On this page, you'll find a few important facts about each chapter. The book also contains a summary and vocabulary terms after each chapter. We will take quizzes written by the publisher every few sections. You will find the sections those quizzes cover immediately under the chapter name. Scroll down or choose the chapter here:

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 11

Quizzes covering sections: 1-3, 4-6, 7-9

Functions are difficult to define because they are used for so many things in so many ways. One good definition of a function is, a relationship between two variables such that each first component is paired with exactly one second component. The set of all possible replacements for the first component (or independent variable) is called the domain of the function. The range is the set of values the second component (or dependent variable) could equal.

If the domain is discrete (a disconnected set of points), it is possible to list the members, like {1, 2, 3, 4, ...}. They may also be described, like {the set of natural numbers}. When stating domain from a continuous graph, list all values graphed left to right with this symbolism: {x: minimum x maximum} or {x: x 0}. When stating range from a continuous graph, list all values from bottom to top. Use the same symbolism as with domain, but replace x with y.

Our text uses Euler's notation: f(x) = ... and mapping notation: f: x ... to write sentences for functions. If no vertical line passes through two or more points, the graph of a relation represents a function. This is called the vertical line test.

Another important topic of this chapter is writing expressions and sentences with variables in them to represent situations. Algebraic means it contains a variable. The verb in a math sentence is usually is, is equal to, is not equal to, is greater than, is greater than or equal to, etc. The language of math is just a compilation of accepted shortcuts for terms like this that occur often, like =, >, etc.

Please Excuse My Dear Aunt Sally is a mnemonic that may help you remember the order of operations (the order steps are performed in when you simplify an expression).

1. Parenthesis or grouping symbols
2. Exponents or powers
3. Multiplication and Division from left to right
4. Addition and Subtraction from left to right

You will solve equations that contain one variable for the value of that variable. You will also solve for one variable in terms of others. Both processes are similar, but people seem to have much more trouble when there's more than one variable. Here is a set of steps to use when solving for a variable:

1. Clear fractions and decimals (this is optional)
2. Simplify each side of the sentence
3. Move all the variables you're solving for to the same side
4. Undo addition and subtraction from the variable
5. Undo multiplication and division from the variable
6. Check

A sequence is an ordered list. In other words, the order of the terms matters. Explicit means you can find any term by putting that number in for the independent variable.

Here is an example: t_n = 5n - 2. To find any term, replace n with the location of that term in the sequence.

The first term: t₁ = 5(1) - 2 = 3.

The second term: t₂ = 5(2) - 2 = 8.

The third term: t₃ = 5(3) - 2 = 13, etc.

So, the sequence described by t_n is 3, 8, 13, 18, 23, ...

To find the 25th term, for instance, replace the independent variable with 25.

The twenty-fifth term: t₂₅ = 5(25) - 2 = 123.

A recursive formula is one in which the first term or terms is listed, along with a rule for obtaining the next term from the term or terms before it.

Here's an example: . It says the first term is -6. Read r_n as "the next term" and r_n-1 as "the previous term" rather than replacing n with the number of the term. The n 2 just tells us to use the second line of the formula to find the 2nd term, and all those terms after it.

r₁ = -6.

r₂ = the previous term (which is -6) + 5 = -1.

r₃ = the previous term (which is -1) + 5 = 4, etc.

So, the sequence described by r_n is -6, -1, 4, 9, 14, ...

To find the 25th term of a sequence described recursively, you'd need to know the 24th term.

Quizzes covering sections: 1-3, 4-6, 7-9

A direct variation is one that can be described by y = kxⁿ. It would be said that y varies directly as the nth power of x. k is the constant of variation. An inverse variation is one that can be described by y = ^k/_xn. It would be said that y varies inversely as the nth power of x. Again, k is the constant of variation.

One of the common problems in the chapter will require you to find, for instance, the value of y₂ given the corresponding value for x₂ in a variation relationship. You will be given an ordered pair x₁ and y₁. Plug them into the formula to find k, and use it along with x₂ to find y₂.

Generally, in a direct variation, if the independent variable increases, so does the dependent variable. If one decreases, so does the other. In an inverse variation, as one variable goes up, the other goes down and vice versa. This assumes the constant of variation is positive, which normally is the case with measurable quantities.

