Algebra Class: Functions and Polynomials

The distance of a number from zero on the number line.

A function described by different formulas for different parts of its domain.

A function where f(-x) = f(x) for all x in its domain. Its graph is symmetric across the y-axis.

A function where f(-x) = -f(x) for all x in its domain. Its graph is symmetric about the origin.

The absolute value function, defined as |x| = x if x >= 0 and |x| = -x if x < 0.

A function that increases or decreases in steps. Its graph looks like a series of horizontal lines.

The symbol used to represent the absolute value of a number.

A function that represents the cost of mailing a large envelope, where the cost depends on the weight of the envelope.

Reflecting a function's graph about the x-axis changes the sign of the y-coordinate for every point on the graph. Think of folding the graph along the x-axis.

Multiplying the y-coordinate of every point on a graph by a constant 'c' results in either stretching or shrinking the graph vertically, depending on the value of 'c'. If 'c' is greater than 1, the graph is stretched. If 'c' is between 0 and 1, the graph is shrunk.

The graph of 'y = |f(x)|' is obtained by reflecting the part of the graph of 'y = f(x)' that lies below the x-axis about the x-axis. The part of the graph that lies above the x-axis remains the same.

The limit of a function f(x) as x approaches a is the value that f(x) gets closer and closer to as x gets closer and closer to a, without actually reaching 'a'. It indicates the behavior of the function near a point.

A mathematical relationship that represents the change in temperature with respect to height, where the temperature decreases linearly as the height increases.

The rate at which the temperature changes with respect to height. It is the slope of the linear function.

A mathematical function that describes a curved line, determined by its coefficients, which can be used to represent various relationships.

The highest power of the variable in a polynomial function determines its degree. For example, a quadratic function has a degree of 2.

A specific type of polynomial function with a degree of 2, represented by the equation P(x) = ax^2 + bx + c.

The graph of a quadratic function, which is always a symmetric, U-shaped curve. It opens upwards if the leading coefficient is positive and downwards if negative.

A mathematical model that is based on observed data and seeks to capture the general trend of the data points.

A mathematical concept used to represent a linear relationship where the y-intercept is the value of the function when the independent variable is zero.

A way to represent the limit of a function f(x) as x approaches a when the function grows infinitely large.

A line that the graph of a function approaches as the input approaches a certain value, but never actually touches.

A way to express that a function f(x) grows infinitely large (either positive or negative) as x approaches a particular value.

The limit of f(x) as x approaches 'a' from the left side (values of x less than 'a').

The limit of f(x) as x approaches 'a' from the right side (values of x greater than 'a').

A specific type of limit where the function's output becomes infinitely large and negative as x approaches a certain value.

The limit of a function exists if and only if both the left-hand and right-hand limits exist and are equal.

The limit of f(x) as x approaches a from either the left or the right side of 'a' is infinity.

A function where the limit exists and equals the function value at the point. In simple terms, you can find the limit by simply plugging in the number.

The derivative of a constant times a function is equal to the constant multiplied by the derivative of the function.

The derivative of a sum of functions is equal to the sum of the derivatives of each function.

Calculates the limit of a function as x approaches a number 'a', but from the smaller side (values less than 'a').

The derivative of a difference of functions is equal to the difference of the derivatives of each function.

Calculates the limit of a function as x approaches a number 'a', but from the larger side (values greater than 'a').

The derivative of a product of two functions is the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.

The derivative of a constant is always zero.

A theorem stating that a two-sided limit exists only if both the left-hand and right-hand limits exist and are equal.

The derivative of x raised to a power (n) is equal to n times x raised to the power (n-1).

Any function can be broken down into simpler functions, and its derivative can be calculated by applying the appropriate rules for each simpler function.

A function squeezed between two other functions with the same limit, also has the same limit. Think of a sandwich.

A function that can be evaluated by simply plugging in the value of x. It does not have any holes or breaks.

The derivative is a measure of the instantaneous rate of change of a function. It tells us how fast a function is changing at a specific point.