Rectangular Coordinates and Graphs

Ordered Pairs

In mathematics, an ordered pair is a pair of elements written in a specific order, usually as (x, y). Ordered pairs are used to represent points in the coordinate plane, where the first element is the x-coordinate and the second is the y-coordinate.

• Key Point: Ordered pairs express relationships between two quantities, such as categories and amounts spent.

• Example: The table below shows categories of spending and the corresponding amounts, which can be written as ordered pairs (category, amount).

The Rectangular Coordinate System

The rectangular coordinate system (also called the Cartesian coordinate system) consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin (0, 0), dividing the plane into four quadrants.

• Key Terms: origin, x-axis, y-axis, quadrants, coordinates, coordinate plane (xy-plane)

• Application: Used to plot points, graph equations, and analyze geometric relationships.

The Distance Formula

The distance formula calculates the distance between two points P$(x_1, y_1)$ and Q$(x_2, y_2)$ in the coordinate plane:

$ d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $

• Key Point: This formula is derived from the Pythagorean Theorem.

• Example: Find the distance between $P(-8, 4)$ and $Q(3, -2)$.

The Midpoint Formula

The midpoint formula finds the point exactly halfway between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$:

$ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $

• Key Point: The midpoint is the average of the x-coordinates and the y-coordinates.

• Example: Estimate the midpoint of fast-food revenue between 1990 and 2014 using the graph below.

Equations in Two Variables

An equation in two variables relates x and y. The graph of such an equation is the set of all points (x, y) that satisfy the equation.

• Key Point: Solutions to the equation are represented as ordered pairs.

• Example: For $y = 4x - 1$, some solutions are (0, -1), (1, 3), (2, 7).

Graphing an Equation by Point Plotting

To graph an equation:

1. Find the intercepts (where the graph crosses the axes).

2. Find additional ordered pairs as needed.

3. Plot the points.

4. Connect the points with a smooth line or curve.

Functions and Relations

Relations and Functions

A relation is a set of ordered pairs. A function is a relation in which each input (x-value) corresponds to exactly one output (y-value).

• Key Point: In a function, no two ordered pairs have the same first component with different second components.

• Example: The diagram below shows a relation that is not a function, since one input maps to multiple outputs.

Domain and Range

The domain of a relation is the set of all possible input values (x-values). The range is the set of all possible output values (y-values).

• Key Point: The domain and range can be found from a set of ordered pairs, a table, or a graph.

• Example: The diagram below illustrates domain and range on a graph.

Determining Whether Relations Are Functions: The Vertical Line Test

The vertical line test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.

• Key Point: If a vertical line crosses the graph more than once, the relation is not a function.

Function Notation

Function notation expresses the output of a function f for an input x as $f(x)$. For example, if $f(x) = 2x + 1$, then $f(3) = 7$.

• Key Point: $f(x)$ is not multiplication; it denotes the value of the function at x.

Increasing, Decreasing, and Constant Functions

A function is increasing on an interval if its output rises as x increases, decreasing if its output falls, and constant if its output remains the same.

• Key Point: These properties are determined by comparing $f(x_1)$ and $f(x_2)$ for $x_1 < x_2$.

• Example: The graph below shows intervals where a function is increasing, decreasing, or constant.

Linear Functions and Equations of Lines

Linear Functions

A linear function has the form $f(x) = ax + b$, where a and b are real numbers. Its graph is a straight line.

• Key Point: The slope a determines the steepness and direction; b is the y-intercept.

• Example: The graph below shows a linear function.

Standard Form of a Line

The standard form of a line is $Ax + By = C$, where A, B, and C are real numbers and A and B are not both zero.

Slope

The slope m of a line through points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$ m = \frac{y_2 - y_1}{x_2 - x_1} $

• Key Point: Slope measures the rate of change of y with respect to x.

Point-Slope and Slope-Intercept Forms

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Slope-Intercept Form: $y = mx + b$

Parallel and Perpendicular Lines

• Parallel lines have the same slope.

• Perpendicular lines have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$.

Graphs of Basic Functions

Basic Function Types and Their Graphs

• Identity Function: $f(x) = x$

• Squaring Function: $f(x) = x^2$

• Cubing Function: $f(x) = x^3$

• Square Root Function: $f(x) = \sqrt{x}$

• Cube Root Function: $f(x) = \sqrt[3]{x}$

• Absolute Value Function: $f(x) = |x|$

Piecewise-Defined and Greatest Integer Functions

A piecewise-defined function is defined by different expressions over different intervals of its domain. The greatest integer function (step function) assigns to each x the greatest integer less than or equal to x.

Graphing Techniques

Transformations of Functions

• Vertical and horizontal shifts: $f(x) + k$ shifts up/down; $f(x - h)$ shifts right/left.

• Stretching/Shrinking: $af(x)$ stretches vertically if $a > 1$, shrinks if $0 < a < 1$.

• Reflections: $-f(x)$ reflects across the x-axis; $f(-x)$ reflects across the y-axis.

Symmetry, Even and Odd Functions

• Even function: $f(-x) = f(x)$ (symmetric about the y-axis)

• Odd function: $f(-x) = -f(x)$ (symmetric about the origin)

Function Operations and Composition

Arithmetic Operations on Functions

• Sum: $(f + g)(x) = f(x) + g(x)$

• Difference: $(f - g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x)g(x)$

• Quotient: $(f/g)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

Composition of Functions

The composition of functions $f$ and $g$ is $(f \circ g)(x) = f(g(x))$. The domain of $f \circ g$ consists of all x in the domain of g such that $g(x)$ is in the domain of f.

Quadratic Functions and Models

Quadratic Functions

A quadratic function has the form $f(x) = ax^2 + bx + c$, $a \neq 0$. Its graph is a parabola.

• Vertex: The vertex is at $\left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right)$.

• Axis of symmetry: $x = -\frac{b}{2a}$

• Opens upward if $a > 0$; downward if $a < 0$.

Completing the Square and Vertex Formula

Completing the square rewrites $ax^2 + bx + c$ in vertex form: $a(x - h)^2 + k$.

Quadratic Models

Quadratic functions are used to model projectile motion and other real-world phenomena where the relationship between variables is parabolic.