The fundamental theorem of variation states that it doesn't matter what the value of k is, if the independent variable is multiplied by a number, the dependent variable increases or decreases by the change in the independent variable raised to the power in the variation. An example: in a direct-square variation, the independent variable is multiplied by 3. The dependent variable will be multiplied by 3². If it were an inverse-square variation, the dependent variable would be divided by 9.

The graphs of all direct-variation formulas go through the origin. The graph of y = kx is a line with slope k. If k is positive, the line rises to the right. If negative, to the left. The graph of y = kx² is a parabola. k is not the rate of change because the rate of change varies in a parabola. The larger the magnitude of k, the tighter the parabola. If k is positive, the parabola opens up. If negative, down.

Inverse-variation graphs do not pass through the origin. The graph of y = ^k/_x is a hyperbola in opposite quadrants. It will lie in I and III if k is positive, and in II and IV if k is negative. The graph of y = ^k/_x2 lies in quadrants I and II if k is positive, and in III and IV if k is negative.

You will be asked to find a variation formula given a set of values. The inverse and inverse-square graphs look very much the same in quadrant I, so you must test points to determine which one matches the data. If there are more that two variables, all but two must be held constant to determine how the others relate.

Combined variation means both a direct and inverse relationship exists. A joint variation is the product of two or more direct variations.

Quizzes covering sections: 1-3, 4-6, 7-9

A constant increase or decrease situation results in a graph that is linear (a line). An equation for such a situation (a model) is y = mx + b. m is the constant change that's taking place in the situation. It is also the slope of the graph. b is the initial value of the dependent variable in the situation. It is also the y-intercept of the graph.

Slope is usually denoted by m. m = ^rise/_run = (y₂ - y₁)/(x₂ - x₁).

A piecewise-linear graph results when change is constant for a period of time, but then something causes rate of change to become a different constant value. The graph consists of two or more segments or rays.

Two lines are parallel if and only if they have the same slope.

A linear combination is of the form Ax + By where x is the independent variable, y is the dependent variable, and A and B are real numbers. The graph of Ax + By = C where A and B are not both 0 is a line. The slope of the line is -^A/_B. The x-intercept can be found by replacing y with 0. The y-intercept can be found by replacing x with 0.

In a vertical line, B is 0, the slope is undefined, and the equation is of the form x = (a constant). In a horizontal line, A is 0, the slope is 0, and the equation is of the form y = (a constant).

Here are three forms for linear equations:

• slope-intercept form: y = mx + b
• standard form: Ax + By = C
• point-slope form: y - y₁ = m(x - x₁)

Arithmetic sequences are ones that have a constant change from one term to the next. In other words, they're linear. Here are the two forms of equations for arithmetic formulas:

• explicit: a_n = a₁ + (n - 1)d
• recursive: a₁ = _, a_n = a_n-1 + d for n 2

Step functions are ones in which the graph is a horizontal segment for an interval, then jumps up to another horizontal segment.

Quizzes covering sections: 1-3, 4-7, 8-10

A Matrix consists of objects arranged in rows and columns. Each object is called an element. The dimensions of a matrix are given in the form m x n where m is the number of rows and n is the number of columns.

Matrices can store raw data, points or sets of points in the coordinate plane, be used to solve systems (chapter 5), or they can represent transformations. Ordered pairs are always listed in columns. Matrices that contain three or more points represent polygons. Transformation matrices are ones that, when multiplied by a figure, perform some operation like a size change, scale change, reflection, rotation, or translation.

Matrices must have the same dimensions in order to add them. Corresponding elements are added. Scalar multiplication consists of multiplying every element in a matrix by a constant.

Matrix multiplication is more complicated. In order for matrices to be multiplied, the number of columns of the first must equal the number of rows of the second. Elements in the first row of the first matrix are aligned with those those in the first column of the second. Corresponding elements are multiplied and the results added. This value becomes the element in the first row and first column of the product matrix. This process continues until each row of the first matrix is multiplied by each column of the second.

The original figure in a transformation is called the preimage. The result of the transformation is called the image. Size changes result in a preimage and image that are similar. Reflections, rotations, and translations result in a preimage and image that are congruent. Scale changes result in a preimage and image that are neither similar or congruent.

