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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.
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 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 \(d \ge 0\).
A correlation coefficient is a number between -1 and 1 that describes the strength and direction of a linear relationship between two numerical variables.
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 \(DEF\):
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 \(x\).
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 \(ABC\).
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 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.
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 \(x\).
When a figure is inscribed, it is completely inside another figure so that their sides, edges, vertices, or curves touch.
Two functions are inverses to each other if their input-output pairs are reversed.
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 non-statistical question is a question that can be answered by a specific measurement or procedure where no variability is expected.
For example:
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:
In this box plot, the minimum, 0, and the maximum, 44, are both outliers.
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 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 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.
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 translation is a rigid transformation that is defined by a directed line segment. It takes one point to another point so that:
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 |
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.
