Chapter 1 Overview

Graphing utilities, like handheld calculators or computer software, don't actually draw continuous lines. Instead, they plot thousands of tiny dots called pixels.
Resolution refers to how densely these pixels are packed. A higher resolution (more pixels per inch) results in a smoother-looking curve, whereas low resolution can make a graph look "jagged."

Think of the viewing window as a camera lens. If you are looking at the wrong part of the plane, you won't see your graph.
Xmin and Xmax: These define the left and right boundaries of your screen.
Ymin and Ymax: These define the bottom and top boundaries.
Xscl and Yscl: These are "scales." If $Xscl = 1$ , there is a tick mark every 1 unit. If you are looking at a huge range (e.g., 0 to 1000), you might set $Xscl = 100$ so the axis isn't cluttered with marks.

### 1.3 Graphing Equations by Plotting Points
- An equation like $y = 2x + 1$ is a rule. A point $(x, y)$ is a "solution" if plugging the numbers in makes the rule true.
- To graph by hand, we choose several values for $x$ , calculate the resulting $y$ , and plot those points. Connecting them reveals the shape of the equation.

The Cartesian plane is created by two intersecting number lines: the horizontal x-axis and the vertical y-axis.
They divide the plane into four Quadrants:
- Quadrant I ( $x > 0, y > 0$ ): Top-right (all positive).
- Quadrant II ( $x < 0, y > 0$ ): Top-left.
- Quadrant III ( $x < 0, y < 0$ ): Bottom-left (all negative).
- Quadrant IV ( $x > 0, y < 0$ ): Bottom-right.

### 2.2 Plotting Points
- Every point is an ordered pair $(x, y)$ . The first number tells you how far to move horizontally from the Origin $(0, 0)$ , and the second tells you how far to move vertically.

This formula is actually just the Pythagorean Theorem ( $a^2 + b^2 = c^2$ ) in disguise. The distance $d$ is the hypotenuse of a right triangle formed by two points.
$d(P1, P2) = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$
Because we square the differences, the order of the points doesn't matter; distance is always positive.

### 3.2 Midpoint Formula
- The midpoint $M$ is the "average" location. To find it, you simply average the x-coordinates and average the y-coordinates:
- $M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right)$

Slope measures the steepness and direction of a line, defined as "rise over run."
$m = \frac{y2 - y1}{x2 - x1}$
Interpretations:
- $m > 0$ : The line goes uphill from left to right.
- $m < 0$ : The line goes downhill.
- $m = 0$ : The line is perfectly horizontal ( $y = \text{constant}$ ).
- Undefined: The line is perfectly vertical ( $x = \text{constant}$ ).

Point-Slope Form: $y - y1 = m(x - x1)$ . This is most useful when you know one point and the slope and want to build the equation.
Slope-Intercept Form: $y = mx + b$ . This is the best form for quick graphing, where $b$ is the y-intercept (the point $(0, b)$ ).

### 4.3 Parallel and Perpendicular Lines
- Parallel: Slopes are identical ( $m1 = m2$ ). They will never intersect.
- Perpendicular: Slopes are "negative reciprocals." This means they flip and change sign (e.g., $2$ becomes $-\frac{1}{2}$ ). Mathmatically, $m1 \cdot m2 = -1$ .

A circle is the set of all points at a fixed distance $r$ from a center $(h, k)$ .
$(x - h)^2 + (y - k)^2 = r^2$
If the center is at the origin $(0,0)$ , it simplifies to $x^2 + y^2 = r^2$ .

### 5.2 General Form
- $x^2 + y^2 + ax + by + c = 0$
- You can move from General Form to Standard Form using a technique called completing the square.

Slopes are used in construction (the pitch of a roof) and economics (marginal cost). Circles are used in GPS trilateration and engineering.

### 6.2 Key Summary
- Geometry in the Cartesian plane allows us to turn shapes into algebra. By mastering distance, midpoints, and linear slopes, you can model almost any physical path or boundary.