PreCalculus Glossary

Absolute value: for any real number x, the absolute value of x, |x|, is x if x is positive or 0; and is the negative of x if x is less than 0.

Asymptote: If the graph of a function approaches a horizontal line or a vertical line, we call the line an asymptote.

Circle: a point (x,y) lies on a circle of radius r and center (h,k) if and only if

Composition of functions f and g, where the range of g is in the domain of f:

The composition of f with g is f(g(x)). This is read as " f of g of x". The inner function is evaluated first, and then that result is used to evaluate the outer function.

Directed distance between two points a and b on the real line:

Directed distance from a to b is b - a

Directed distance from b to a is a - b

Distance in a plane: between two points (x₁,y₁) and (x₂,y₂) in the plane, the distance is

Distance on the real line: between two points x₁ and x₂, the distance d is

Domain: the collection of all the x-values for a function

Ellipse with center at (h,k), major axis of length 2a and minor axis of length 2b:

Major axis horizontal:

Major axis vertical:

Function with independent variable x and dependent variable y: exactly one y for each x.

Intercepts:

x-intercept: where a graph intersects the x-axis. To find it, let y be 0 and solve the equation for x

y-intercept: where a graph intersects the y-axis. To find it, let x be 0 and solve the equation for y

Inverse functions: f and g are inverses of each other if

f(g(x)) = x for all x in the domain of g, and

g(f(x)) = x for all x in the domain of f.

g is also denoted by f ^-1 called "f inverse"

Line:

Vertical line: x = a

Horizontal line: y = b

Two-point equation:

Point-slope equation: y - y₁ = m(x - x₁)

Slope-intercept equation: y = mx + b

Two intercepts equation: ,

General equation: Ax + By + C = 0

Midpoint of a line segment joining points (x₁, y₁) and (x₂, y₂) is

Midpoint, c, of a segment [a,b] on the real line is located at the average of the endpoints, a,b:

One-to-one: a function is said to be one-to-one if for each y value in its range there corresponds only one x value in its domain.

Parallel lines: have equal slopes

Perpendicular lines: have slopes related by

Point of intersection of two graphs: a solution point which satisfies both equations. The points of intersection can be found by solving the equations simultaneously.

Pythagorean Theorem: for a right triangle with sides A and B and hypotenuse H satisfies

Range: the collection of all the y values of a function

Slope m of the line passing through the points (x₁,y₁) and (x₂,y₂) is

, where .

Symmetry with respect to the

x-axis if replacing y by -y yields an equivalent equation

y-axis if replacing x by -x yields an equivalent equation

the origin if replacing x by -x and y by -y yields an equivalent equation