Except for translations, which are performed with matrix addition, the other transformations are performed with matrix multiplication and use a 2 x 2 matrix. The Matrix Basis Theorem states that the image of the point (1,0) under the given transformation is the first column of the matrix that performs it. The image of (0,1) under the transformation is the second column of the matrix that performs that transformation.

Composites of transformations result when more than one transformation is performed on a preimage. Compositions of functions use the following symbol (o). The transformation that is performed first is written last. Since matrix multiplication is associative (but not commutative), the first matrix can be multiplied by the preimage, then the second. Or, the two transformation matrices can be multiplied to arrive at one matrix that will perform the composite on any preimage.

If a line is rotated 90° or 270°, it is perpendicular to the preimage. Lines are perpendicular if and only if the product of their slopes is -1.

Quizzes covering sections: 1-4, 5-7, 8-10

Systems in math are conditions or sentences joined by and. A sentence in which two clauses are connected by and or or is called a compound sentence. Intersection means and and is denoted by the symbol . Union means or and is denoted by .

Usually, all the solutions to inequalities can not be listed. It is typical to describe solution sets of sentences in one variable with a graph on a number line. Solution sets of sentences in two variables are graphed in a coordinate plane.

The solutions to a system are the points of intersection of the sentences that make up the system. So, estimates of solutions to systems can be made by graphing and by using calculator tables. Better methods of solution are substitution, linear combination, and matrices because they afford exact solutions.

Inverse matrices are ones whose product is the identity matrix. Only square matrices have inverses. If the determinant of a matrix is 0, it has no inverse.

Linear programming is a method of finding the minimum or maximum value of a formula when the system has several constraints. Here is a set of steps that can be used on a linear programming problem:

1. Choose two variables defined by the system.
2. Write a system of inequalities using the two variables.
3. Graph the system of inequalities.
4. Write a formula with the two original variables, plus a third to be minimized or maximized.
5. Find the vertices of the feasible region and test them in the formula.
6. Use the information to answer the question.

Quizzes covering sections: 1-4, 5-7, 8-10

Quadratic functions are ones in which the highest degree of any of the expressions is two. We will be primarily concerned with quadratic expressions, equations and functions in one variable. ax² + bx + c is a quadratic expression. ax² + bx + c = 0 is a quadratic equation. y = ax² + bx + c is a quadratic function.

The absolute value of a number is the distance from the number to zero on a number line. Thus, absolute value is always nonnegative. The absolute value of x is denoted |x|. (x²) = |x|. If x² = k, x = +k.

The Graph-Translation Theorem states that if x is replaced by x - h and y is replaced by y -k, the graph is translated h units right and k units up, written T_h,k. The image of y = ax² under the translation T_h,k is y - k = a(x - h)². y - k = a(x - h)², then, is a parabola with vertex at (h,k).

y - k = a(x - h)² is called the vertex form of a parabola and y = ax² + bx + c is called standard form of a parabola. One must complete the square to convert standard form to vertex form. Completing the square consists of these steps:

1. Move the constant to the other side.
2. Factor out a if it is not 1.
3. Take half of b, square it, and add it on the right.
4. Add a(^b/₂)² on the left.
5. Simplify and write in vertex form.

The quadratic formula is often the easiest way to solve a quadratic equation. The solutions to ax² + bx + c = 0 are x = (-b + (b² - 4ac))/(2a).

The number i is called the imaginary unit. i = -1 and i² = -1. Here are rules for dealing with imaginary numbers:

1. The square root of a negative number involves i.
2. Work with i as if it's a variable.
3. Replace i² with -1.
4. Don't try to replace i with -1.

Complex numbers are ones of the form a + bi where a and b are real. All known numbers are complex numbers. Real numbers are complex numbers where b = 0. If a = 0, the number is called imaginary or pure imaginary. Complex conjugates are imaginary numbers that vary only in the sign of the imaginary part. Example: the conjugate of 3 + 4i is 3 - 4i. Note, they are not opposites.

To simplify expressions that have a complex number in the denominator, multiply numerator and denominator by the conjugate of the denominator. (a + bi)(a - bi) = a² + b², so the product of conjugates is always real.

