Introduction to Graphing: Points, Equations, and Circles

Section 1.1 Introduction to Graphing

Objectives

• Plot points in the Cartesian plane.

• Determine whether an ordered pair is a solution of an equation.

• Find the x- and y-intercepts of equations of the form $Ax + By = C$.

• Graph equations and interpret their solutions.

• Calculate the distance between two points and find the midpoint of a segment.

• Write and graph the equation of a circle given its center and radius.

Cartesian Coordinate System

Definition and Quadrants

The Cartesian coordinate system is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis) that intersect at 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

How to Plot a Point

• The first coordinate (x) tells you how far to move left (if negative) or right (if positive) from the origin.

• The second coordinate (y) tells you how far to move up (if positive) or down (if negative).

Example: To plot (−3, 5), move 3 units left and 5 units up from the origin.

Solutions of Equations in Two Variables

Definition

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

• Example: Is (−5, 7) a solution to $2x + 3y = 18$?

  • $2(−5) + 3(7) = −10 + 21 = 11 \neq 18$ (False)

• Example: Is (3, 4) a solution?

  • $2(3) + 3(4) = 6 + 12 = 18$ (True)

Graphs of Equations

Definition

To graph an equation is to draw all points $(x, y)$ that satisfy the equation. The graph visually represents the set of all solutions.

Intercepts

x-Intercept

• The x-intercept is where the graph crosses the x-axis: $(a, 0)$.

• To find it, set $y = 0$ and solve for $x$.

• Example: For $2x + 3y = 18$, set $y = 0$:

  • $2x + 3(0) = 18 \implies 2x = 18 \implies x = 9$

  • x-intercept: (9, 0)

y-Intercept

• The y-intercept is where the graph crosses the y-axis: $(0, b)$.

• To find it, set $x = 0$ and solve for $y$.

• Example: For $2x + 3y = 18$, set $x = 0$:

  • $2(0) + 3y = 18 \implies 3y = 18 \implies y = 6$

  • y-intercept: (0, 6)

Graphing Linear Equations

Procedure

• Find the x- and y-intercepts.

• Find a third point as a check.

• Plot the points and draw a straight line through them.

x	y	(x, y)
0	6	(0, 6)
9	0	(9, 0)
3	4	(3, 4)

Example: The graph of $2x + 3y = 18$ passes through (0, 6), (9, 0), and (3, 4).

Graphing Quadratic Equations

Example: $y = x^2 - 9x - 12$

• Select values for $x$ and compute corresponding $y$ values.

• Plot the points and connect them to form a parabola.

x	y	(x, y)
-3	24	(-3, 24)
-2	14	(-2, 14)
0	-12	(0, -12)
2	-32	(2, -32)
4	-36	(4, -36)
8	-20	(8, -20)
10	8	(10, 8)
12	24	(12, 24)

The Distance Formula

Definition

The distance $d$ between two 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

Definition

The midpoint of a segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Example: Find the midpoint of (−4, −2) and (2, 5):

• $\left( \frac{−4 + 2}{2}, \frac{−2 + 5}{2} \right) = (−1, \frac{3}{2})$

Circles

Definition

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

• Using the distance formula, the equation for a circle is:

  • $r = \sqrt{(x - h)^2 + (y - k)^2}$

• Squaring both sides gives the standard form:

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

Example: Equation of a Circle

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

• Using the standard form:

• $(x - 3)^2 + (y + 7)^2 = 25$