Precalculus Foundations: Coordinate Geometry, Symmetry, and Graphs

Coordinate Geometry Basics

The Cartesian Plane and Quadrants

The Cartesian coordinate plane is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). The intersection point is called the origin $(0,0)$. The plane is divided into four quadrants:

• Quadrant I (Q1): $(+, +)$

• Quadrant II (Q2): $(-, +)$

• Quadrant III (Q3): $(-, -)$

• Quadrant IV (Q4): $(+, -)$

Each point in the plane is represented as an ordered pair $(x, y)$.

Distance and Midpoint Formulas

Distance Between Two Points

The distance between two points $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$ is always positive and can be found using the Pythagorean Theorem:

• Formula:

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

• Example: Find the distance between $(5,6)$ and $(-5,-3)$:

$d = \sqrt{(5 - (-5))^2 + (6 - (-3))^2} = \sqrt{(10)^2 + (9)^2} = \sqrt{100 + 81} = \sqrt{181}$

Midpoint of a Line Segment

The midpoint $M$ of the segment from $P_1 = (x_1, y_1)$ to $P_2 = (x_2, y_2)$ is:

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

• Example: Find the midpoint from $(-5,5)$ to $(3,1)$:

$x = \frac{-5 + 3}{2} = -1, \quad y = \frac{5 + 1}{2} = 3$ $M = (-1, 3)$

Right Triangles in the Coordinate Plane

To determine if three points form a right triangle, use the Pythagorean Theorem:

• Given $A(-2,1)$, $B(3,1)$, $C(2,3)$, calculate the lengths of sides and check if $a^2 + b^2 = c^2$.

• If true, the triangle is a right triangle.

Equations and Graphs

Linear Equations

A linear equation in two variables has the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Example: $y = 2x + 3$

• To graph, pick values for $x$ and solve for $y$.

Quadratic Equations

A quadratic equation has the form $y = ax^2 + bx + c$.

• Example: $y = 2x^2 - 1$

• Graph is a parabola; calculate $y$ for several $x$ values to plot points.

Intercepts

• x-intercepts: Points where $y = 0$.

• y-intercepts: Points where $x = 0$.

• To find intercepts algebraically, set the other variable to zero and solve.

Example: For $y = x^2 + 2x - 3$:

• y-intercept: $x = 0 \implies y = -3$

• x-intercepts: $0 = x^2 + 2x - 3 \implies (x + 3)(x - 1) = 0 \implies x = -3, 1$

Symmetry of Graphs

Types of Symmetry

• y-axis symmetry: If $(x, y)$ is on the graph, so is $(-x, y)$.

• x-axis symmetry: If $(x, y)$ is on the graph, so is $(x, -y)$.

• Origin symmetry: If $(x, y)$ is on the graph, so is $(-x, -y)$.

To test for symmetry:

• Replace $x$ with $-x$ for y-axis symmetry.

• Replace $y$ with $-y$ for x-axis symmetry.

• Replace both $x$ with $-x$ and $y$ with $-y$ for origin symmetry.

• If the equation remains unchanged, the graph has that symmetry.

Example: $y = 2x^2 - 1$ is symmetric about the y-axis, but not the x-axis or origin.

Slope of a Line

Definition and Calculation

The slope $m$ of a line measures its steepness and is calculated as:

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

• Positive slope: Line rises from left to right.

• Negative slope: Line falls from left to right.

• Undefined slope: Vertical line ($x_1 = x_2$).

Example: For points $(-1, -3)$ and $(4, 2)$:

$m = \frac{2 - (-3)}{4 - (-1)} = \frac{5}{5} = 1$

Graphing and Interpreting Equations

Graphing Linear and Quadratic Equations

• To graph, create a table of values for $x$ and calculate corresponding $y$ values.

• Plot the points and connect them smoothly.

• For a complete graph, show all intercepts and relevant points.

Testing Points on a Graph

• If a point $(x, y)$ satisfies the equation, it lies on the graph.

• Substitute $x$ and $y$ into the equation to check.

Summary Table: Symmetry Tests

Type of Symmetry	Test	Condition
y-axis	Replace $x$ with $-x$	Equation unchanged
x-axis	Replace $y$ with $-y$	Equation unchanged
Origin	Replace $x$ with $-x$, $y$ with $-y$	Equation unchanged

Key Formulas

• Distance: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

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

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