The discriminant (b² - 4ac) determines how many real roots a quadratic equation has and how many real x-intercepts a quadratic function has. If:

• b² - 4ac > 0, there are two real solutions and two real x-intercepts.
• b² - 4ac < 0, there are no real solutions (The solutions are complex conjugates.) and no real x-intercepts.
• b² - 4ac = 0, there is one real double solution and one x-intercept. The vertex lies on the x-axis.

Furthermore, if the discriminant is a square number, the solutions are not only real, but rational.

Quizzes covering sections: 1-3, 4-6, 7&8

In the expression bⁿ, b is the base, n is the exponent, and the entire expression is called a power. It means b * b * b ... until there are n b's.

y = xⁿ where n is a positive integer is called the nth power function. These things are true for all powering functions:

1. The graph passes through the origin.
2. The domain is {all real numbers}.
3. The range is {all real numbers} if n is odd, and {y: y 0} if n is even.
4. The graph has symmetry with respect to the y axis if n is even, and symmetry with respect to the origin if n is odd.

The following are properties of powers:

1. b^m * bⁿ = b^m+n. example: x² * x³ = (x * x)(x * x * x) = x⁵.
2. (b^m)ⁿ = b^mn. example: (x²)³ = (x²)(x²)(x²) = x⁶.
3. (ab)^m = a^mb^m. example: (2x)³ = (2x)(2x)(2x) = 2³x³. We'd simplify to write it as 8x³.
4. (b^m)/(bⁿ) = b^m-n. example: (x⁵)/(x²) = ^{(x * x * x * x * x)}/_{(x * x)} = x³.
5. (^a/_b)^m = (a^m)/(aⁿ). example: (^x/₂)³ = (^x/₂)(^x/₂)(^x/₂) = (x³)/(2³). We'd simplify to write it as (x³)/8.
6. b⁰ = 1 for all nonzero values of b.

Generally, raising to negative powers results in the reciprocal. If something is in the numerator raised to a negative, it's the same as having it in the denominator raised to a positive. If it's in the denominator raised to a negative, put it in the numerator raised to a positive. The basic rule states b^-n = ¹/_bn. If you have a fraction raised to a negative power, the easiest thing to do is take the reciprocal raised to a positive power.

The most basic interest formula is I = PRT where I is interest, P is principal, R is interest rate per year as a decimal, and T is time in years. We won't work much with this formula. It works for simple interest only. Simple interest means interest is computed only one time, at the end of the time period.

Most interest is compounded. This means the interest is added more that one time in the time period. So, each time interest is computed, there is a little more money in the account and the interest is a little larger.

A = P(1 + r)^t is used for annual compounding (once a year). A is the amount in the account, P is principal, r is the interest rate per year as a decimal and t is time in years.

Most interest is figured more that once a year. The formula, then, is A = P(1 + ^r/_n)^nt where the n is times compounded a year. Credit card interest is compounded monthly (n = 12). Many stocks and bonds have dividends compounded quarterly (n = 4), etc. Most bank accounts and loans are compounded continuously. Continuous change is a topic of chapter 9.

Geometric sequences occur when the previous term is continuously multiplied by a constant to get the next term in the sequence. Following are formulas for geometric sequences. explicit: g_n = g₁r^n-1. recursive: g₁ = _, g_n = rg_n-1 for n 2.

If n is an integer greater than 1, b is defined as the nth root of x if and only if bⁿ = x. Example: 2 is the fourth root of 16 because 2⁴ = 16.

If x 0 and n is an integer greater than 1, x^(1/n) is an nth root of x. Example: 16^(1/4) means the fourth root of 16, and is equal to 2. Though both 2 and -2 raised to the fourth power equal 16, 16^(1/4) equals only 2. It is defined as the nonnegative fourth root of 16 if there are two that exist. Note, also, we don't attempt to raise negative numbers to fractional powers.

The following table demonstrates the number of real nth roots a real number has.

Note that zero has 1 nth root (namely, 0) no matter what n is. Also, every real number has exactly 1 real odd root. The confusion comes with even roots. This is because, for example, 2⁴ and (-2)⁴ both equal 16. So, 16 has two fourth roots. But, no real number raised to the fourth power will equal -16. So, -16 has no real fourth roots.

When a number is raised to a fractional power (rational exponent), raise it to the power in the numerator and take the root of the denominator. The order doesn't matter. For example: 8^(2/3) means the cube root of 8, squared. Or, it means square 8, then take the cube root. Both equal 4. Again, negative numbers raised to fractional powers are undefined.

