Glossary
Integrated Math 1
An angle bisector is a line through the vertex of an angle that divides it into two congruent angles.
In this diagram, the dashed line is the angle bisector.
There is an association between two variables if they are statistically related to each other. This means that the value of one variable can be used to estimate the value of the other. An association can apply to categorical data or numerical data.
An auxiliary line is an extra line drawn on a figure to show hidden structure.
- The auxiliary line drawn on this isosceles triangle shows a line of symmetry.
- The auxiliary lines drawn on this parallelogram suggest a way of rearranging it into a rectangle.
The average rate of change of a function is a ratio that describes how fast one quantity changes with respect to another.
The average rate of change for function \(f\) between inputs \(a\) and \(b\) is the change in the outputs divided by the change in the inputs: \(\frac{f(b)-f(a)}{b-a}\). It is the slope of the line that connects \((a,f(a))\) and \((b, f(b))\) on the graph.
In a bell-shaped distribution, most of the data cluster near the center and fewer points are farther from the center. The dot plot or histogram for the data has the form of a bell.
This dot plot shows a bell-shaped distribution.
In a bimodal distribution, there are two very common data values. The dot plot or histogram for the data has two distinct peaks.
This dot plot shows a bimodal distribution. The two common data values are 2 and 7.
Categorical data are data where the values are divided into groups, or categories.
For example, the breeds of 10 different dogs are categorical data. Another example is the colors of 100 different flowers.
A categorical variable is a variable that takes on values that are divided into groups, or categories.
For example, color is a categorical variable that can take on the values, red, blue, green, and so on.
In a causal relationship, a change in one of the variables causes a change in the other variable.
A circle of radius \(r\) with center \(O\) is the set of all points that are a distance \(r\) units from \(O\).
To draw a circle of radius 3 units and center \(O\), use a compass to draw all the points at a distance 3 units from \(O\).
Two figures are congruent if there is a rigid motion or a sequence of rigid motions (translations, rotations, and reflections) that takes one figure onto the other.
In this figure, Triangle A is congruent to Triangle D.
- A translation takes Triangle A onto Triangle B.
- A rotation takes Triangle B onto Triangle C.
- A reflection takes Triangle C onto Triangle D.
A constraint is a limitation on the possible values of variables in a model. It is often expressed by an equation or inequality or by specifying that the value must be an integer.
For example, distance above the ground \(d\), in meters, might be constrained to be non-negative, expressed by \(d \ge 0\).
The converse of an if-then statement is the statement that switches the hypothesis and the conclusion.
Example:
- Statement: “If it is raining, then the game is canceled.”
- Converse: “If the game is canceled, then it is raining.”
A correlation coefficient is a number between -1 and 1 that describes the strength and direction of a linear relationship between two numerical variables.
- The sign of the correlation coefficient is the same as the sign of the slope of the best-fit line.
- The closer the correlation coefficient is to 0, the weaker the linear relationship.
- When the correlation coefficient is closer to 1 or -1, the linear model fits the data better.
Correlation coefficient is close to 1.
Correlation coefficient is positive, and closer to 0.
Correlation coefficient is close to -1.
Corresponding parts are the matching parts of an original figure and its scaled copy that are in the same relative positions. The parts could be points, segments, angles, or distances. When two figures are congruent, all of their corresponding parts are congruent.
For example, in triangles \(ABC\) and \(DEF\):
- Point \(B\) corresponds to point \(E\).
- Segment \(AB\) corresponds to segment \(DE\).
- Angle \(BCA\) corresponds to angle \(EFD\).
A function is decreasing if its outputs get smaller as the inputs get larger. This results in a downward sloping graph as it goes from left to right. A function can also be decreasing just for a restricted range of inputs.
This graph shows the function \(f\) given by \(f(x)=3−x^2\). It is decreasing for \(x \geq 0\) because the graph slopes downward to the right of the vertical axis.
A dependent variable is a variable that represents the output of a function.
For example, the equation \(y = 6-x\) defines \(y\) as a function of \(x\).
