What are the place values to the left of the decimal point and their associated powers of ten?


 Roy Warner
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1 The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything in the prealgebra memorization guide should be memorized as well.) Number Basics Starting with the counting numbers and ending with the real numbers, what are the different classifications of numbers and their associated descriptions? Counting numbers The numbers used for counting (,,, 4,, 6,...) Whole numbers The counting numbers and zero (0,,,, 4,, 6,...) Integers Negative and positive counting numbers and zero (...,,,, 0,,,,...) 0 Rational numbers Numbers that can be written as a ratio of two integers (i.e.,, ) 7 Irrational numbers Numbers that cannot be written as a ratio of two integers (i.e., π, ) Real numbers All rational and irrational numbers (i.e.,,, 7, π, ) 4 What are the place values to the left of the decimal point and their associated powers of ten? 4 0,... 0 Ones ( ) 0, tens ( 0 ), hundreds ( 0 ), one thousands ( 0 ), ten thousands ( ) What are the place values to the right of the decimal point and their associated powers of ten? 4 0,... Tenths ( ) 0, hundredths ( 0 ), thousandths ( 0 ), ten thousandths ( ) What is the rule for order of operations? PEMDAS. Parenthesis first, exponents next, then multiplication and division left to right, finally addition and subtraction left to right = + 6 = = + 6 = 7 example: ( ) Algebra Basics What is a variable? A variable is a letter or symbol used to represent a number (often unknown). example: x and y are variables in the expression x 4y. How do you tell if something is a solution to an equation? You plug the value in and see if the equation is true. example: Is x = 7 a solution of the equation x + = 8. Yes, since 7 + = 8. What are domain and range? The domain is the permitted inputs (typically x) and the range is the associated outputs (typically y). What are the restrictions on domain? (The restrictions an algebra student should know) You cannot divide by zero. You cannot have a negative expression in a square root ( x ). Algebra (page /0)
2 What is a function? How can a function relationship be represented? A function is a relationship that maps inputs (domain) to outputs (range). Each input has exactly one output. Functions can be represented verbally, numerically (tables, lists of ordered pairs), graphically, and algebraically (using an equation). What is meant by the distributive property? When an item is multiplied by a quantity, the item needs to be multiplied by all the items in the quantity x + y + z = x + xy + xz examples: ( + ) = + = and ( ) What is a term? What are like terms? A term is set of numbers and/or variables that are all multiplied and/or divided by each other. Terms are separated by addition or subtraction. Like terms are terms that have the same variable portion. example: In the expression "x + 4xy + 8 ", the terms are x, 4 xy, 8, and x x example: xy and 0 xy are like terms since the variable portion xy is the same for both terms. How are distances and rates related? Distance equals rate times time. D = RT example: If a driver drove 7 miles at 0mph, how long did the trip take? 7 = 0 x x =. hours What is direct variation and what is its model? Direct variation is a model where outputs get larger when inputs get larger and outputs get smaller when inputs get smaller. The input and outputs are directly related. The model is y = kx. What is inverse variation and what is its model? Inverse variation is a model where outputs get larger when inputs get smaller and outputs get smaller when inputs get larger. The input and outputs are inversely related. The model is y = k x. How do you find xintercepts? Set y = 0 and solve for x. How do you find yintercepts? Set x = 0 and solve for y. Solving equations How does order of operations affect solving equations? Solving equations is undoing steps. Essentially it is PEMDAS backwards. Undo the addition and subtraction, then undo the multiplication and division, etc. How do you solve linear equations? Get all the variables to one side, everything else to the other, factor if necessary, and then divide. examples: x + 8 = x 8 + = x x = x x = 7 π π ( x + ) = x π x + π = x π = x π x π = x ( π ) x = π Algebra (page /0)
