The maximum distance of the values of a periodic function above or below the midline.

Cavalieri’s Principle states that if two solids of equal height are cut into cross-sections by parallel planes, and the corresponding cross-sections on each plane always have equal areas, then the two solids have the same volume.

A logarithmic expression can be rewritten using logarithms with any positive base other than 1 using the equation \(\log_a (b) = \frac{\log_c (b)}{\log_c (a)}\).

A cross-section is the two-dimensional figure formed by intersecting a solid with a plane.

These diagrams show that the shape of the cross-section depends on how the plane intersects the solid.

The cube root of a number \(x\), written \(\sqrt[3]{x}\), is the number \(y\) whose cube is \(x\).

That is, \(y^3 = x\). So, \(\sqrt[3]{x}=y\).

Example: \(2^3 = 8\). So, \(\sqrt[3]{8}=2\).

The degree of a polynomial in \(x\) is the highest exponent occuring on \(x\) when you write the polynomial out as a sum of non-zero constants times powers of \(x\) (with like terms collected).

Density is a measure of how tightly the amount of matter in a substance is packed into the space it takes up. That is, density is the mass of a substance per unit volume.

\(\text{density}=\frac{\text{mass}}{\text{volume}}\)

The number \(e\) is an irrational number with an infinite decimal expansion that starts \(2.71828182845\ .\ .\ .\), which is used in finance and science as the base for an exponential function.

How the outputs of a function change as we look at input values further and further from 0.

A function \(f\) that satisfies the condition \(f(x) = f(\text-x)\) for all inputs \(x\). You can tell an even function from its graph: Its graph is symmetric about the \(y\)-axis.

An experimental study collects data by directly influencing something to determine how another thing is changed.

Any flat surface on a three-dimensional figure is a face.

• A cube has 6 faces that are all squares.
• A cylinder has 2 faces that are both circles.

A sequence in which each term is a constant times the previous term.

The time it takes for a material to decay to half of its original value. For example, the half-life of Carbon-14 is about 5,730 years. This means that if an object started with 10 micrograms of Carbon-14, then after 5,730 years it will have 5 micrograms of Carbon-14 left.

The line \(y =c\) is a horizontal asymptote of a function if the outputs of the function get closer and closer to \(c\) as the inputs get larger and larger in either the positive or negative direction. This means the graph gets closer and closer to the line as you move to the right or left along the \(x\)-axis.

An equation which is true for all values of the variables in it.

The logarithm to base 10 of a number \(x\), written \(\log_{10}(x)\), is the exponent you raise 10 to get \(x\), so it is the number \(y\) that makes the equation \(10^y = x\) true. Logarithms to other bases are defined the same way with 10 replaced by the base, e.g. \(\log_2(x)\) is the number \(y\) that makes the equation \(2^y = x\) true. The logarithm to the base \(e\) is called the natural logarithm, and is written \(\ln(x)\).

A logarithmic function is a constant multiple of a logarithm to some base, so it is a function given by \(f(x) = k \log_{a}(x)\) where \(k\) is any number and \(a\) is a positive number (10, 2, or \(e\) in this course). The graph of a typical logarithmic function is shown. Although the function grows very slowly, the graph does not have a horizontal asymptote.

The maximum expected difference between an estimate for a population characteristic and the actual value of the population characteristic. 

The value halfway between the maximum and minimum values of a period function. Also the horizontal line whose \(y\)-coordinate is that value.

The power to which a factor occurs in the factored form of a polynomial. For example, in the polynomial \((x-1)^2(x+3)\), the factor \(x-1\) has multiplicity 2 and the factor \(x+3\) has multiplicity 1.

The natural logarithm of \(x\), written \(\ln(x)\), is the log to the base \(e\) of \(x\). So it is the number \(y\) that makes the equation \(e^y = x\) true.

A specific distribution in statistics whose graph is symmetric and bell-shaped, has an area of 1 between the \(x\)-axis and the graph, and has the \(x\)-axis as a horizontal asymptote. 

An oblique solid is not exactly upright—it seems to lean over at an angle.

• Prisms and cylinders are said to be oblique if the directed line segment that defines the translation between the bases is not perpendicular to the bases.
• A cone is said to be oblique if a line drawn from its apex at a right angle to the plane of its base does not intersect the center of the base. The same definition applies to pyramids whose bases are figures with a center point, such as a square or a regular pentagon.

An observational study collects data without influencing the subjects directly.

