Glossary
Integrated Math 2
The addition rule states that given events A and B, the probability of either A or B is given by \(P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B})\).
An altitude in a triangle is a line segment from a vertex to the opposite side that is perpendicular to that side.
In this diagram, the dashed line segments show the altitude of each triangle.
An arc is a part of a circle’s circumference between two points on the circle.
In this diagram, the blue highlighting shows one arc of the circle.
Arccosine is a relationship used to find an acute angle measure in a right triangle when two side lengths are known.
The arccosine of a number between 0 and 1 is the measure of an acute angle whose cosine is that number.
\(\arccos \left( \frac{\text{adjacent}}{\text{hypotenuse}} \right)=\theta\)
Arcsine is a relationship used to find an acute angle measure in a right triangle when two side lengths are known.
The arcsine of a number between 0 and 1 is the measure of an acute angle whose sine is that number.
\(\arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right) = \theta\)
Arctangent is a relationship used to find an acute angle measure in a right triangle when two side lengths are known.
The arctangent of a positive number is the measure of an acute angle whose tangent is that number.
\(\arctan \left( \frac{\text{opposite}}{\text{adjacent}} \right) = \theta\)
A central angle is an angle formed by two radii of a circle that each have an endpoint at the center of the circle.
In this diagram, the angle arc symbol shows one central angle of the circle.
A chance experiment is something that can be done over and over again, and what will happen each time is not known.
For example, each time the spinner is spun, it could land on red, yellow, blue, or green.
The circumcenter of a triangle is the intersection point of all three perpendicular bisectors of the triangle’s sides. It is the center of the triangle’s circumscribed circle.
In this diagram, the circumcenter is point \(D\).
When a figure is circumscribed, it is completely surrounded by another figure, so their sides, edges, vertices, or curves touch.
- A polygon is circumscribed by a circle if it fits inside the circle and every vertex of the polygon is on the circle.
- A circle is circumscribed by a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.
Completing the square is a method of rewriting a quadratic expression or equation.
- A quadratic expression is rewritten in the form \(a(x+p)^2-q\), where \(a\), \(p\), and \(q\) are constants and \(a\neq0.\)
- A quadratic equation is rewritten in the form \(a(x+p)^2=q\).
A number in the complex plane. It can be written as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i^2 = \text-1\).
Conditional probability is the likelihood that one event occurs given that another event occurs.
The cosine of an acute angle in a right triangle is the ratio (quotient) of the length of the adjacent leg to the length of the hypotenuse.
In this diagram, \(\cos(x)=\frac{b}{c}\).
If a circle can be drawn around a quadrilateral so that it passes through every vertex, the quadrilateral is called a cyclic quadrilateral.
That is, a cyclic quadrilateral can be circumscribed by a circle.
Events are called dependent when they are from the same experiment and where the outcome of one event affects the probability of another.
A dilation is a transformation that can reduce or enlarge a figure.
A dilation with center \(P\) and positive scale factor \(k\) takes a point \(A\) along the ray \(PA\) to another point whose distance is \(k\) times farther away from \(P\) than \(A\) is.
Triangle \(A'B'C'\) is the result of applying a dilation with center \(P\) and scale factor 3 to triangle \(ABC\).
A directrix is the line that, together with a point called the focus, defines a parabola.
This diagram shows a parabola is the set of points equidistant from the focus and directrix.
An event is a set of 1 or more outcomes in a chance experiment.
For example, if a number cube is rolled, there are 6 possible outcomes.
Some events are “rolling a number less than 3,” “rolling an even number,” or “rolling a 5.”
A quadratic expression is in factored form when it is written as the product of a constant times two linear factors.
- \(2x^2 + 4x - 6\) written in factored form is \(2(x-1)(x+3)\).
- \(15x^2 + x - 2\) written in factored form is \((5x + 2)(3x-1)\).
A focus is the point that, together with a line called the directrix, defines a parabola.
This diagram shows a parabola is the set of points equidistant from the focus and directrix.
