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Operations on Functions | Purplemath

Details: Demonstrates how to add, subtract, multiply, and divide functions. Includes an example of evaluation. Demystifies the notation.

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Condensing Log Expressions | Purplemath

Details: Uses worked examples to demonstrate how to condense (or "simplify" or "compress") logarithmic expressions, converting strings of logs into one log expression.

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Solving Quadratic Equations: Picking a Method | Purplemath

Details: Before I panic, I think about the one method of "solving" that doesn't involve an actual quadratic equation: solving by graphing. When they want me to solve a quadratic equation by graphing, they're actually asking me to find the x-intercepts of the associated quadratic function.And, by "find", they mean "from the pretty picture".

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Basic Number Properties: Associative, Commutative, and

Details: The Distributive Property is easy to remember, if you recall that "multiplication distributes over addition". Formally, they write this property as "a(b + c) = ab + ac".In numbers, this means, for example, that 2(3 + 4) = 2×3 + 2×4.Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses (or factor something out); any time a

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Composition of Functions: Composing Functions with Functions

Details: Composition of Functions: Composing Functions with Functions (page 3 of 6). Sections: Composing functions that are sets of point, Composing functions at points, Composing functions with other functions, Word problems using composition, Inverse functions and composition

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Geometric Series | Purplemath

Details: So this is a geometric series with common ratio r = –2. (I can also tell that this must be a geometric series because of the form given for each term: as the index increases, each term will be multiplied by an additional factor of –2.). The first term of the sequence is a = –6.Plugging into the summation formula, I get:

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Solving Log Equations: Calculator Considerations

Details: The base of the natural logarithm is the number e (which has a value of about 2.7).. This equation has a strictly numerical term (being the 3 on the right-hand side). So, to solve this, I'll use The Relationship to convert the log equation to its corresponding exponential form, keeping in mind that the base of this log is "e ":

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Number Bases: Base 4 and Base 7 | Purplemath

Details: Purplemath Base 4. In base four, each digit in a number represents the number of copies of that power of four. That is, the first digit tells you how many ones you have; the second tells you how many fours you have; the third tells you how many sixteens (that is, how many four-times-fours) you have; the fourth tells you how many sixty-fours (that is, how many four-times-four-times-fours) you

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2-by-2 Determinants

Details: Just as absolute values can be evaluated and simplified to get a single number, so can determinants. The process for evaluating determinants is pretty messy, so let's start simple, with the 2×2 case.

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Composition of Functions: Inverse Functions and Composition

Details: The lesson on inverse functions explains how to use function composition to verify that two functions are inverses of each other. However, there is another connection between composition and inversion: Given f (x) = 2x – 1 and g(x) = (1 / 2)x + 4, find f –1 (x), g –1 (x), (f o g) –1 (x),

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The Quadratic Formula Explained | Purplemath

Details: For the Quadratic Formula to work, you must have your equation arranged in the form "(quadratic) = 0".Also, the "2a" in the denominator of the Formula is underneath everything above, not just the square root.And it's a "2a" under there, not just a plain "2".Make sure that you are careful not to drop the square root or the "plus/minus" in the middle of your calculations, or I can guarantee that

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Vertical Asymptotes | Purplemath

Details: You can see how the graph avoided the vertical lines x = 6 and x = –1.This avoidance occurred because x cannot be equal to either –1 or 6.In other words, the fact that the function's domain is restricted is reflected in the function's graph.

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Translating Word Problems: Examples | Purplemath

Details: This next one is very important; it crops up in many different word-problem contexts, but isn't usually pointed out to students.It's kinda hoped that you'll somehow figure it out on your own. It's the "how much is left" construction, and you'll usually need it when you're working with two things, like two legs of a journey, or two ingredients in one mix.

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Graphing Exponential Functions: Step-by-Step Instructions

Details: The exponential, remember, will get (and stay) very close to zero on the left-hand side, so I will draw the graph "skinnying along" the top of the x-axis on the left-hand side:

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Graphing with Slope and y-Intercept: Examples | Purplemath

Details: Purplemath. We now know that, given a line equation in the form y = mx + b (if the values of m and b are reasonably "nice"), we can quickly and easily do the graph by starting with the y-intercept at b on the y-axis, and then counting "up and over" to the next point by using the slope.So, for these next graphs, let's not do any other "computations"; let's just work straight from the equation.

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Descartes' Rule of Signs | Purplemath

Details: Descartes' Rule of Signs will not tell me where the polynomial's zeroes are (I'll need to use the Rational Roots Test and synthetic division, or draw a graph, to actually find the roots), but the Rule will tell me how many roots I can expect, and of which type.

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Solving Quadratic Equations with the Quadratic Formula

Details: Purplemath. Somebody (possibly in seventh-century India) was solving a lot of quadratic equations by completing the square.At some point, he (and, yes, it would have been a guy back then) noticed that he was always doing the exact same steps in the exact same order for every equation.

