Lesson 7.4: Coordinate Geometry

Introduction

In this lesson, we will explore the fascinating world of coordinate geometry, specifically focusing on slopes, intercepts, equations of lines, and the graphical representation of functions, equations, and inequalities. Understanding these concepts is crucial for solving a variety of problems on the GRE. By the end of this lesson, you will be able to:

• Calculate slopes, intercepts, and equations of lines.
• Graph functions, equations, and inequalities accurately.
• Apply distance and midpoint formulas in the coordinate plane.
• Match equations to their respective graphs and vice versa.

What is Coordinate Geometry?

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through the use of a coordinate system. It allows us to connect algebra and geometry, making it possible to use algebraic equations to describe curves, lines, and shapes in a plane.

Slopes of Lines

Definition of Slope

The slope of a line is a measure of its steepness, usually denoted by the letter $m$. The slope is calculated as the ratio of the vertical change to the horizontal change between any two points on the line.

Slope Formula

The formula to calculate the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$$

m = $\frac{y_2 - y_1}{x_2 - x_1}$.

$$

Example 1

Consider the points $A(1, 2)$ and $B(4, 6)$. To find the slope of the line passing through these points:

1. Identify the coordinates:

• $x_1 = 1,\; y_1 = 2$
• $x_2 = 4,\; y_2 = 6$

1. Use the slope formula:

$$

m = $\frac{6 - 2}{4 - 1}$ = $\frac{4}{3}$.

$$

The slope of the line between points $A$ and $B$ is $\frac{4}{3}$. This means for every 3 units moved horizontally, the line rises 4 units vertically.

Common Misconceptions

1. Zero Slope: A slope of $0$ means the line is horizontal. The equation of such a line can be expressed as $y = b$, where $b$ is the y-intercept.
2. Undefined Slope: If a line is vertical, the slope is undefined because you cannot divide by zero when $x_2 - x_1 = 0$.

Intercepts

Definition of Intercepts

Intercepts are points where the line crosses the axes. There are two types:

• Y-intercept ($b$): The point where the line crosses the y-axis, occurring when $x = 0$.
• X-intercept: The point where the line crosses the x-axis, occurring when $y = 0$.

Finding Intercepts

To find the y-intercept of a line represented by the equation $y = mx + b$, the y-intercept is simply the value of $b$. To find the x-intercept, set $y = 0$ and solve for $x$ in the equation.

Example 2

Consider the line given by the equation:

$$

$ y = \frac{2}{3}x + 4.$

$$

1. To find the y-intercept:

• Directly from the equation, $b = 4$. Thus, the y-intercept is the point $(0, 4)$.

1. To find the x-intercept, set $y = 0$:

$$

$ 0 = \frac{2}{3}x + 4.$

$$

Solve for $x$:

$$

$\frac{2}{3}$x = -4 \Rightarrow x = -$4 \cdot$ $\frac{3}{2}$ = -6.

$$

Therefore, the x-intercept is $( -6, 0)$.

Equations of Lines

Slope-Intercept Form

A popular form of the equation of a line is the slope-intercept form, given by:

$$

$ y = mx + b.$

$$

Point-Slope Form

Another useful form is the point-slope form:

$$

y - y_1 = m(x - x_1),

$$

where $(x_1, y_1)$ is a known point on the line.

Example 3

Determine the equation of a line with a slope of $-2$ that passes through the point $(3, 5)$ using the point-slope form:

1. Substitute the values into the point-slope equation:

$$

y - 5 = -2(x - 3).

$$

1. Distributing $-2$:

$$

y - 5 = -2x + 6.

$$

1. Solve for $y$:

$$

$ y = -2x + 11.$

$$

The equation of the line is $y = -2x + 11$.

General Form

The general form of the equation of a line is given by:

$$

Ax + By + C = 0,

$$

where $A$, $B$, and $C$ are constants.

Graphing Lines

Steps to Graph a Line

To graph a line given its equation:

1. Start by identifying the intercepts.
2. Plot the intercepts on the coordinate plane.
3. Use the slope to find a second point, starting from one intercept.
4. Draw the line through the two points.

Example 4

Graph the line given by $y = \frac{1}{2}x - 3$:

1. Identify the y-intercept ($b = -3$, plot point $(0, -3)$).
2. The slope is $\frac{1}{2}$, so from $(0, -3)$, move up 1 unit and right 2 units to get the point $(2, -2)$.
3. Plot $(2, -2)$ and draw the line through both points.

Distance and Midpoint

Distance Formula

The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$$

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

$$

Midpoint Formula

The midpoint $M$ between two points is given by:

$$

M = $\left($$\frac{x_1 + x_2}{2}$, $\frac{y_1 + y_2}{2}$

ight).

$$

Example 5

Find the distance and midpoint between the points $(1, 2)$ and $(4, 6)$:

1. Distance:

$$

d = $\sqrt{(4 - 1)^2 + (6 - 2)^2}$ = $\sqrt{3^2 + 4^2}$ = $\sqrt{9 + 16}$ = $\sqrt{25}$ = 5.

$$

1. Midpoint:

$$

M = $\left($$\frac{1 + 4}{2}$, $\frac{2 + 6}{2}$

$ight) = \left(\frac{5}{2}, 4$

$ight) = \left(2.5, 4$

ight).

$$

Conclusion

In this lesson, we have examined the essential components of coordinate geometry, including slopes, intercepts, and the equations of lines. We also explored how to graph lines and understand their relationship in the coordinate plane. The distance and midpoint formulas provided additional tools for analyzing geometric relationships. Mastery of these topics is crucial for success in the quantitative reasoning sections of the GRE.

Study Notes

• Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$
• Y-intercept: The point where $x = 0$ in the line equation.
• X-intercept: The point where $y = 0$ in the line equation.
• Slope-intercept form: $y = mx + b$; Point-slope form: $y - y_1 = m(x - x_1)$.
• Distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Midpoint formula: $M = $\left($$\frac{x_1 + x_2}{2}$, $\frac{y_1 + y_2}{2}

ight).