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Math Review

Math Review. Algebra and Functions Review. Operations on Algebraic Expressions. You will need to be able to apply the basic operations of arithmetic – addition, subtraction, multiplication, and division - to algebraic expressions In examples like this: 4x+5x=9x 10z-3y-(-2z)+2y = 12z-y

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Math Review

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  1. Math Review Algebra and Functions Review

  2. Operations on Algebraic Expressions • You will need to be able to apply the basic operations of arithmetic – addition, subtraction, multiplication, and division - to algebraic expressions • In examples like this: • 4x+5x=9x • 10z-3y-(-2z)+2y = 12z-y • (x+3)(x-2) = x^2+x-6

  3. Factoring • Difference of 2 squares • a^2-b^2 = (a+b)(a-b) • Finding common factors • x^2+2x = x(x+2) • 2x+4y = 2(x+2y) • Factoring quadratics • x^2-3x-4 = (x-4)(x+1) • X^2+2x+1 = (x+1)(x+1) = (x+1)^2

  4. Exponents • Definitions • a^3 = a x a x a • P^-4 = 1/P x 1/P x 1/P x 1/P • X^0 = 1 • X ^ (a/b) = (b square root of x)^a • Y^ (1/2) = (square root of y)

  5. Exponents • Three Points to Remember • When multiplying expressions with the same base, add the exponents • a^2 * a^5 = a^7 • t^5 * t^-2 = t^3 • When dividing expressions with the same base, subtract exponents • r^5/r^3 = r^2 • a^m/a^n = a^ (m-n) • When a number raised to an exponent is raised to a second exponent, multiply the exponents • (n^3)^6 = n^18 • (a^m)^n = a^mn

  6. Evaluating Expressions with Exponents and Roots • You will need to know how to evaluate expressions involving exponents and roots • If y = 8, what is y^(2/3) • y^(2/3) = 3rd square root of 8^2 = 4 • If x^(3/2) = 64, what is the value of x? • x^(3/2) = (x^3)^1/2 = 64 • Cube both sides, and x^1/2 = 4 • Square both sides and x = 16

  7. Solving Equations • Working with “Unsolvable” Equations • Though you can’t solve the equation, you can answer the question. • If a+b=5, what is the value of 2a+2b? • You can’t solve for a or b, but it doesn’t ask that • Factor: 2(a+b), so 2(5) = 10

  8. Solving Equations • Solving for One Variable in Terms of Another • You will not always be able to find a specific, numerical value for all the variables, but you can still solve for one variable in terms of another one • If 3x+y= z, what is x in terms of y and z? • Put X on one side of equation by itself • 3x=z-y, then x = (z-y)/3

  9. Solving Equations • Involving Radical Expressions • 5(square root of x) is a radical expression because it involves a root • A radical equation is one that involves a radical expression • 5(square root of x) +14 = 29 • 5(square root of x) = 15 • (square root of x)=3 • X=9

  10. Absolute Value • Absolute Value of a number is its distance from zero on the number line. Denoted: IxI • I6.5I=6.5, I-32I=32 • Think of it as its “size” of the number, disregarding pos or neg • Example • I7-tI=10 • Can be split up into two equations: • 7-t=10 and –(7-t)=10 • t=-3 or t=17

  11. Direct Translation into Mathematical Expressions • Many word problems require you to translate the verbal description of a mathematical fact or relationship into mathematical terms. • Always read the word problem carefully and double-check that you have translated it exactly. • “3 times the quantity” (4x+6) is 3(4x+6) • “A number y decreased by 60” is y-60 • “20 divided by n” is (20/n) • “20 divided into a number y” is (y/20)

  12. Inequalities • An inequality is a statement that one quantity is greater than or less than another • Symbols: • Greater than: > (5>3) • Less than: < (-7 < -6) • A line under the symbol means “or equal to” • Example: • 2x+1>11 • 2x>11-1, 2x>10 • So, x>5

  13. Systems of Linear Equations and Inequalities • You may be asked to solve systems of two or more linear equations or inequalities. • Example: • For what values of a and b are the following equations true? a+2b=1, -3a-8b=1 • Eliminate one of the variables, let’s do b. so multiply both sides by 4 to get the b’s equal. • 4a+8b=4 + -3a-8b=1 • Add the two equations: a=5, then put back into first equation and you get b=-2