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Function Notation: Definitions & Evaluating at a Number

Details: If you try to express the above, or something more complicated, using variously-shaped boxes, you'd quickly run out of shapes. Besides, you know from experience that "A" stands for "area", "h" stands for "height", and "a" and "b" stand for the lengths of the parallel top and bottom sides.Heaven only knows what a square box or a triangular box might stand for!

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Long Polynomial Division | Purplemath

Details: Since the remainder on the division above was zero (that is, since there wasn't anything left over), the division "came out even". When you do regular division with numbers and the division "comes out even", it means that the number you divided by is a factor of the number you're dividing.

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Adding and Subtracting Matrices - Purplemath

Details: Demonstrates how to add and subtract matrices, explains why the addition or subtraction sometimes can't be done, and gives an example of how matrix addition is used in homework problems.

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Special Factoring: Sums and Differences of Cubes

Details: First, I note that they've given me a binomial (a two-term polynomial) and that the power on the x in the first term is 3 so, even if I weren't working in the "sums and differences of cubes" section of my textbook, I'd be on notice that maybe I should be thinking in terms of those formulas.. Looking at the other variable, I note that a power of 6 is the cube of a power of 2, so the other

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The Change-of-Base Formula | Purplemath

Details: Purplemath. There is one other log "rule", but it's more of a formula than a rule. You may have noticed that your calculator only has keys for figuring the values for the common (that is, the base-10) log and the natural (that is, the base-e) log.There are no keys for any other bases.

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Set Notation | Purplemath

Details: Purplemath. You never know when set notation is going to pop up. Usually, you'll see it when you learn about solving inequalities, because for some reason saying "x < 3" isn't good enough, so instead they'll want you to phrase the answer as "the solution set is { x | x is a real number and x < 3 }".How this adds anything to the student's understanding, I don't know.

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Systems of Non-Linear Equations: Using the Quadratic Formula

Details: As previously mentioned, sometimes you'll need to use old tools in new ways when solving the more advanced systems of non-linear equations. The example below demonstrates how the Quadratic Formula is sometimes used to help in solving, and shows how involved your computations might get.. Solve the system:

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Factoring Quadratics: The Weird Case | Purplemath

Details: Purplemath "The weird case" of quadratic factoring is where it doesn't seem like we're factoring a quadratic, but we kind-of are. We need to be clever with these, but they reduce to little more than pattern-recognition, once you catch on to how to do them.

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Asymptotes: Worked Examples | Purplemath

Details: Next I'll turn to the issue of horizontal or slant asymptotes. Since the degrees of the numerator and the denominator are the same (each being 2), then this rational has a non-zero (that is, a non-x-axis) horizontal asymptote, and does not have a slant asymptote.The horizontal asymptote is found by dividing the leading terms:

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Polynomial Graphing: Degrees, Turnings, and "Bumps

Details: Demonstrates the relationship between the turnings, or "bumps", on a graph and the degree of the associated polynomial. Shows that the number of turnings provides the smallest possible degree, but that the degree could be larger, by multiples of two.

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Graphing Trigonometric Functions: Examples | Purplemath

Details: The graph for tan(θ) – 1 is the same shape as the regular tangent graph, because nothing is multiplied onto the tangent.. But this graph is shifted down by one unit. In other words, instead of the graph's midline being the x-axis, it's going to be the line y = -1.. Rather than trying to figure out the points for moving the tangent curve one unit lower, I'll just erase the original

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Solving Exponential Equations with Logarithms | Purplemath

Details: If this equation had asked me to "Solve 2 x = 32", then finding the solution would have been easy, because I could have converted the 32 to 2 5, set the exponents equal, and solved for "x = 5".But, unlike 32, 30 is not a power of 2 so I can't set powers equal to each other. I need some other method of getting at the x, because I can't solve with the equation with the variable floating up there

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"Investment" Word Problems - Purplemath

Details: Investment problems usually involve simple annual interest (as opposed to compounded interest), using the interest formula I = Prt, where I stands for the interest on the original investment, P stands for the amount of the original investment (called the "principal"), r is the interest rate (expressed in decimal form), and t is the time.

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Venn Diagrams: Set Notation | Purplemath

Details: The tilde ("TILL-duh") is the wiggly "~" character at the beginning of ~ A; on your keyboard, the tilde is probably located at or near the left-hand end of the row of numbers.The tilde, in the set-relation context, says that I now want to find the complement (in a sense, the opposite) of whatever is being negated or "thrown out"; in this case, that's the set A.

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Converting Between Decimals, Fractions, and Percents

Details: Purplemath Fraction to Percent. The process of converting from fractions to percentages starts out the same as does the process for converting from fractions to decimals, but the final answer can come in a couple different formats.You always start by doing the long division (because fractions are division, remember!), and then (usually) you move the decimal point two places to the right and

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Solving Absolute-Value Equations: A Special Case | Purplemath

Details: I got too many answers from using the previous method. That method does not work for equations of this particular type. The previous method allowed us to avoid some very nasty algebra, but for an equation with two (or more) un-nested absolute values, and where there is also a loose number (or some other variable, etc), we have no choice but to get technical.

