Algebra and functions review
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This document provides an overview of key algebra and functions concepts covered on the SAT, including operations on algebraic expressions, factoring, exponents, evaluating expressions, solving equations, inequalities, systems of equations, quadratic equations, rational expressions, direct and inverse variation, word problems, functions, and translations of functions. Key points are that the SAT will not include solving quadratic equations using the quadratic formula or complex numbers. Factoring may be required to solve or evaluate expressions. Functions are represented using function notation and defined by their domain and range.
![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](https://image.slidesharecdn.com/algebraandfunctionsreview-140825100631-phpapp01/85/Algebra-and-functions-review-7-320.jpg)
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)