Algebra Functions and Graphs

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

A special type of relation where each x-value has only one corresponding y-value. Its graph passes the vertical line test.

The set of all possible x-values for which the function is defined.

The set of all possible y-values (outputs) that the function can produce.

A special notation used to represent a function. f(x) refers to the output of the function at a particular value of x.

Identifying sections of a graph where the function's output is consistently increasing, decreasing, or staying the same.

A function where the output is decided by different rules depending on the input value.

A function where all the powers of x are even numbers. The graph is symmetrical with respect to the y-axis.

A function where the powers of x are odd numbers. The graph is symmetrical with respect to the origin.

Expressing the domain and range of a function using intervals, where each interval represents a continuous set of x or y values.

They have symmetry to the origin, meaning if you rotate the graph 180 degrees, it will look exactly the same.

This is NOT an odd function because it contains a term with an even exponent (x^2).

Constant terms (like 3, -5, etc., without any variables) are considered to have an even power (x^0). Therefore, a function with a constant term is NOT an odd function.

A function that has only constant terms (no variables) is classified as an even function, NOT an odd function.

y = mx + b is the equation of a line where 'm' represents the slope and 'b' represents the y-intercept.

The slope of a line is its steepness; how much the line rises or falls for every unit of horizontal change.

The point where the line crosses the y-axis, represented by the value of 'b' in y = mx + b.

A line that goes straight up and down, with an undefined slope. Its equation is x = #.

A line that goes horizontally across the graph, with a slope of 0. Its equation is y = #.

Lines that have the same slope. They never intersect, and they always maintain the same distance apart.

Lines that intersect at a right angle (90 degrees). Their slopes are opposite and reciprocal.

The slope of a line that is perpendicular to another line. To find it, flip the original slope and change its sign.

A negative sign outside the function, like -f(x), flips the graph vertically, reflecting it across the x-axis.

A negative sign inside the function, like f(-x), flips the graph horizontally, reflecting it across the y-axis.

Adding a number outside the function, like f(x) + #, shifts the graph vertically. Upward if positive, downward if negative.

Adding a number inside the function, like f(x + #), shifts the graph horizontally. Solve for x using the inside (x + # = 0).

Multiplying the function by a number outside (like ay = f(x)) changes the graph's height and width. If a > 1, it stretches vertically. If 0 < a < 1, it compresses vertically.

The distance between two points is calculated using the distance formula: √((x₂ - x₁)² + (y₂ - y₁)²).

The midpoint of a line segment is the point exactly halfway between the two endpoints: ((x₁ + x₂)/2 , (y₁ + y₂)/2).

A circle is defined as all points that are the same distance away from a central point. The standard equation for a circle is (x - h)² + (y - k)² = r², where (h, k) represents the center and r represents the radius.

The center of a circle is the point from which all points on the circle are equidistant.

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

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

A math operation where you take the output of one function and use it as the input of another function. It's like plugging one function into the other.

The process of finding the domain of a function involves identifying and excluding values that would lead to undefined results. This typically includes values that cause division by zero, taking the square root of a negative number, or other operations that result in an undefined output.

To find the domain when a function involves a square root, set the radicand (the expression under the square root) greater than or equal to zero and solve the inequality. The solution represents the allowable domain values.

To find the domain of a function with a fraction, set the denominator equal to zero and solve for x. Exclude these x-values from the domain, as they would result in division by zero.