  14. Solving Quadratic Equations by Factoring • You may be asked to solve quadratic equations that can be factored. You will not be asked to use the quadratic formula. • Example: • For what values of x is x^2-10x+20=-4 • Add 4 to both sides to get the standard quad. Formula: x^2-10x+24=0 • Now factor: (x-4)(x-6)=0 • So, either x-4=0 and x-6=0, so x=4 and x=6

  15. Rational Equations and Inequalities • A rational algebraic expression is the quotient of two polynomials. • Example: • For what value of x is the following equation true? 3=(x-1)/(2x+3) • Multiply both sides by (2x+3) • 6x+9=x-1 • 5x=-10 • X=-2

  16. Direct and Inverse Variation • The quantities x and y are directly proportional if y=kx for some constant k. • If x and y are directly proportional, when x is 10, y is equal to -5. If x=3, then what is y? • y=kx, so solve for k with 10 and -5 • -5=k(10), k=(-1/2) • So then plug in k with 3: y=3(-1/2) • y=(-3/2) • Inversely proportional: y=k/x • If xy=4, show the x and y are inversely proportional • So y=4/x, so with k as 4, then they are inversely prop.

  17. Word Problems • Some math questions are presented as word problems to apply math skills to everyday situations. With word problems you need to: • 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. When checking word problems, don’t substitute your answer into your equations, because they may be wrong. Instead, check word problems by checking your answer with the original words.

  18. Word Problems • As you read the problems, translate the words into mathematical expressions and equations. • Is, Was, Has means = (equals) • More than, older than, farther than, greater than, sum of means + (addition) • Less than, difference, younger than, fewer means – (subtraction) • Of means x (multiplication, percent) • For, per means / (ratio or division)

  19. Word Problems • Example • The price of a sweater went up 20% since last year. If last year’s price was x, what is this year’s price in terms of x? • So, last year’s price is 100% of x • This year’s price is 100% of x plus 20% of x • x+(20% * x) = x+.2x = 1.2x

  20. Functions • Function Notation and Evaluation • A function can be thought of as a rule or formula that tells how to associate the elements in one set (or the domain) with the elements in another set (or the range) • Example: The “Squaring Function” can be thought of as the rule “taking the square of x” or the rule x^2. • Function notation lets you write complicated functions much more easily. • g is defined by g(x)=3^x+(1/x). Then g(2)=3^2+(1/2)= 9 (1/2)

  21. Functions • Domain and Range • The domain of a function is the set of all the values for which the functions is defined. • The range of a function is the set of all values that are the output, or result, of applying the function. • Example: What are the domain and range of f(x)=1+(square root of x) • Domain of f is the set of all values of x for which the formula is defined. This formula makes sense if x is 0 or a positive number (can’t be negative): so domain is all nonnegative numbers x • Range would be the set of all possible values of the equation. It must be at least 1 (b/c of the square root). Is r greater than or equal to 1, though? See if it could be higher: • 1+(square root of x)=9, (square root of x)=8, x=64, so it would be greater than or equal to 1

  22. Using New Definitions • For functions, especially those involving more than one variable, a special symbol is sometimes introduced and defined. They will have unusual looking signs to not be confused with standard mathematical symbols. • The key to these questions is to make sure you read the definition carefully. • Example: = wy-xz, where w,y,x, and z are integers. What is the value of

  23. Functions as Models • Functions can be used as models of real life situations. • The temp. in City X is W(t) degrees Fahrenheit t hours after sundown at 5 pm. The function W(t) is given by: • W(t)=.1(400-40t+t^2) for 0 is less than or equal to t is less than or equal to 12.

  24. Linear Functions: Their Equations and Graphs • Linear Function: y=mx+b, where m and b are constants • m is the slope, and y intercept is b

  25. Quadratic Functions: Their Equations and Graphs • y=x^2-1 • Graphs will look like a u (going up or down). A negative answer (once solved) will go down and positive will go up. Largest number of y will occur when x equals zero.

  26. Qualitative Behavior of Graphs and Functions • Know how properties of a function and its graph are related. • The zeros of the function f are given by the points where the graph of f(x) in the xy plane intersects the x axis.

  27. Translations and their Effects on Graphs of Functions • If you are given the graph g(x) you should be able to identify the graph of g(x+3) • You can plug in numbers to find out specific points.

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