- The variable \(x\) is called the independent variable because its value is chosen.
- The variable \(y\) is called the dependent variable because it depends on \(x\). Once a value is chosen for \(x\), the value of \(y\) is determined.
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 directed line segment is a line segment that has distance (length) and direction.
The arrow on this directed line segment shows that it starts at point \(A\) and ends at point \(B\).
Elimination is a method of solving a system of two equations in two variables. A multiple of one equation is added to or subtracted from another to get an equation with only one of the variables. (The other variable is eliminated.)
An exponential function is a function that has a constant growth factor. This means that it grows by equal factors over equal intervals.
For example, \(f(x)=2 \boldcdot 3^x\) defines an exponential function. Any time \(x\) increases by 1, \(f(x)\) increases by a factor of 3.
The five-number summary is one way to describe the distribution of a data set. The five numbers are the minimum, the three quartiles, and the maximum.
This box plot represents a data set with the following five-number summary: The minimum is 2, the three quartiles are 4, 4.5, and 6.5, and the maximum is 9.
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.
Function notation is a way of writing the relationship between the inputs and outputs of a function.
For example, a function is named \(f\) and \(x\) is an input. Then \(f(x)\) denotes the corresponding output in function notation.
In an exponential function, the output is multiplied by the same factor every time the input increases by 1. This multiplier is called the growth factor.
In an exponential function, the growth rate is the fraction or percentage of the output that gets added every time the input is increased by 1.
For example, if the growth rate is 20%, or 0.2, then the growth factor is 1.2.
A horizontal intercept of a graph is a point where the graph crosses the horizontal axis. If the axis is labeled with the variable \(x\), a horizontal intercept is also called an \(x\)-intercept. The term can also refer to only the \(x\)-coordinate of the point where the graph crosses the horizontal axis.
For example, the horizontal intercept of the graph of \(2x+4y=12\) is \((6,0)\), or just 6.
An image is the result of a transformation. Every part of the original figure moves in the same way to match up with a part of the image.
This diagram shows a transformation that takes \(A\) to \(A’\).
\(A\) is the original, and \(A’\) is the image.
A function is increasing if its outputs get larger as the inputs get larger. This results in an upward sloping graph as it goes from left to right. A function can also be increasing just for a restricted range of inputs.
This graph shows the function \(f\) given by \(f(x)=3−x^2\). It is increasing for \(x \leq 0\) because the graph slopes upward to the left of the vertical axis.
An independent variable is a variable that represents the input of a function.
For example, the equation \(y=6−x\) defines \(y\) as a function of \(x\).
- The variable \(x\) is called the independent variable because its value is chosen.
- The variable \(y\) is called the dependent variable because it depends on \(x\). Once a value is chosen for \(x\), the value of \(y\) is determined.
When a figure is inscribed, it is completely inside another figure so that their sides, edges, vertices, or curves touch.
- A polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle.
- A circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.
Two functions are inverses to each other if their input-output pairs are reversed.
- If one function takes \(a\) as input and gives \(b\) as an output, then the other function takes \(b\) as an input and gives \(a\) as an output.
- An inverse function can sometimes be found by reversing the processes that define the first function in order to define the second function.
A line of symmetry is a line that divides a figure into two parts that are mirror images of each other. When a figure is reflected across one of its lines of symmetry, it takes the figure onto itself.
These dashed lines show two lines of symmetry for a regular hexagon, and two lines of symmetry for the capital letter I.
A linear function is a function that has a constant rate of change. This means that it grows by equal differences over equal intervals.
For example, \(f(x)=4x-3\) defines a linear function. Any time \(x\) increases by 1, \(f(x)\) increases by 4.
A maximum of a function is a value of the function that is greater than or equal to all the other values. The maximum of the function’s graph is the highest point on the graph.
A minimum of a function is a value of the function that is less than or equal to all the other values. The minimum of the function’s graph is the lowest point on the graph.
A model is a mathematical or statistical representation of information from science, technology, engineering, work, or everyday life, that is used to understand the situation and make decisions.
