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, start at the origin, move x units along the x-axis, then y units parallel to the y-axis.

• 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.

• Example: Graph 2x + 3y = 18.

• To determine if a point is a solution, substitute its coordinates into the equation and check if the equation is satisfied.

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

x-Intercept and y-Intercept

• An intercept is a point where a graph crosses an axis.

• x-intercept: Set y = 0 and solve for x.

• y-intercept: Set x = 0 and solve for y.

Solving for y and Creating Tables

• For equations like 3x − 2y = −10, solve for y in terms of x, then create a table of values to plot points.

• Example: Graph 3x − 2y = −10 by finding several (x, y) pairs.

Graphing Quadratic Equations

• Quadratic equations like y = x2 − 8x − 12 produce parabolic graphs.

• Plot several points to reveal the shape of the parabola.

The Distance Formula

The distance between any two points (x1, y1) and (x2, y2) is given by:

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

• 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

• Finding the center and radius from the equation of a circle.

• Finding the equation of a circle given endpoints of a diameter.

• Graphing circles tangent to an axis.

Additional info: These notes cover foundational graphing concepts in College Algebra, including the coordinate plane, graphing linear and quadratic equations, and the geometry of circles.