Introduction to Graphing and the Coordinate Plane

Section 1.1: Introduction to Graphing

Cartesian Coordinate System

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

• Quadrant I: (+, +)

• Quadrant II: (−, +)

• Quadrant III: (−, −)

• Quadrant IV: (+, −)

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

Plotting Points

To plot a point, locate its x-coordinate on the horizontal axis and its y-coordinate on the vertical axis. The intersection of these values is the location of the point.

• Example: Plot and label the points (−3, 5), (4, 3), (1, 4), (−4, −2), (3, −4), (6, 4), (−2, 8), and (0, 0).

Graphing Equations

To graph an equation, create a drawing that represents all solutions of that equation. This is typically done by plotting points that satisfy the equation and connecting them appropriately.

• Example: Graph 2x + 3y = 18.

x-Intercept and y-Intercept

• x-intercept: The point where the graph crosses the x-axis (set y = 0).

• y-intercept: The point where the graph crosses the y-axis (set x = 0).

Solving for Ordered Pairs

To determine if a point is a solution to an equation, substitute the x and y values into the equation and check if the equation is satisfied.

• Example: Is (−1, 7) a solution to 2x + 3y = 18?

Graphing Linear and Quadratic Equations

• Linear Equations: Often solved for y in terms of x, then values are substituted to find points to plot.

• Quadratic Equations: The graph is a parabola. Example: y = x2 − 8x + 12.

The Distance Formula

The distance formula calculates the distance between two points (x1, y1) and (x2, y2):

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

• Example: Find the distance between (−2, 2) and (4, −4).

The Midpoint Formula

The midpoint of a segment with endpoints (x1, y1) and (x2, y2) is:

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

• Example: Find the midpoint of the segment with endpoints (−4, −2) and (1, 1).

The Equation of a Circle

The standard form of the equation of a circle with center (h, k) and radius r is:

$(x - h)^2 + (y - k)^2 = r^2$

• Example: Find the equation of a circle with center (5, −7) and radius 3.

• Example: Graph the circle (x + 3)2 + (y − 2)2 = 16.

Applications and Additional Examples

• Find the equation for a circle with a diameter whose endpoints are (4, 6) and (−1, −12).

• Find the center of a circle given a point on the circle and a tangent to the y-axis.

Additional info: These notes cover foundational graphing concepts in College Algebra, including coordinate systems, graphing equations, and basic geometric formulas relevant to algebraic graphing.