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Venn Diagrams: Set Notation | Purplemath

Details: An illustration of a use of these set relationships would be the manner in which some search engines process searches: If you type "cats AND dogs" into the search box, a search engine using this syntax (called "Boolean" logic) will return all web pages that contain both the word "cats" and the word "dogs".

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Finding Quadratics from Their Zeroes - Purplemath

Details: Explains the connection between a quadratic's zeroes and its equation; also demonstrates how to find the quadratic from the zeroes.

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Special Angle Values: 30-60-90 and 45-45-90 - Purplemath

Details: The base angle, at the lower left, is indicated by the "theta" symbol (θ, THAY-tuh), and is equal to 45°.So how does knowing this triangle help us? It helps us because all 45-45-90 triangles are similar. Therefore, every "evaluation" or "solve the triangle" question involving a 45-45-90 triangle or just a 45° angle can be completed by using this triangle.

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"Age" Word Problems | Purplemath

Details: Purplemath explains how to decode "age" types of word problems (for instance, where the ages of two or more people are compared), and provides many worked examples. It's very important to start with clear definitions.

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Factoring "In Pairs" (or "By Grouping") | Purplemath

Details: Any time you encounter such a situation, you should try factoring in pairs. It's a pretty safe bet, especially when you're doing factoring before quadratics, that the four-term polynomial they've given you is factorable, and that the method they're expecting you to use is "in pairs".. Factor xy – 5y – 2x + 10

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The Distance Formula: Worked Examples | Purplemath

Details: Purplemath. The most common mistake made when using the Formula is to accidentally mismatch the x-values and y-values.Be careful you don't subtract an x from a y, or vice versa; make sure you've paired the numbers properly.. Also, don't get careless with the square-root symbol.

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Solving Simple (to Medium-Hard) Trig Equations | Purplemath

Details: Purplemath. Solving trig equations use both the reference angles and trigonometric identities that you've memorized, together with a lot of the algebra you've learned. Be prepared to need to think in order to solve these equations.. In what follows, it is assumed that you have a good grasp of the trig-ratio values in the first quadrant, how the unit circle works, the relationship between

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Fractions Review: Adding and Subtracting - Purplemath

Details: Reviews how to add and subtract fractions. Includes how to find common denominators.

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Converting Between Decimals, Fractions, and Percents

Details: Purplemath Decimal to Fraction. Converting any terminating decimal into a fraction is fairly straightforward. You count the number of decimal places, move the decimal point that number of places to the right, and put the resulting number over "1" followed by that number of zeroes.For instance:

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Higher-Index Roots | Purplemath

Details: Purplemath. Operations with cube roots, fourth roots, and other higher-index roots work similarly to square roots, though, in some spots, we'll need to extend our thinking a bit.

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Partial-Fraction Decomposition: Repeated and Irreducible

Details: Sometimes a factor in the denominator occurs more than one. For instance, in the fraction 13 / 24, the denominator 24 factors as 2×2×2×3.The factor 2 occurs three times.To get the 13 / 24, there may have been a 1 / 2 or a 1 / 4 or a 1 / 8 that was included in the original addition. You can't tell by looking at the final result.

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Slant (Oblique) Asymptotes | Purplemath

Details: The graphs show that, if the degree of the numerator is exactly one more than the degree of the denominator (so that the polynomial fraction is "improper"), then the graph of the rational function will be, roughly, a slanty straight line with some fiddly bits in the middle. Because the graph will be nearly equal to this slanted straight-line equivalent, the asymptote for this sort of rational

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Regression Models, Interpolation, and Extrapolation

Details: Depending on your calculator, you may need to memorize what the regression values mean. On my old TI-85, the regression screen would list values for a and b for a linear regression. But I had to memorize that the related regression equation was "a + bx" (instead of the "ax + b" that I would otherwise have expected) because the screen didn't say.If you need to memorize this sort of information

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Polynomials: Combining "Like" Terms | Purplemath

Details: Looking at these two terms, I see that each contains the variable x, and the variable has the same (understood) power of 1 in each term. So these are like terms, and I can combine them. Back in grade-school arithmetic, "three apples plus four apples" got combined into "seven apples" by adding the three and the four to get seven, and bringing the "apples" along for the ride.

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Completing the Square: Ellipse Equations

Details: The process for hyperbolas is the same, except that the signs on the x-squared and y-squared terms will be opposite; that is, while both the x-squared and y-squared terms are added in the case of ellipses (and circles), one or the other will be subtracted in the case of hyperbolas.. Top | Return to Index

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Solving Harder Absolute-Value Equations | Purplemath

Details: The quadratic inside the absolute-value bars might be positive and might be negative, depending on where I look. This is a positive quadratic, so it's an upward-opening parabola.

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Graphing Absolute-Value Functions | Purplemath

Details: Purplemath. Most functions are not straight lines, so you should expect to need to know how to graph the absolute values of quadratics, cubics, and other types of functions.

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