A function is a rule that takes inputs from one set and assigns them to outputs from another set. Each input is assigned exactly one output.
A number on the imaginary number line. It can be written as \(bi\), where \(b\) is a real number and \(i^2 = \text-1\).
The incenter of a triangle is the intersection point of all three of the triangle’s angle bisectors. It is the center of the triangle’s inscribed circle.
In this diagram the incenter is point \(D\).
Events are called independent when they are from the same experiment and the outcome of one event does not affect the probability of another.
An irrational number is a number that is not rational. This means it cannot be expressed as a positive fraction, a negative fraction, or zero. It cannot be written in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\).
For example, the numbers \(\pi\) and \(\text{-}\sqrt{2}\) are irrational numbers.
A linear term of an expression has a variable raised to the first power.
- In the expression, \(5x+2\), the linear term is \(5x\).
- In the expression, \(20-x^2+x\), the linear term is \(x\).
- In the expression \(ax^2+bx+c\), where \(a\), \(b\), and \(c\) are constants, the linear term is \(bx\).
A median is a line drawn from a vertex of a triangle to the midpoint of the opposite side.
Each dashed line segment in this image is a median.
An outcome of a chance experiment is one of the things that can happen.
For example, the possible outcomes of tossing a coin are heads and tails.
A parabola is the set of points that are equidistant from a given point, called the focus, and a given line, called the directrix.
A perfect square is a number or an expression that is the result of multiplying a number or an expression to itself. In general, the multiplied number is rational and the multiplied expression has rational coefficients.
The probability of a chance event is a number from 0 to 1 that expresses the likelihood of the event occurring, with 0 meaning it will never occur and 1 meaning it will always occur.
A quadratic equation is an equation that is equivalent to one of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).
A quadratic expression is an expression that is equivalent to one of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).
The quadratic formula is \(x = {\text-b \pm \sqrt{b^2-4ac} \over 2a}\) and gives the solutions of the quadratic equation \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).
A quadratic function is a function where the output is given by a quadratic expression in the input.
For example, \(f(x) =ax^2+bx+c\), where \(a\), \(b\), and \(c\) are constants and \(a\ne0\), is a quadratic function.
A radian is a unit of measurement for angles, based on the radius of a circle.
The radian measure of any central angle is this ratio: \(\frac{\text{length of intercepted arc}}{\text{radius}}\)
A rational number is a number that can be written as a positive fraction, a negative fraction, or zero. It can be written in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\).
- 0.7 is a rational number because it can be written as \(\frac{7}{10}\).
- The numbers 0, 3, \(\text{-}\frac{3}{4}\), and 6.7 are all rational numbers.
The sample space is the list of every possible outcome for a chance experiment.
For example, the sample space for tossing two coins is:
| heads-heads | tails-heads |
| heads-tails | tails-tails |
A sector is the region inside a circle between two radii.
In this diagram, the shaded region shows one sector of the circle.
One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second.
Triangle \(A'B'C'\) is similar to triangle \(ABC\) because a rotation with center \(B\) followed by a dilation with center \(P\) takes \(ABC\) to \(A'B'C'\).
The sine of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the hypotenuse.
In this diagram, \(\sin(x) = \frac{a}{c}.\)
The standard form of a quadratic expression is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a\) \(\ne\) 0.
The tangent of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the adjacent leg.
In this diagram, \(\tan(x) = \frac{a}{b}.\)
A line is tangent to a circle if the line intersects the circle at exactly one point and is perpendicular to the radius.
Trigonometric ratios relate the angles and sides of right triangles.
Three trigonometric ratios are sine, cosine, and tangent.
\(\sin(\theta)=\dfrac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\theta)=\dfrac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\theta)=\dfrac{\text{opposite}}{\text{adjacent}}\)
The vertex of the graph of a quadratic function or of an absolute value function is the point where the graph changes from increasing to decreasing, or vice versa. It is the highest or lowest point on the graph.
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 \((h,k)\).
A zero of a function is an input that results in an output of 0. In other words, if \(f(a) = 0\), then \(a\) is a zero of \(f\).