Unit 5
Glossary
Algebra 1
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.
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 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.
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 \(x\).
- 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.
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 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 piecewise function is a function defined using different expressions for different intervals in its domain.
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.