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Algebra and functions review

- 1. Algebra and Functions Review
- 2. The SAT doesn’t include: • Solving quadratic equations that require the use of the quadratic formula • Complex numbers (a +b i) • Logarithms
- 3. Operations on Algebraic Expressions Apply the basic operations of arithmetic—addition, subtraction, multiplication, and division—to algebraic expressions: 4x + 5x = 9x 10z -3y - (-2z) + 2 y = 12z - y (x + 3)(x - 2) = x2 + x - 6 x yz z x y z xy 3 5 3 4 3 2 2 24 = 8 3
- 4. Factoring Types of Factoring • You are not likely to find a question instructing you to “factor the following expression.” • However, you may see questions that ask you to evaluate or compare expressions that require factoring.
- 5. Exponents x4 = x × x × x × x y - 3 = 1 3 y a xb = b xa = ( b x ) a 1 x2 = x Exponent Definitions: a0 = 1
- 6. • To multiply, add exponents x2 × x3 = x5 xa × xb = xa+b • To divide, subtract exponents x 5 x 2 x x x x x x x x = 3 = - 3 = = - 1 2 5 3 m m n n • To raise an exponential term to an exponent, multiply exponents (3x3 y4 )2 = 9x6 y8 (mxny )a = maxnay
- 7. Evaluating Expressions with Exponents and Roots Example 1 2 If x = 8, evaluate x3 . 2 83 = 3 82 = 3 64 = 4 or use calculator [ 8 ^(2/3)] Example 2 If , what is x ? 3 x2 = 64 2 æ 3 ö 3 2 ç x2 ÷ = (64)3 ® è ø x = 3 642 ® (( ) )2 x = 3 43 × 3 43 ® 3 x = 3 4 → x = 4× 4® x = 16
- 8. Solving Equations • Most of the equations to solve will be linear equations. • Equations that are not linear can usually be solved by factoring or by inspection.
- 9. "Unsolvable" Equations • It may look unsolvable but it will be workable. Example If a + b = 5, what is the value of 2a + 2b? • It doesn’t ask for the value of a or b. • Factor 2a + 2b = 2 (a + b) • Substitute 2(a + b) = 2(5) • Answer for 2a + 2b is 10
- 10. Solving for One Variable in Terms of Another Example If 3x + y =z, what is x in terms of y and z? • 3x = z – y • x = z - y 3
- 11. Solving Equations Involving Radical Expressions Example 3 x + 4 = 16 3 x = 12 3 12 3 3 x = x = 4 ®( )2 x = 42 → x = 16
- 12. Absolute Value Absolute value • distance a number is from zero on the number line • denoted by • examples x -5 = 5 4 = 4
- 13. • Solve an Absolute Value Equation Example 5 - x = 12 first case second case 5 - x = 12 5 - x = -12 -x = 7 -x = -17 x = -7 x = 17 thus x=-7 or x=17 (need both answers)
- 14. Direct Translation into Mathematical Expressions • 2 times the quantity 3x – 5 • a number x decreased by 60 • 3 less than a number y • m less than 4 • 10 divided by b Þ 4 - m • 10 divided into a number b Þ x - 60 10 b Þ Þ 2(3x - 5) Þ y - 3 Þ b 10
- 15. Inequalities Inequality statement contains • > (greater than) • < (less than) • > (greater than or equal to) • < (less than or equal to)
- 16. Solve inequalities the same as equations except when you multiply or divide both sides by a negative number, you must reverse the inequality sign. Example 5 – 2x > 11 -2x > 6 x -2 > 6 -2 -2 x < -3
- 17. Systems of Linear Equations and Inequalities • Two or more linear equations or inequalities forms a system. • If you are given values for all variables in the multiple choice answers, then you can substitute possible solutions into the system to find the correct solutions. • If the problem is a student produced response question or if all variable answers are not in the multiple choice answers, then you must solve the system.
- 18. Solve the system using • Elimination Example 2x – 3y = 12 4x + y = -4 Multiply first equation by -2 so we can eliminate the x -2 (2x - 3y = 12) 4x + y = -4 -4x + 6y = -24 4x + y = -4
- 19. Example 2x – 3y = 12 4x + y = -4 continued Add the equations (one variable should be eliminated) 7y = -28 y = -4 Substitute this value into an original equation 2x – 3 (-4) = 12 2x + 12 = 12 2x = 0 x = 0 Solution is (0, -4)
- 20. Solving Quadratic Equations by Factoring Quadratic equations should be factorable on the SAT – no need for quadratic formula. Example x2 - 2x -10 = 5 x2 - 2x -15 = 0 subtract 5 (x – 5) (x + 3) = 0 factor x = 5, x = -3
- 21. Rational Equations and Inequalities Rational Expression • quotient of two polynomials • 2 x 3 x 4 Example of rational equation - + 3 4 x x + = Þ - 3 2 x + 3 = 4(3x - 2) x + 3 = 12x - 8 Þ 11x = 11Þ x = 1
- 22. Direct and Inverse Variation Direct Variation or Directly Proportional • y =kx for some constant k Example x and y are directly proportional when x is 8 and y is -2. If x is 3, what is y? Using y=kx, Use , 2 - = k ´8 1 4 k = - k = - (- 1)(3) 1 4 y = 3 4 4 y = -
- 23. Inverse Variation or Inversely Proportional • y k = for some constant k x Example x and y are inversely proportional when x is 8 and y is -2. If x is 4, what is y? • Using y = k , -2 = k x 8 • Using k = -16, -16 4 y = k = -16 y = - 4
- 24. Word Problems With word problems: • Read and interpret what is being asked. • Determine what information you are given. • Determine what information you need to know. • Decide what mathematical skills or formulas you need to apply to find the answer. • Work out the answer. • Double-check to make sure the answer makes sense. Check word problems by checking your answer with the original words.
- 26. Functions Function • Function is a relation where each element of the domain set is related to exactly one element of the range set. • Function notation allows you to write the rule or formula that tells you how to associate the domain elements with the range elements. f (x) = x2 g(x) = 2x +1 Example Using g(x) = 2x +1 , g(3) = 23 + 1 = 8+1=9
- 27. Domain and Range • Domain of a function is the set of all the values, for which the function is defined. • Range of a function is the set of all values, that are the output, or result, of applying the function. Example Find the domain and range of f (x) = 2x -1 2x – 1 > 0 x > 1 2 domain 1 or 1 , = ìí x ³ üý êé ¥ö¸ î 2 þ ë 2 ø range = { y ³ 0} or [ 0,¥)
- 28. Linear Functions: Their Equations and Graphs • y =mx + b, where m and b are constants • the graph of y =mx + b in the xy -plane is a line with slope m and y -intercept b • rise slope slope= difference of y's run difference of x's =
- 29. Quadratic Functions: Their Equations and Graphs • Maximum or minimum of a quadratic equation will normally be at the vertex. Can use your calculator by graphing, then calculate. • Zeros of a quadratic will be the solutions to the equation or where the graph intersects the x axis. Again, use your calculator by graphing, then calculate.
- 30. Translations and Their Effects on Graphs of Functions Given f (x), what would be the translation of: 1 f ( x ) 2 shifts 2 to the left shifts 1 to the right shifts 3 up stretched vertically shrinks horizontally f (x +2) f (x -1) f (x) + 3 2f (x)