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STEM 9 Algebra 2 Summer Review 2016 Welcome to STEM 9! I am Ms. Blasko and I will be your STEM Algebra 2 teacher. This coming school year we will build on the skills you learned in Algebra 1 and Geometry. The purpose of the summer review is to remind you and reinforce what you learned in previous math classes. I suggest working on this review in the few weeks before returning to school. This material will not be graded. I will provide an answer key and can review your work with you the first week of school. For each topic addressed, there are examples, explanations, and/or references, followed by a short set of practice problems. Topics: 1. 2. 3. 4. 5. 6. 7. 8. Fractions Simplify Polynomial Expressions Solve Equations Rules for Exponents Radicals Slope/Rate of Change Graphing Lines Right Triangles Note that there is a mini lesson and a link to an online resource for each topic. If you do not remember how to complete a problem you are expected to research and review the topic. Have a wonderful summer and see you in August! Ms. Blasko [email protected] 1. Fractions Multiplying fractions: a c ac b d bd 2 5 10 3 7 21 Dividing fractions: a c a d ad b d b c bc 1 2 1 7 7 4 7 4 2 8 Adding or subtracting fractions without a common denominator: a c d a c b da cb da cb b d d b d b db db db 3 2 3 3 2 4 9 8 9 8 17 4 3 3 4 3 4 12 12 12 12 Practice Set I: Perform the following operations. Write answers in the lowest terms. 1. 12 x 3xz 5y 4y 2. 3a 7b 5b 5c 3. 5 6 x y 4. 4a 6a b c http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut3_fractions.htm 2. Simplify Polynomial Expressions Combine like terms. “Like” terms have the same variable to the same power. Example: 8x2 + 10x3 – 5x2 + 3x3 8x2 – 5x2 + 10x3 + 3x3 3x2 + 13x3 Apply the Distributive Property: Example: 5(4x – 6) 5 • 4x – 5 • 6 20x – 30 Combine Like Terms and Apply the Distributive Property Example: 4(6x – 3y) + 5(2x + 7y) 4 • 6x – 4 • 3y + 5 • 2x + 5 • 7y 24x – 12y + 10x + 35y 34x + 23y Practice Set 2. Simplify. Show all work. 1. 3(5a – b) + 4(2a – 2b) 2. -5(4x – 7) + 13 – 6x 3. 8(3x + 5y – 6z) – 2(2x + 4z). 4. 2(3x – 4) – (12x + 3) http://www.khanacademy.org/math/algebra/polynomials/polynomial_basics/v/simply-a-polynomial 3. Solve Equations Simplify both sides of the equation. Use addition/subtraction to move variables to one side, constants to the other. Use multiplication/division to solve for the variable. Example: 4(x + 7) = 22 – 2x 4x + 28 = 22 – 2x distribute the 4 +2x add 2x to both sides +2x 6x + 28 = 22 - 28 -28 6x subtract 28 from both sides = -6 x = -1 divide both sides by 6 Problem Set 3. Solve each equation. Show all work. 1. 45x – 720 + 15x = 60 2. 8(3x – 4) = 232 2. -131 = -5(3x – 8) + 6x 4. – 7x – 22= 18 + 3x 5. 6. http://www.purplemath.com/modules/solvelin.htm http://www.purplemath.com/modules/solvelin3.htm 4. Rules for Exponents Practice Set 4. Simplify each expression. 1. (-3m2n)4 2. (x2y4)(x3y5) 12a 6 b 9 3. 6a 4 b 3 c 3x 2 4. 4 12 x 3 http://www.mathsisfun.com/algebra/exponent-laws.html 5. Radicals Practice Set 5. Simplify each radical. Show your work. 1. 486 3. 475 2. 500 http://hotmath.com/help/gt/genericalg1/section_8_1.html 6. Slope/Rate of Change The slope of a line describes its steepness, or how it angles away from the horizontal. The slope of a line is a rate of change and can expressed as a relationship between two variables, such as miles per gallon or cost per pound. slope m change in y vertical change rise y 2 y1 change in x horizontal change run x2 x1 Find the slope of the line passing through points (3, -1) and (-2, 5) m 5 (1) 6 23 5 Practice Set 6. Find the slope of the line passing through each pair of points. Show your work. 1. (7, -9) (-1, 5) 2. (4, 0) (-6, 6) F ind the slope of the lines represented on the graphs below. y 3. 9 4. y 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 x x 1 2 3 4 5 6 7 8 9 1 2 3 4 http://www.regentsprep.org/Regents/math/ALGEBRA/AC1/Rate.htm http://www.purplemath.com/modules/slopyint.htm 5 6 7 8 9 7. Write Linear Equations/Graph Lines You can find the equation of a line from two points, or from one point and the slope. Slopeintercept form, y = mx +b, where m Is the slope and b is the y-intercept, is one form of a linear equation that is particularly easy to graph. Example: Write the equation of a line with a slope of 3 and passing through the point (5, 7) y = mx + b 7 = 3(5) + b Equation: y = 3x – 8 -8 = b Example: Write the equation of a line that passes through (2, 9) and (-1, 3) m 39 6 2 1 2 3 y mx b 9 2( 2) b b5 y 2x 5 Problem Set 7. Write an equation, in slope-intercept form, using the given information. 1. m = -⅓ (-3, 6) 2. (-4, -1) (4, 5) Graph the lines. 3. y = (2/3)x - 4 4. y = -3x + 3 y y 6 5 4 3 2 1 6 5 4 3 2 1 x -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6 7 8 9 x -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6 7 8 9 http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut14_lineargr.htm 8. Right Triangles Pythagorean Theorem : a 2 b 2 c 2 SOHCAHTOA opp adj opp cos tan hyp hyp adj a b a sin A cos A tan A c c b b b b sin B cos B tan B c c a sin Example: Given that a = 6 and b = 8, find the length of the hypotenuse. c 2 a 2 b 2 so c 2 6 2 8 2 36 64 100 c 2 100, c 10 Problem Set 8. Given right triangle ABC, where C = 90º, solve for the missing side. Show all work. 1. a = 12, b = 5, find c. 2. b = 15, c = 17, find a. Given that a = 9, b = 40, and c = 41, find the trig ratio. 3. sin B 4. tan A http://www.mathsisfun.com/algebra/trig-finding-angle-right-triangle.html http://www.purplemath.com/modules/basirati.htm