Introduction to Graphing: Cartesian Coordinates, Intercepts, and Circles

Section 1.1: Introduction to Graphing

Overview

This section introduces fundamental graphing concepts in algebra and trigonometry, including the Cartesian coordinate system, plotting points, solutions of equations, intercepts, and equations of circles. Mastery of these topics is essential for understanding more advanced topics in precalculus.

Cartesian Coordinate System

Definition and Structure

• Cartesian Coordinate System is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis), intersecting at the origin (0, 0).

• The plane is divided into four quadrants:

  • Quadrant I: (+, +)

  • Quadrant II: (−, +)

  • Quadrant III: (−, −)

  • Quadrant IV: (+, −)

• An ordered pair (x, y) represents a point, where x is the horizontal distance from the origin and y is the vertical distance.

Example:

• To plot (−3, 5): Move 3 units left (negative x-direction), then 5 units up (positive y-direction).

Solutions of Equations in Two Variables

Ordered Pairs as Solutions

• An equation in two variables, such as $2x + 3y = 18$, has solutions that are ordered pairs (x, y) making the equation true when substituted.

• To check if (x, y) is a solution, substitute x and y into the equation and verify if the resulting statement is true.

Example:

• Is (−5, 7) a solution to $2x + 3y = 18$? $2(-5) + 3(7) = -10 + 21 = 11 eq 18$ Not a solution.

• Is (3, 4) a solution? $2(3) + 3(4) = 6 + 12 = 18$ Is a solution.

Graphs of Equations

Definition

• To graph an equation is to draw all points (x, y) that satisfy the equation, visually representing its solutions.

x-Intercept and y-Intercept

Definitions

• x-intercept: The point where the graph crosses the x-axis. It has the form (a, 0). To find it, set y = 0 and solve for x.

• y-intercept: The point where the graph crosses the y-axis. It has the form (0, b). To find it, set x = 0 and solve for y.

Example:

• Find the x-intercept of $2x + 3y = 18$: $2x + 3(0) = 18$ $2x = 18$ $x = 9$ x-intercept is (9, 0)

• Find the y-intercept: $2(0) + 3y = 18$ $3y = 18$ $y = 6$ y-intercept is (0, 6)

Finding Additional Solutions

Checking Solutions

• To graph a line, find at least two points (often the intercepts), and a third point as a check.

• For $2x + 3y = 18$, if $x = 3$: $2(3) + 3y = 18$ $6 + 3y = 18$ $3y = 12$ $y = 4$ (3, 4) is a solution.

Distance Formula

Finding Distance Between Two Points

• The distance $d$ between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Example:

• Find the distance between (−2, 2) and (3, −6): $d = \\sqrt{(3 - (-2))^2 + (-6 - 2)^2} = \\sqrt{5^2 + (-8)^2} = \\sqrt{25 + 64} = \\sqrt{89} \approx 9.4$

Midpoint Formula

Finding the Midpoint of a Segment

• The midpoint $M$ of a segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is: $M = \\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right)$

Example:

• Find the midpoint of (4, −2) and (2, 5): $M = \\left( \\frac{4 + 2}{2}, \\frac{-2 + 5}{2} \\right) = (3, 1.5)$

Circles in the Coordinate Plane

Definition and Equation

• A circle is the set of all points $(x, y)$ in a plane that are a fixed distance $r$ (radius) from a center $(h, k)$.

• The standard form of the equation of a circle is: $(x - h)^2 + (y - k)^2 = r^2$

Example:

• Find the equation of a circle with center (3, −7) and radius 5: $(x - 3)^2 + (y + 7)^2 = 25$

Applications

• Given the equation of a circle, you can identify its center and radius by comparing to the standard form.

• To graph a circle, plot the center and use the radius to mark points at equal distance in all directions.

Summary Table: Key Formulas

Concept	Formula	Example
Distance between points	$d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	Between (−2, 2) and (3, −6): $\\sqrt{89} \approx 9.4$
Midpoint of segment	$M = \\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right)$	Between (4, −2) and (2, 5): $(3, 1.5)$
Equation of a circle	$(x - h)^2 + (y - k)^2 = r^2$	Center (3, −7), radius 5: $(x - 3)^2 + (y + 7)^2 = 25$
x-intercept	Set $y = 0$ and solve for $x$	For $2x + 3y = 18$: $x = 9$
y-intercept	Set $x = 0$ and solve for $y$	For $2x + 3y = 18$: $y = 6$

Additional info: The notes cover foundational graphing skills essential for all subsequent topics in precalculus, including functions, lines, and conic sections.