Two numerical variables have a negative relationship if an increase in the data for one variable tends to be paired with a decrease in the data for the other variable.
This scatter plot shows a negative relationship.
A non-statistical question is a question that can be answered by a specific measurement or procedure where no variability is expected.
For example:
- How high is that building?
- If I run at 2 meters per second, how long will it take me to run 100 meters?
- Who was the first woman appointed to the Supreme Court?
Numerical data are data where the values are numbers, measurements, or quantities. Numerical data is also called measurement data or quantitative data.
For example, the weights of 10 different dogs are numerical data.
An outlier is a data value that is far from the other values in the data set. A value is considered an outlier if it is:
- More than 1.5 times the interquartile range greater than Q3.
- More than 1.5 times the interquartile range less than Q1.
In this box plot, the minimum, 0, and the maximum, 44, are both outliers.
Two lines that never intersect are called parallel. Line segments can also be parallel if they extend into parallel lines.
This figure shows two parallel line segments.
The perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to the segment.
In this diagram, the dashed line is the perpendicular bisector of segment \(AB\).
A piecewise function is a function defined using different expressions for different intervals in its domain.
Point-slope form is one way to write the equation of a line using its slope and the coordinates of one point on the line.
For a line with slope \(m\) through the point \((h,k)\), point-slope form is usually written as \(y-k=m(x-h)\). It can also be written as \(y = k+m(x-h)\).
Two numerical variables have a positive relationship if an increase in the data for one variable tends to be paired with an increase in the data for the other variable.
This scatter plot shows a positive relationship.
Two numbers that multiply to equal 1 are reciprocals.
If \(p\) is a rational number that is not 0, then the reciprocal of \(p\) is the number \(\frac{1}{p}\).
A reflection is a rigid transformation that is defined by a line. It takes one point to another point that is the same distance from the given line, but on the other side. The segment from the original point to its image is perpendicular to the line of reflection.
In this figure, \(A\) is reflected across line \(m\), and \(A′\) is the image of \(A\) under the reflection.
Reflect \(A\) across line \(m\).
A figure has reflection symmetry if there is a reflection that takes the figure onto itself.
In this diagram, the letter X has reflection symmetry over each of the dashed lines.
A regular polygon is a polygon where all of the sides are congruent and all of the angles are congruent.
A relative frequency table is a version of a two-way table that shows how often data values occur in relation to a total. Each entry in the table shows the frequency of one response divided by the total number of responses in the entire table or by the total number of responses in a row or a column.
Each entry in this relative frequency table represents the proportion of all the textbooks that have the characteristics given by its row and column. For example, out of all 1,000 textbooks, the proportion of textbooks that are new and \\$10 or less is 0.025, or 2.5%.
frequency table
| $10 or less | more than \\$10 but less than \\$30 | $30 or more | total | |
|---|---|---|---|---|
| new | 25 | 75 | 225 | 325 |
| used | 275 | 300 | 100 | 675 |
| total | 300 | 375 | 325 | 1,000 |
relative frequency table
| $10 or less | more than \\$10 but less than \\$30 | $30 or more | |
|---|---|---|---|
| new | \(0.025 = \frac{25}{1000}\) | 0.300 | 0.225 |
| used | 0.275 | 0.300 | 0.100 |
A residual is the difference between an actual data value and its value predicted by a model. It can be found by subtracting the \(y\)-value predicted by the linear model from the \(y\)-value for the data point.
On a scatter plot, the residual can be seen as the vertical distance between a data point and the best-fit line.
The lengths of the dashed segments on this scatter plot show the residuals for each data point.
A rigid transformation is a move that does not change any measurements of a figure.
Translations, rotations, and reflections are rigid motions. So is any sequence of any of these.
A rotation is a rigid transformation that is defined by a center, an angle, and a direction. It takes one point on a circle to another point, using a given center. The two radii—the one from the center to the original point and the one from the center to the image—make the angle of rotation.
