Lecture Notes: Rectangular Coordinates, Intercepts, Distance, and Graphing

Quick mindset and course logistics

Math is challenging for many students; it's normal to struggle at first. If you find topics hard, take breaks, ask for help from peers, tutoring centers, or the course resources. Reading the book ahead of time can help you recognize concepts during lecture.
Take-clear notes and consider paraphrasing the instructor’s explanations to fit your understanding. Lectures will be posted after class (often by the next day or before the next class).
The instructor emphasizes that many homework questions mirror quiz/test problems; pay attention to how problems are structured to recognize patterns.

A rectangular (Cartesian) coordinate is an ordered pair (x,y) on the plane where:
- $x$ is the horizontal coordinate (distance left/right from the $x$-axis).
- $y$ is the vertical coordinate (distance up/down from the $y$-axis).
Sign convention:
- Positive $x$ moves to the right; negative $x$ moves to the left.
- Positive $y$ moves up; negative $y$ moves down.
Example points:
- (3,4) means 3 units to the right and 4 units up.
- (-2,-5) means 2 units to the left and 5 units down.
Why called rectangular coordinates: moving along $x$ and $y$ forms the sides of a rectangle with the point as a corner.

The plane is divided into four quadrants (QI to QIV) arranged counterclockwise starting from the top-right:
- Quadrant I: x>0, y>0
- Quadrant II: x
- Quadrant III: x<0, y<0
- Quadrant IV: x>0, y<0
These conventions help with quick mental location of points and signs of coordinates.

Consider a linear equation in two variables, e.g., 3x - 2y = 4.
To find a point on the line, substitute a value for one variable and solve for the other.
- If x=0: 3(0) - 2y = 4 \ -2y = 4 \ y = -2
- Point on the line: (0,-2) .
- If y=4: 3x - 2(4) = 4 \ 3x - 8 = 4 \ 3x = 12 \ x = 4
- Point on the line: (4,4) .
Note: these two points define the line; the entire set of solutions is all points
that satisfy 3x - 2y = 4. In general, a line is the graph of a linear equation; two points suffice to draw it.
When graphing, choose convenient whole-number coordinates when possible to make plotting easier.
Terminology: a point on the graph is an ordered pair (also called a coordinate or border pair).
Quick tip from the instructor: when connecting two plotted points to form a line, you can improve your line by focusing on the target point and drawing toward it without watching your pen.

If you have two points, you can draw the line that passes through both; that line contains all other solutions to the equation.
The line may extend to infinity in both directions (arrows on the ends indicate it continues indefinitely).
In many math platforms (e.g., MyOpenMath), you only need two plotted points to graph the line for homework/quizzes.

Definitions:
- x-intercept: the point where the graph crosses the x-axis (y = 0). The corresponding x-value is the x-intercept.
- y-intercept: the point where the graph crosses the y-axis (x = 0). The corresponding y-value is the y-intercept.
Example: For 7x + 12y = 14
- x-intercept: set y=0 → 7x = 14 \ x = 2 → x-intercept (2,0) .
- y-intercept: set x=0 → 12y = 14 \ y = rac{14}{12} = rac{7}{6} → y-intercept (0, rac{7}{6}) .
Important note: The intercepts are points on the axes; listing them as (rac{7}{6},0) would be the wrong interpretation for this equation.

The distance between two points P1=(x1,y1) and P2=(x2,y2) is given by the distance formula:
- d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}
- Equivalently, d = \sqrt{(x1 - x2)^2 + (y1 - y2)^2}
Rationale: This comes from the Pythagorean theorem. If you connect the points, you form a right triangle with legs |x2 - x1| and |y2 - y1|, and the distance is the hypotenuse.
Pythagorean reminder: a^2 + b^2 = c^2 where c is the hypotenuse.
Example: Find the distance between P1 = (-3,-1) and P2 = (2,3).
- Use differences: x2 - x1 = 2 - (-3) = 5, \, y2 - y1 = 3 - (-1) = 4
- Then d = \sqrt{5^2 + 4^2} = \sqrt{25 + 16} = \sqrt{41}
- Alternative computation using the negative differences (as presented in class): d = \sqrt{((-3) - 2)^2 + ((-1) - 3)^2} = \sqrt{(-5)^2 + (-4)^2} = \sqrt{41}
Note: The distance is symmetric; exchanging the points does not change the result.

Ordered pair / coordinate / border pair are interchangeable terms for a point on the plane.
The line representing a linear equation is called a linear relation; it is a straight line in the plane.
When graphing, using two or more points helps verify the line, but two points suffice to determine the line.

Given the equation 3x - 2y = 4, find two easy points on the line by choosing convenient values for one variable.
For the equation 7x + 12y = 14, determine the x-intercept and the y-intercept, and verify the intercept points on the axes.
Compute the distance between (-3,-1) and (2,3) using the distance formula.

Understanding coordinates helps with graphing data, solving systems of equations, and visualizing relationships between two variables.
The distance formula is foundational for geometry problems, navigation, and any scenario involving spatial relationships.
Intercepts are often used to quickly sketch lines and understand how equations relate to axis crossings.

The instructor will introduce different forms of linear equations: standard form, slope-intercept form, and point-slope form.
We’ll connect distance, slope, and intercepts to build a full toolkit for graphing and solving linear systems.