3 How do you solve equations with fractions? Multiply every term by the common denominator of all the fractions to eliminate them and then solve the new equation that no longer has fractions. example: x + = 4 x x + = 4 x 0x + 4 = x 4 = x x = 4 How do you solve quadratic equations? Get everything to one side. Set the equation equal to zero. Factor and set the factors equal to zero. If the equation doesn t factor, use the quadratic formula. x = 6x 8 x 6x + 8 = 0 x x 4 = 0 x = or 4 examples: ( )( ) ± ± x + x = x x + 0 = x x + x = x = 4 ( ) ( ) 4 7 Why do we set quadratic equations equal to zero and factor them when we solve them? If two numbers are multiplied and the answer is zero then one of the numbers must be zero. This is the ZeroProduct Property. What is the quadratic formula? When can it be used? Negative b plus or minus the square root of b squared minus four times a times c all over two times a. ± b b 4ac a. This formula can be used on quadratic equations in the form How do you solve equations with square roots? Get the square root all by itself. Square both sides to undo the square root. example: ( ) ax bx c + + = 0. x + 4 = x + = 9 x + = 9 x + = 8 x = 78 How do you solve equations with absolute value? Since the expression inside the absolute value can be either positive or negative, you must solve two equations. Solve it as is and then make the expression inside the absolute value negative and solve it again. Both answers are solutions. example : x + = a) x + = x = 0 x = b) (x + ) = x + = x = x = 6 How does solving inequalities differ from solving equalities? When you multiply or divide by negative numbers, you must flip the inequality symbol. example: x + 8 < x < 6 x > How do you solve inequalities with absolute value? Since the expression inside the absolute value can be either positive or negative, you must solve two equations. Solve it as is and then make the expression inside the absolute value negative and solve it again. If it is < or the keyword is and. If it is > or the keyword is or. example : x + a) x + x 0 x b) (x + ) x + x x 6 so the answer will x 6 and x which can also be written as 6 x Algebra (page /0)
4 Lines What is the slopeintercept form for the equation of a line? y = mx + b where m is the slope and b is the yintercept. What is the pointslope form for the equation of a line? y y = m x x where m is the slope and x, y is the point. ( ) ( ) What two things do you need in order to write an equation of a line? A point and a slope. How do you write an equation of a line? Find a point on the line and the slope of the line. If you are using pointslope, just plug everything in. If you are using slopeintercept, plug in the point and the slope and solve for b. Write out your answer. example: Write the equation of a line in slopeintercept format going through (, ) with a slope of. y = mx + b = + b = 6 + b = b y = x What are the relationships between parallel and perpendicular lines? Parallel lines have the same slope and perpendicular lines have negative reciprocal slopes. examples: The two lines y = x + and y = x 4 are parallel. 4 The two lines y = x and y = x + are perpendicular. 4 Explain the process of solving a system of equations using substitution. You solve one of the equations for one of the variables and substitute it into the other equation. Then this new equation that has only one variable in it is solved. Finally, substitute the value of the solved variable back into the st equation you solved to find the value of the other variable. example: x + y = 8 x + ( 4x ) = 8 x = (, ) 4x y = y = 4x y = 4 y = Explain the process of solving a system of equations using elimination. The original equations are multiplied by numbers to get one of the variables equal and opposite. These new equations are added together to eliminate that variable. This new equation containing only one variable is solved. That answer is plugged back into one of the original equations to find the value of the other variable. x + y = 8 x + y = 8 example: ( ) 4x y = x y = 6 4 y = y =, Exponents and Radicals 4 x = 4 x = A nonzero number raised to the zero power is? One 0 example: = Algebra (page 4/0)
5 Zero raised to a nonzero power is? Zero example: 0 = 0 If you move a base with a negative exponent from the numerator to the denominator or vice versa what happens? The exponent becomes positive. (The opposite is true as well. If you move a base with a positive exponent from the numerator to the denominator or vice versa, the exponent becomes negative.) 4 examples: a) = b) = 4 What happens when identical bases with exponents are multiplied? The exponents are added example: = What happens when identical bases with exponents are divided? The exponents are subtracted. 9 example: = What happens when a base with an exponent is raised to another exponent? The exponents are multiplied. example: ( ) = 4 What happens when more than one item is raised to an exponent? Each item will be raised to the exponent. 4 4 example: ( ) ( ) xy = x y = 8x y How do you simplify a square root of a number? Find the prime factorization of the number. Whenever there are two identical factors, they can be taken outside of the square root as a single item. (Alternatively, you can look for perfect squares. Once found, the square root of the perfect squares are brought outside.) examples: = =, 90 = = 0, and 00 = = 0 alternative method examples: = 4 =, 90 = 9 0 = 0, and 00 = 00 = 0 In order to simplify a square root, what must be true of the numbers inside the square root? All the numbers in the square root must be multiplied or divided by each other. examples: 6 = =, = = + 4, and x + 9 is already fully simplified. How do you simplify expressions where the denominator is the square root of a number? If the denominator is a square root, then the numerator and the denominator are multiplied by that square root. example: = = = Algebra (page /0)
