Problem 37 - Introductory and Intermediate Algebra for College Students 4th

Question

On the same set of axes, draw lines passing through the origin with slopes \(-1,-\frac{1}{2}, 0, \frac{1}{3},\) and 2.

Step-by-Step Solution

Answer

The lines are drawn through the origin with the respective slopes of -1, -1/2, 0, 1/3, and 2.

Step 1: Drawing a Line with Slope of -1

Start by drawing a line through the origin that makes an angle of 135 degrees with the positive x-axis. This line will have a slope of -1. This is because the tangent of 135 degrees is -1.

Step 2: Drawing a Line with Slope of -1/2

Next, draw another line through the origin making an angle such that the tangent of the angle is -1/2. This line's slope is -1/2.

Step 3: Drawing a Line with Slope of 0

The line with a slope of 0 is perfectly horizontal. Draw this line through the origin.

Step 4: Drawing a Line with Slope of 1/3

Draw another line through the origin making an angle such that the tangent of the angle is 1/3. This line's slope is 1/3.

Step 5: Drawing a Line with Slope of 2

Lastly, draw a line through the origin making an angle such that the tangent of the angle is 2. This line's slope is 2.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in mathematics that describes the line's steepness and direction. It is an essential part of understanding linear equations and graphing them. Slope can be thought of as "rise over run," meaning it is the ratio between the vertical change (rise) and the horizontal change (run) between two points on a line.

For instance, if you move from one point to another along a line, the slope tells you how far you need to move up or down (rise) for each step you take to the right (run). In mathematical terms, the slope \( m \) can be calculated as:

\[ m = \frac{\Delta y}{\Delta x} \]

where \( \Delta y \) is the change in the y-coordinates, and \( \Delta x \) is the change in the x-coordinates. Here, positive slope indicates that the line rises as it moves from left to right, while a negative slope means it falls.

• A slope of 0 results in a horizontal line.
• An undefined slope corresponds to a vertical line.
• Slope values like -1 or 2 indicate the steepness and direction of the line.

Understanding the slope is crucial in graphing linear equations as it visually represents how the line behaves across the coordinate plane.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe the positions and relationships of points, lines, and shapes on the coordinate plane. The coordinate plane consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin \((0, 0)\).

Using this system, every point in the plane can be described by an ordered pair \((x, y)\). These coordinates allow us to easily calculate distance, midpoints, and the slope of a line that passes through two points.

• The origin is the central point where the x-axis and y-axis intersect.
• Points on the plane can be positive or negative, indicating their position relative to the axes.
• Coordinate geometry is used to accurately graph lines and curves.

Applying this knowledge, lines passing through the origin with given slopes can be graphed accurately. The slope will determine the angle of the line as it crosses the axes, providing an intuitive understanding of linear equations.

Equation of a Line

The equation of a line is a mathematical description that represents the line on a coordinate plane. There are several ways to express this equation, but the most common is the slope-intercept form:

\[ y = mx + b \]

In this equation, \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis. When the line passes through the origin, the y-intercept \( b \) is zero, simplifying the equation to \( y = mx \).

• The slope-intercept form makes it easy to quickly identify both the slope and the y-intercept of a line.
• Understanding the equation of a line helps in predicting the line's behavior and how it interacts with the coordinate plane.
• Different forms like point-slope or standard form are also used, depending on what information is available.

Mastering the equation of a line enables students to graph lines quickly and understand the relationship between algebraic expressions and geometric figures.