Quizzes covering sections: 1-3, 4-6, 7&8

As stated when working with matrices, the composite of two functions can be written with the following symbol (o). These are all ways to write the composite of functions: g o f(x) and g o f: x and (g(f(x)). In this case, f is evaluated at the value x, then that result is used to replace the independent variable in function g. The order is important because composition of functions is not commutative. The function listed second is the first to be performed.

The inverse of a relation is obtained by replacing the independent variable with the dependent variable, and vice versa. The domain of a function is the range of its inverse, and vice versa. The graph of an inverse is a reflection of the graph of the original relation through the line y = x. We discussed the vertical line test in chapter 1. It tells us if the graph of a relation is a function. If no horizontal line passes through two or more points on a graph, the inverse of the relation will be a function. This is called the horizontal line test.

If both f o g(x) and g o f(x) equal x for all x in the domain of both functions, the functions are inverses. f^-1(x) is notation to indicate that this function is the inverse of f(x).

If n is an integer 2 or larger, and x is real and not negative, ⁿx = x^(1/n). Remember from last chapter that 16^(1/4) is equal to only 2, though (-2)⁴ also equals 16. Likewise, ⁴16 is equal to only 2. The symbol is called a radical sign. Any expression containing it is called a radical. Possibly the biggest advantage of radicals is that, for instance, ³(-8) = -2. Technically, (-8)^(1/3) is undefined.

ⁿ(xy) = ⁿx * ⁿy where x and y are nonnegative reals and n is an integer 2 or larger. Note that this theorem only applies when we are dealing with real numbers. The geometric mean of a set of n numbers is the nth root of the product of the numbers.

We don't like to have radicals in denominators. When it happens, we rationalize the denominator. Here's a simple example: 1/2 is the expression we want to simplify. Multiply by the well-chosen one, 2/2. Multiplying by 1 doesn't change the value of the expression, but there is now no radical in the denominator because (1/2)(2/2) = ²/₂. If the denominator involves addition or subtraction, , the well-chosen one is the conjugate of the denominator over itself.

If you raise both sides of an equation to a power or take the nth root of both sides of an equation, remember to check all solutions. Sometimes, these techniques produce extraneous roots. These are solutions that do not check. Do not report extraneous roots.

Quizzes covering sections: 1-3, 4-7, 8-10

A function of the form y = ab^x where a isn't zero and b is greater than zero, but not 1, is called an exponential function. The graph is an exponential curve. If b > 1, the function represents exponential growth. If 0 < b < 1, the function represents exponential decay. b is called the growth factor and a is the initial value of the function.

All exponential functions have these properties:

1. The domain is {all real numbers}.
2. The range is {y: y > 0}.
3. a is the y-intercept of the graph. There is no x-intercept.
4. The graph is increasing if it is growth and decreasing if it is decay.
5. The line y = 0 is an asymptote.

If a value is said to be changing continuously, the formula used is A = Pe^rt where A is the final amount, P is the initial amount, r is the rate of change written as a decimal and t is time. e is not a variable. e is an irrational number, a little like . e 2.71828. Lots of things are thought to be under continuous change. Population change, radioactive decay, and the compounding of investments and loans are a few we'll work with.

When r is positive in A = Pe^rt, it represents exponential growth. When r is negative, it is exponential decay.

Logarithms are another way to write things that are in exponential form. If b is positive, but not 1, then n is the logarithm of m to the base b (or log base b of m), written n = log_b m if and only if bⁿ = m. Here's an example: 2³ = 8 can be written log₂ 8 = 3. Logarithms are exponents.

We can not take the log of a negative number. Bases of logarithms must be positive. Though a positive fractional base for a logarithm is defined, it is typical to use only small positive integers as bases.

The inverse of y = 10^x is y = log₁₀ x. Logarithms base 10 are called common logs. They are usually written without the 10 as just "log x". The inverse of y = e^x is y = log_e x. Logarithms base e are called natural logs. They are usually written as "ln x".

The following facts are true of all logarithmic graphs:

1. The domain is {x: x > 0}.
2. The range is {all real numbers}.
3. The x-intercept is 1. There is no y-intercept.
4. The line x = 0 is an asymptote.