6 How do you multiply two numbers written in scientific notation? Multiply the numbers. Add the exponents. Convert back to scientific notation if necessary = = = 0 7 example: ( )( ) How do you divide two numbers written in scientific notation? Divide the numbers. Subtract the exponents. Convert back to scientific notation if necessary = 0. 0 = 0 0 = 0 example: ( ) ( ) Polynomials What is the degree of a polynomial? The degree of a polynomial is the same as the largest exponent of a polynomial. example: What is the degree of f ( x) = 4x 8x + 0x? f ( x ) is a rd degree polynomial. How do you add or subtract polynomials? The coefficients of the like terms are combined together. (Do not forget to distribute the negative sign for subtraction.) x 4x + 8 x + x + 0 = x 4x + 8 x x 0 = x 9x example: ( ) ( ) What is the process for multiplying binomials? Explain this process. FOIL. The first terms are multiplied, the outer terms are multiplied, the inner terms are multiplied, and the last terms are multiplied. The polynomial is then simplified. example: ( x + )( x ) = 6x 0x + 9x = 6x x Explain the process for multiplying polynomials. Each term of the first polynomial is multiplied by every term of the second polynomial. example : x + x + x 4x + = x x 4x + + x x 4x + + x 4x + ( )( ) ( ) ( ) ( ) How do you multiply ( a + b) and ( a b) ( ) ( ) +? = x 8x 4 x x x 6 x x 4x = x x 7x x a + b = a + ab + b and a b = a ab + b example: ( ) x y = x 0xy + 4 y How do you multiply ( a + b)( a b) a b +? example: ( )( ) x + y x y = 4x 9 y Explain the process of factoring (algebra topics only)? Factor out the greatest common factor first. Then look for the quadratic factoring patterns of a b, x + bx + c, and ax + bx + c. Finally, look for factoring by grouping. Algebra (page 6/0)
7 How do you factor the pattern of a + b a b ( )( ) a b? example: 6y 4 x = ( 4 y + x)( 4 y x) How do you factor the pattern x + bx + c? You look for two numbers that multiply to the constant term c and add to the middle term b. The factors are then written out as x plus those numbers. x x 8 = x + 4 x + = x 4 x + example: ( )( ) ( )( ) How do you factor the pattern ax + bx + c? You look for two numbers that multiply to the leading coefficient a times the constant term c and add to the middle term b. The factors are then written out as ax plus those numbers and with everything also divided by a. Simplify. ( x + )( x + ) example: x + x = = ( x + )( x ) Explain the process of factoring by grouping? Factoring by grouping works when there are an even number of terms. A common factor is taken out of the st set of terms and another common factor is taken out of the nd set of terms. If the expression that is left over is the same, you can factor by grouping. Factor out the expression that was left over. 4x + 8x 9x 8 = 4x x + 9 x + = 4x 9 x + = x + x x + example: ( ) ( ) ( )( ) ( )( )( ) Explain the process for completing the square on a quadratic expression. (algebra level expressions with a coefficient of one on the x squared term that look like x + bx + c ) Half of the coefficient on the x term (middle term) is squared. This quantity ( ) b is both added and subtracted from the expression. The positive term is used to factor the variable portion of the equation into a perfect square. The negative term is combined with the original constant term. example: 8 8 ( ) ( ) ( ) y = x + 8x y = x + 8x + y = x + 8x y = x How do you simplify rational expressions? Factor all of the rational expressions. Cancel any factors in both the numerator and denominator. x + x 4 x + 7x + 6 ( x + 4)( x ) ( x + )( x + 6) ( x + 6) example: = = x + x + 4 x x + ( x + )( x + 4) x ( x ) Statistics and Probability ( ) What is the difference between theoretical and experimental probability? Theoretical probability is the expected ratio of the number of successful outcomes to the number of possible outcomes. Experimental probability is the probability that an outcome occurs based upon repeated trials. Algebra (page 7/0)
8 What is the complement of an event in a probability problem? The complement of an event is the opposite of the event occurring. How do you find the probability for the complement of an event? The probability of an event and its complement both happening is one so the probability of the complement happening is one minus the probability of the event happening. example: If the probability of winning a particular drawing is 00, what is the probability of not winning the drawing? Probability of not winning = Probability of winning = 00 = How do you find the probability of an independent compound event? The probability of an independent compound event is the product of the probabilities of the two events. example: The 6 letters of the alphabet are placed in a bag. What is the probability of picking a vowel and then a consonant if the st letter is returned to the bag before picking the nd letter? 