y- intercept, and typically no x-intercept. Ex: y = 8 Vertical lines have the equation x = #, the graph goes straight up and down, have no slope (or undefined slope), an x-intercept, and typically no y-intercept. Ex. x = -6 14

15 PRACTICE SET 7 15

16 IV. Finding Equations of Lines Slope Formula Slope-Intercept Form Point-Slope Form m = y 2 y 1 x 2 x 1 y = mx + b y y 1 = m(x x 1) where m=slope and (x 1, y 1) where m=slope and b = y-intercept where m=slope and (x 1, y 1) is a and (x 2, y 2) are points on the line. point on the line. Parallel lines have the same slope. Perpendicular lines have opposite reciprocal slope: (ex. if m = 2/3 for a line, then m = -3/2 for the perpendicular line) PRACTICE SET 8 Find the equation of the lines described. State your final answer in slope-intercept form. Do not graph. 1. Slope=3; y-intercept = Slope = 2/3 and passes through (6,8) 3. Passes through points (2,-7) and (-6,-3) 4. Parallel to y = ½ x -2 and passes through (-4,6) 5. Perpendicular to y = ½ x -2 and passes through (-4,6) 16

17 E. Systems of Equations I. Solve by Graphing One Solution No Solution Infinite Solutions 17

18 II. Solving Systems by Substitution Method The method of solving "by substitution" works by solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other. Then you back-solve for the first variable. Solve the following system by substitution: 2x 3y = 2 4x + y = 24 The idea here is to solve one of the equations for one of the variables, and plug this into the other equation. It does not matter which equation or which variable you pick. For instance, in this case, can you see that it would probably be simplest to solve the second equation for "y =". 4x + y = 24 y = 4x + 24 Now I'll plug this in ("substitute it") for "y" in the first equation, and solve for x: 2x 3( 4x + 24) = 2 2x + 12x 72 = 2 14x = 70 x = 5 Copyright Elizabeth Stapel All Rights Reserved Now I can plug this x-value back into either equation, and solve for y. y = 4(5) + 24 = = 4. Then the solution is (x, y) = (5, 4). III. Solving Systems by Elimination/Adding Method The addition method of solving systems of equations is also called the method of elimination. Solve the following system using addition. 2x + y = 9 3x y = 16 Note that, if I add down, the y's will cancel out. So I'll draw an "equals" bar under the system, and add down: 2x + y = 9 3x y = 16 5x = 25 Now I can divide through to solve for x = 5, and then substitute the answer for x back into one of the equations. You can use either of the original equations, to find the value of y. The first equation has smaller numbers, so I'll use that one: 2(5) + y = y = 9 y = 1 Then the solution is (x, y) = (5, 1). 18

19 PRACTICE SET 9 6. At the homecoming football game, the senior class officers sold slices of pizza for $0.75 each and hamburgers for $1.35 each. They sold 40 more slices of pizza than hamburgers, and sales totaled $ How many slices of pizza did they sell? 19

20 F. Multiplying Binomials and Factoring 20

21 PRACTICE SET 10 21

22 Factoring 22

23 III. Factoring Trinomials a*c = 6*4 = 24 23

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25 PRACTICE SET x 2-11x x x 2 +9x x

26 I. Graphing Quadratics G. Quadratics Steps for Graphing a Quadratic Equation in Standard Form 1. Determine if the graph will open up or down. Opens up if "a" is positive (the vertex point will be the minimum point). Opens down if "a" is negative (the vertex point is the maximum point). 2. Find the vertex point. Find the x-value by x = -b/(2a). Find the y-value by substituting the x-value into the equation and solving for "y". 3. Find more points to determine the graph. Choose two integers larger than the x-value of the vertex point. Choose two integers smaller than the x-value of the vertex point. Substitute these values in place of "x" in the equation and solve for "y". Four ordered pairs have been found. 4. Graph and connect all points that have been found. Ex. 1: y = 2x² - 8x + 6 Axis of symmetry = x = b 2a X= ( 8) 2(2) = 8 4 = 2 Y= 2(2) 2 8(2) + 6 = = -2 Vertex (2, -2) X Y vertex: (2,-2) II. Identifying parts of a quadratic graph Parabola - the graph of a quadratic equation, which is u-shaped Standard Form - y = ax² + bx + c Vertex - it's the highest or lowest point on the graph Axis of Symmetry - the vertical line that goes through the vertex point X-Intercept- Where the graph crosses the x-axis Y-Intercept- Where the graph crosses the y-axis 26

27 PRACTICE SET 12 For each of the following, fill out the table of values and graph the parabola. 1. y = x X Y 2. y = 2x 2 3 X Y 27

28 Identify the coordinates of the Vertex, the axis of symmetry, any x-intercept(s) and y-intercept(s). Vertex: AOS: x= x-int: y-int: Vertex: AOS: x= x-int: y-int: Vertex: AOS: x= x-int: y-int: Vertex: AOS: x= x-int: y-int: 28

29 II. Solving Quadratic Equations Solve by graphing: Get the whole quadratic equation on one side of the equal sign and graph the function. The solutions are the x-intercepts on the graph. Solve with square roots: Get x 2 on one side of the equals sign and the numbers on the other. Take the square root and simplify. Remember that when you take a square root there are two possible answers, one positive and one negative. Also, you cannot take the square root of a negative number to get a real solution. Ex.1: x 2 +5 = 20 Ex. 2: x 2 = 49 x 2 = 15 x = ± 7 x= ± 15 Solve by factoring: To solve a quadratic equation by factoring, 1. Put all terms on one side of the equal sign, leaving zero on the other side. 2. Factor. 3. Set each factor equal to zero. 4. Solve each of these equations. 5. Check by inserting your answer in the original equation. Example : Solve x 2 6 x = 16. x 2 6 x = 16 becomes x 2 6 x 16 = 0 (x 8)( x + 2) = 0 x-8=0 and x+2 = 0 x=8 and x=-2 Then to check, (8) 2 6(8) = 16 and (-2) 2 6(-2) = = = 16 Both values, 8 and 2, are solutions to the original equation. 29

30 Solve by completing the square: 1. Put the equation into the form ax 2 + bx = c. 2. Make sure that a = 1 (if a 1, multiply through the equation by 1 before proceeding). a 3. Using the value of b from this new equation, add ( b 2 )2 to both sides of the equation to form a perfect square on the left side of the equation. 4. Find the square root of both sides of the equation. 5. Solve the resulting equation. Solve with Quadratic formula: Get the whole quadratic equation on one side of the equal sing and plug the coefficients into the quadratic formula. Remember, it will yield two answers, one with adding and one with subtraction. Ex. x 2-5x + 3 a=1, b= -5, c=3 x = ( 5) ± ( 5)2 4(1)(3) 2(1) x = 5 ± (1) x = 5 ± ,

31 PRACTICE SET 13 Solve the following by square roots or factoring. Please show your work. 1) x 2 + 7x = 0 2) x 2 9 = 0 3) x 2 7x + 12 = 0 4) x 2 + 4x + 3 = 0 5) 3x 2 = -30 6) x 2 + 8x = 20 7) 2x = -7x 8) 6x x +10 = 0 Solve the following using the quadratic formula or by completing the square. Please show your work. 31

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