In this figure, \(P′\) is the image of \(P\) after a counterclockwise rotation of \(t^\circ\) using the point \(O\) as the center.
In this figure, quadrilateral \(ABCD\) is rotated \(120^\circ\) counterclockwise using the point \(D\) as the center.
A figure has rotation symmetry if there is a rotation that takes the figure onto itself. (This does not include rotations using angles, such as \(0^\circ\) and \(360^\circ\), that take every point on a figure back to its original position.)
This hexagon has rotation symmetry 60 degrees clockwise or counterclockwise around its center.
In a skewed distribution, one side has more values farther from the bulk of the data than the other side. The mean is usually not equal to the median. The dot plot or histogram for the data shows only one peak leaning to one side.
This dot plot shows a skewed distribution. The data values on the left, such as 1, 2, and 3, are farther from the bulk of the data than the data values on the right.
A solution to a system of equations is the values for the variables that make all the equations true.
This graph shows a system of two equations. The solution of the system is a coordinate pair that makes both equations true. On the graph, the solution is shown as the point where the two lines intersect.
The solutions to a system of inequalities are all the values for the variables that make all of the inequalities true.
This graph shows a system of two inequalities. The solutions of the system are all the coordinate pairs that make both inequalities true. On the graph, the solution is shown as all the points in the region where the graphs of the two inequalities overlap.
A graph of two intersecting inequalities on a coordinate plane. The line x + y < 5 goes through the points 0 comma 5 and 5 comma 0. The region above the dashed line is shaded. The line 3 x - 2 y < 0 goes through the points 0 comma 0 and 4 comma 6. The region below the dashed line is shaded.
The standard deviation is a measure of the variability, or spread, of a distribution. It is calculated by a method similar to the method for calculating the MAD (mean absolute deviation). The exact method is studied in more advanced courses.
A statistic is a quantity that is calculated from sample data, such as mean, median, or MAD (mean absolute deviation).
A statistical question is a question that can only be answered by using data in which variability is expected.
For example:
- Who is the most popular musical artist at our school?
- When do students in our class typically eat dinner?
- Which classroom in our school has the most books?
Two numerical variables have a strong relationship if the data is tightly clustered around the best-fit line.
Substitution is the action of replacing a variable with a number or expression it is equal to.
In a symmetric distribution, the data values on each side of the center mirror each other. The dot plot or histogram for the data has a vertical line of symmetry in the center, where the mean is equal to the median.
This dot plot shows a distribution that is symmetric about the data value 5.
A figure has symmetry if there is a rigid transformation that takes it onto itself. (This does not include transformations that take every point back to its original position.)
Two or more equations that represent the constraints in the same situation form a system of equations.
Two or more inequalities that represent the constraints in the same situation form a system of inequalities.
A translation is a rigid transformation that is defined by a directed line segment. It takes one point to another point so that:
- The directed line segment from the original point to its image is parallel to the given directed line segment.
- Both directed line segments have the same length and direction.
In the figure shown here, \(A'\) is the image of \(A\) under the translation given by the directed line segment \(t\).
A two-way table is a way of organizing data from two categorical variables in order to investigate the association between them.
This two-way table can be used to study the relationship between age group and cell phone ownership.
| has a cell phone | does not have a cell phone | |
|---|---|---|
| 10–12 years old | 25 | 35 |
| 13–15 years old | 38 | 12 |
| 16–18 years old | 52 | 8 |
In a uniform distribution, the data values are evenly spread out across the range of the data. The dot plot or histogram for the data shows no peaks.
This dot plot shows a uniform distribution.
A variable is a characteristic of individuals in a population that can take on different values.
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.
A vertical intercept of a graph is a point where the graph crosses the vertical axis. If the axis is labeled with the variable \(y\), a vertical intercept is also called a \(y\)-intercept. The term can also refer to only the \(y\)-coordinate of the point where the graph crosses the vertical axis.
For example, the vertical intercept of the graph of \(y=3x−5\) is \((0,\text{-}5)\), or just -5.