Algebra Class: Functions and Graphs

A set of ordered pairs (x, y) representing points on a graph.

A special type of relation where each x-value has only one corresponding y-value.

The set of all possible x-values for a relation or function.

The set of all possible y-values for a relation or function.

The point where the x and y axes intersect, represented by the coordinates (0, 0).

A function where the graph is divided into two or more separate pieces, each with its own equation and defined over specific intervals.

A function that is symmetrical about the y-axis. This means that if you reflect the graph over the y-axis, it will look exactly the same.

A function that is symmetrical about the origin. This means that if you rotate the graph 180 degrees around the origin, it will look exactly the same.

A function that is increasing over an interval if its graph goes up from left to right. It is decreasing if the graph goes down from left to right and it is constant if the graph doesn't change.

A notation used to describe a set of numbers that a function's input (x-values) can take. It includes two symbols: () for open intervals and [ ] for closed intervals.

A function that has symmetry with respect to the origin. This means that if you rotate the graph of the function 180 degrees around the origin, the graph will look exactly the same.

In a function, a constant term (like +5 or -8) is considered to have an exponent of 0 (x^0 = 1). For example, in the function f(x) = 2x^2 + 5, the constant term '5' has an exponent of 0.

A mathematical expression with terms that have both odd and even exponents for the variable. For example, f(x) = x^2 - 2x + 1 is not an odd function because it has both an even exponent (x²), and an odd exponent (x).

A way to represent a line using its slope (m) and y-intercept (b). It has the formula y = mx + b.

The steepness of a line, calculated as the change in y divided by the change in x: (y2 - y1) / (x2 - x1).

The point where a line crosses the y-axis. It has a coordinate of (0, b).

A line with a slope of 0. Its equation is y = #, where # is the y-coordinate of all points on the line.

A line with an undefined slope. Its equation is x = #, where # is the x-coordinate of all points on the line.

Lines that have the same slope. They never intersect and run parallel to each other.

Lines that intersect at a 90-degree angle. Their slopes are opposite reciprocals of each other.

The measure of how steep a line is. It's calculated by dividing the change in y (the rise) by the change in x (the run).

Two numbers that are the inverse of each other. One is flipped and the sign is changed.

The slope of a line is the same throughout its entire length. This means that the line has a constant steepness.

A function where a negative sign is added outside f(x) (like -f(x)). This causes the graph to flip vertically across the x-axis.

A function where a negative sign is added inside the input (like f(-x)). This causes the graph to flip horizontally across the y-axis.

A function where a number is added to the output (like f(x) + 3). This shifts the graph up or down along the y-axis.

A function where a number is added inside the input (like f(x + 2)). This shifts the graph left or right along the x-axis.

A function where a number greater than 1 is multiplied by f(x) (like 2f(x)). This stretches the graph vertically, making it taller and skinnier.

The distance between two points in a coordinate plane, calculated using the formula: √((x₂ - x₁)² + (y₂ - y₁)²)

The midpoint of a line segment, calculated using the formula: ((x₁ + x₂)/2, (y₁ + y₂)/2)

A set of all points that are the same distance from a central point (the center).

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

The radius of a circle is the distance from the center to any point on the circle.

The set of all possible input values (x-values) for which a function is defined. It excludes values that would lead to undefined results, such as division by zero or taking the square root of a negative number.

A process of combining two functions, where the output of one function becomes the input of the other. It involves replacing the 'x' in the outer function with the entire inner function.

Values that need to be excluded from the domain of a function due to division by zero. To find them, set the denominator of the fraction equal to zero and solve for 'x'.

Values that need to be excluded from the domain of a function due to taking the square root of a negative number. To find them, set the expression under the square root greater than or equal to zero and solve.

A step-by-step process for simplifying a function, especially when dealing with compositions. It involves substituting, simplifying using the order of operations (PEMDAS), and avoiding solving for 'x'.