A function \(f\) that satisfies \(f(x) = \text-f(\text-x)\) for all inputs \(x\). You can tell an odd function from its graph: Its graph is taken to itself when you reflect it across both the \(x\)- and \(y\)-axes. This can also be seen as a 180\(^\circ\) rotation about the origin.

The length of an interval at which a periodic function repeats. A function \(f\) has a period, \(p\), if \(f(x+p) = f(x)\) for all inputs \(x\).

A function whose values repeat at regular intervals. If \(f\) is a periodic function then there is a number \(p\), called the period, so that \(f(x + p) = f(x)\) for all inputs \(x\).

A polynomial function of \(x\)  is a function given by a sum of terms, each of which is a constant times a whole-number power of \(x\). The word “polynomial” is used to refer both to the function and to the expression defining it.

A logarithm with an argument raised to a power is equivalent to the power multiplied by the logarithm of the argument with a power of 1.

\(\log_a \left( b^c \right) = c \boldcdot \log_a b\)

A prism is a three-dimensional figure composed of two parallel, congruent faces (called bases) connected by parallelograms. A prism is named for the shape of its bases. For example, if a prism’s bases are pentagons, it is called a pentagonal prism.

The sum of two logarithms with the same base is equivalent to a logarithm with the same base of the product of the arguments.

\(\log_a (b) + \log_a (c) = \log_a (b \boldcdot c)\)

A pyramid is a three-dimensional figure that has one special face called the base. All of the other faces are triangles that meet at a single vertex called the apex. A pyramid is named for the shape of its base. For example, if a pyramid’s base is a hexagon, it is called a hexagonal pyramid.

The identity \(\sin^2(x) + \cos^2(x) = 1\) relating the sine and cosine of a number. It is called the Pythagorean identity because it follows from the Pythagorean theorem.

The difference of logarithms with the same base is equivalent to a logarithm of the quotient of the arguments.

\(\log_a (b) - \log_a (c) = \log_a \left( \frac{b}{c} \right)\)

A selection process in which each item in a set has an equal probability of being selected.

A randomization distribution is a distribution created to analyze the results of an experiment. The data from the control and treatment groups are randomly redistributed into two groups, and the statistic of interest is calculated for each group. The distribution of the statistics of interest from each redistribution is the randomization distribution.

A rational function is a function defined by a fraction with polynomials in the numerator and denominator. Rational functions include polynomials because a polynomial can be written as a fraction with denominator 1.

Histogram from 16.5 to 25.5 by 0 point 5’s. Handspan, centimeters. Beginning at 16.5 up to but not including 17, height of bar at each interval is .001, .007, .013, .016, .052, .059, .087, .132, .159, .132, .104, .098, .069, .032, .023, .010, .004, .002, 0.

A histogram where the height of each bar is the fraction of the entire data set that falls into the corresponding interval (that is, it is the relative frequency with which the data values fall into that interval).

A point on the graph of a function that is higher than any of the points around it.

A point on the graph of a function that is lower than any of the points around it.

A right solid is exactly upright—it does not seem to lean over at an angle.

• Prisms or cylinders are said to be right if the directed line segment that defines the translation between the bases is perpendicular to the bases.
• A cone is said to be right if a line drawn from its apex at a right angle to the plane of its base passes through the center of the base. The same definition applies to pyramids whose bases are figures with a center point, such as a square or a regular pentagon.

A sample is a subset of a population. For example, a population could be all the students at one school in grade 11. One sample of that population is all the grade 11 students who are taking Spanish.

A sampling distribution is a distribution of statistics obtained from samples drawn from a specific population. For example, a group of students draw several samples from a population that has 70% of items that Pass. The proportion of papers from each sample that are marked Pass is shown in this dot plot.

A solid of rotation is a three-dimensional figure formed by rotating a two-dimensional figure around a line called the axis of rotation.

A sphere is the set of all points in a three-dimensional space that are the same distance from a center point. All the cross-sections of a sphere are circles.

A survey is a set of questions given to people to seek their responses.

In an experiment where you are comparing two groups, one of which is being given a treatment and the other of which is the control group without any treatment, the treatment is the value of the variable that is changed for the treatment group.

The circle in the coordinate plane with radius 1 and center the origin.

The vertex form of a quadratic expression is \(a(x-h)^2 + k\), where \(a\), \(h\), and \(k\) are constants and \(a \neq 0\). The vertex of the graph is at the point \((h,k)\).

The line \(x=a\) is a vertical asymptote for a function \(f\) if \(f\) is undefined at \(x=a\) and its outputs get larger and larger in the negative or positive direction when \(x\) gets closer and closer to \(a\) on each side of the line. This means the graph goes off in the vertical direction on either side of the line.