6 6 How do you find the probability of a dependent compound event? The probability of a dependent compound event is the product of the probabilities of the two events taking into account the effects of the st event on the nd event. example: The 6 letters of the alphabet are placed in a bag. What is the probability of picking a vowel and then a consonant if the st letter is not returned to the bag before picking the nd letter? 6 Graphing How do you graph inequalities on a number line? Less than " < " and greater than " > " are open circles. Less than or equal " " and greater than or equal " " are closed circles. Greater " > or " is shaded to the right. Less " < or " is shaded to the left. examples: x 0 x < How do you graph compound inequalities on a number line? The keyword and means that both inequalities must be true so the overlapping region is shaded. The keyword or means that either inequality must be true so both regions are shaded. examples: x 0 and x < x < 0 or x Algebra (page 8/0)
9 In a coordinate plane, describe the origin, xaxis, yaxis, and quadrants. The origin is the point where the xaxis and yaxis meet. The xaxis is the horizontal axis and the yaxis is the vertical axis. The quadrants are numbered I to IV starting in the top right and going counterclockwise. II I origin yaxis III   xaxis IV How is a point (x, y) plotted in a coordinate plane? The x coordinate is plotted left and right. Positive numbers are to the right and negative numbers are to the left. The y coordinate is plotted up and down. Positive numbers are up and negative numbers are down. examples: Plot the points A (, 4) and B (,). How do you know if a graphical relationship is a function? It passes the vertical line test (all vertical lines cross the graph in at most one point). What is the fallback method for graphing? Start graphing points until you understand what the graph is doing. How do you graph a line? Plot two points and connect them with a straight line. If a line is given in slopeintercept format, plot the yintercept and use the slope to find the nd point. If a line is given in pointslope format, plot the point and use the slope to find the nd point. If the line is given in some other format, find two points on the line and then connect them with a straight line. A line in the form of x equals a constant is what type of line? How do you graph this line? It is a vertical line. Draw a vertical line at the given x value. A line in the form of y equals a constant is what type of line? How do you graph this line? It is a horizontal line. Draw a horizontal line at the given y value. B A What does the graph of an absolute value function look like? Absolute value functions are Vshaped graphs. If the leading coefficient is positive, it opens up. If the leading coefficient is negative, it opens down. Absolute Value Function How do you graph inequalities in the coordinate plane? If the inequality is a or a the line is solid. If the inequality is a < or a > the line is dotted. A point is picked to determine which side of the curve makes the inequality true. The true side is shaded. Algebra (page 9/0)
10 What are the important properties of the graph of an exponential function? What does the graph of an exponential function look like? An exponential function has a horizontal asymptote and its slope is always increasing for exponential growth or its slope is always decreasing for exponential decay. Exponential Growth What does the graph of a quadratic function look like? Quadratic functions are parabolas. They are Ushaped graphs. If the leading coefficient is positive, it opens up. If the leading coefficient is negative, it opens down Exponential Decay Quadratic Function How do you graph a quadratic function in standard form?  b The x coordinate of the vertex occurs at negative b over two a. a.  Plug this x coordinate into the original function to find the y coordinate of the vertex. Plot the vertex and two additional points one on each side of the vertex. Draw in a quadratic Ushaped graph. What does the graph of a square root function look like? The graph of a square root function looks like a half parabola that would open left or right instead of up or down. Coordinate Geometry Square Root Function How do you find the slope between two points (, ) and (, ) x y x y? Slope is the difference in the y values over the difference in the x values it is the =. example: What is the slope between (4, ) and (8, )? = = How do you find the distance between two points (, ) and (, ) ( ) + ( ) distance = x x y y x y x y? example: What is the distance between (4, ) and (8, )? ( ) ( ) How do you find the midpoint of two points (, ) and (, ) x y x y? rise run  y x y x = = 0 = x + x y + y The midpoint is the average of the x coordinates and the average of the y coordinates., example: What is the midpoint of (4, ) and (8, )?, = ( 6,4) Algebra (page 0/0)
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