Logarithmic scales are ones in which the units are spaced so that the ratio between successive units are the same. The scales we are used to are called linear scales and the difference between successive units are the same. Logarithmic scales are used when all values are positive, and cover a large range. Common logarithmic scales include the decibel scale for measuring sound intensity, the pH scale for measuring how acidic something is, and the Richter scale for measuring intensity of earthquakes.

The following are properties of logarithms:

1. log_b 1 = 0.
2. log_b bⁿ = n.
3. log_b (xy) = log_b x + log_b y.
4. log_b (^x/_y) = log_b x - log_b y.
5. log_b (xⁿ) = n log_b x.

The Change of Base Property states that log_b a = ^{(log a)}/_{(log b)} = ^{(ln a)}/_{(ln b)}. Actually, you can use a logarithm with any base, but since our calculators can compute base 10 and base e, we'll almost always choose log or ln.

One of the most important uses of logarithms is solving equations with variables in the exponent. Here is a set of steps to apply:

1. Isolate the exponential expression.
2. Take the log of both sides. Since your calculator can handle log base 10 or e, use one of them. If the base of the expression is 10, use log. If it's e, use ln. If it's neither 10 or e, it doesn't matter which you use.
3. Use property 5 above to simplify the expression.
4. Solve the linear equation that results.

Quizzes covering sections: 1-3, 4-6

We won't do all of chapter 11, but the we will work with polynomials, and we will factor. There may be some facts listed here that we don't get to because of time.

A monomial is an expression that contains only the operation of multiplication. A constant, or number, is considered a monomial too. The sum or difference of monomials is called a polynomial. Each monomial in the polynomial is called a term. The coefficient of (or number multiplied by) the term with the highest degree is called the leading coefficient.

When the polynomial contains only one variable, the largest exponent of that variable is the degree of the polynomial. Polynomials in one variable are usually written from largest degree to smallest. If there are two or more variables in the polynomial, the degree is the largest number of times variables are factors in any one term. For example: the monomial 2x²y³z has degree six because in 2 * x * x * y * y * y * z, six variables appear as a factor.

Polynomials can be classified by their degree. Polynomials of degree one are called linear, degree two are called quadratic, and degree three are called cubic. Most higher degrees are not given special names, though fourth degree polynomials are sometimes called quartic.

Polynomials are also classified by the number of terms. If there is only one term, as stated earlier, it is a monomial. Two-term polynomials are called binomials, and three-term polynomials are called trinomials. Larger polynomials are not given special names.

Factoring means to write as things multiplied together. When those factors are multiplied together to get a polynomial, it's called expanded form. Below, the factored form is on the right and the expanded form is on the left. Here are several factoring patterns:

1. a² + 2ab + b² = (a + b)².
2. a² - 2ab + b² = (a - b)².
3. a² - b² = (a + b)(a - b).

Sometimes, all that can be done to factor is factor out the greatest common monomial factor. Other times, a trinomial can be factored, but there is no pattern that applies. When this happens, it is easiest if the leading coefficient is one. Begin with the signs, then read the polynomial backwards. Look for the factors of the last term that added or subtracted yield the coefficient of the middle term.

When the leading coefficient is not one, begin with the signs again. Now, however, there are no shortcuts. Experiment until you place the coefficients of the first and last term in the right spots to yield the middle term. This is called trial and error.

Not all quadratic trinomials factor, so a useful trick is to find the discriminant of the polynomial. If it is a square number, the quadratic trinomial factors into two binomials. If it isn't a square number, the trinomial can not be factored into binomials.

Polynomial functions have solutions that can be estimated by graphing and by using calculator tables. The Zero-Product Theorem says that ab = 0 if and only if a = 0 or b = 0. This is useful when a factored polynomial equals zero. For example, if (x - 3)(x + 5) = 0, then x - 3 = 0 or x + 5 = 0. These are two simple equations to solve to arrive at the solution set {3, -5}.

The Factor Theorem says that x - r is a factor of P(x) if and only if P(r) = 0. This means that if you know, for example, P(x) has x-intercepts of 0, -2, and 5 that (x - 0)(x + 2)(x - 5) is the factored form of P(x). If you wish, you can expand to find the polynomial P(x).

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