Problem 10 - Algebra and Trigonometry

Question

(a) Sketch lines through \((0,0)\) with slopes \(1,0, \frac{1}{2}, 2,\) and \(-1 .\) (b) Sketch lines through \((0,0)\) with slopes \(\frac{1}{3}, \frac{1}{2},-\frac{1}{3},\) and 3

Step-by-Step Solution

Answer

Sketch lines with corresponding slopes using \( y = mx \) through the origin.

Step 1: Understanding Slope and Intercept

Each line can be represented by the equation \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. When a line passes through the origin \((0,0)\), the y-intercept \( b = 0 \). Thus, the equation simplifies to \( y = mx \).

Step 2: Drawing the Line with Slope 1

Using the equation \( y = x \), plot the line through the origin. For every 1 unit you move right along the x-axis, move 1 unit up along the y-axis, following the slope of 1.

Step 3: Drawing the Line with Slope 0

Using the equation \( y = 0 \), plot the line through the origin. This is a horizontal line along the x-axis, as the slope is 0.

Step 4: Drawing the Line with Slope \(\frac{1}{2}\)

Using the equation \( y = \frac{1}{2}x \), plot the line through the origin. For every 2 units you move right along the x-axis, move 1 unit up along the y-axis, following the slope of \( \frac{1}{2} \).

Step 5: Drawing the Line with Slope 2

Using the equation \( y = 2x \), plot the line through the origin. For every 1 unit you move right along the x-axis, move 2 units up along the y-axis, following the slope of 2.

Step 6: Drawing the Line with Slope -1

Using the equation \( y = -x \), plot the line through the origin. For every 1 unit you move right along the x-axis, move 1 unit down along the y-axis, following the slope of -1.

Step 7: Drawing the Line with Slope \(\frac{1}{3}\)

Using the equation \( y = \frac{1}{3}x \), plot the line through the origin. For every 3 units you move right along the x-axis, move 1 unit up along the y-axis, following the slope of \( \frac{1}{3} \).

Step 8: Drawing the Line with Slope \(-\frac{1}{3}\)

Using the equation \( y = -\frac{1}{3}x \), plot the line through the origin. For every 3 units you move right along the x-axis, move 1 unit down along the y-axis, following the slope of \(-\frac{1}{3}\).

Step 9: Drawing the Line with Slope \(\frac{1}{2}\) Again

As this slope has been used in both parts, repeat the plotting from Step 4, using the equation \( y = \frac{1}{2}x \).

Step 10: Drawing the Line with Slope 3

Using the equation \( y = 3x \), plot the line through the origin. For every 1 unit you move right along the x-axis, move 3 units up along the y-axis, following the slope of 3.

Key Concepts

Slope of a Line

The concept of a slope in coordinate geometry is crucial for understanding the behavior of lines in a graph. The slope, often represented by the letter \( m \), describes the inclination or steepness of a line. It's calculated as the "rise over run," or the amount a line goes up or down as you move from left to right across the graph. In mathematical terms, the slope is defined as the change in \( y \) over the change in \( x \) (\( \Delta y / \Delta x \)). For instance:

• When a slope is positive, the line ascends from left to right.
• A negative slope means the line descends.
• If the slope is zero, the line is horizontal, indicating no change in height as \( x \) changes.

In our original exercise, various slopes like 1, 0, and -1 indicate the different directions and steepness of lines through the origin \((0, 0)\). Each slope has its unique characteristic shown by the lines' paths and angles relative to the axes.

Linear Equations

Linear equations play an important role in mathematics, especially in graphing. Typically in the form \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept. This type of equation forms a straight line when graphed on a coordinate plane.

In scenarios where a line passes through the origin, such as in our problem, \( b \) equals zero. Thus, the equation simplifies to \( y = mx \). Here:

• \( y = x \) represents a line with a slope of 1, moving one unit up for each unit moved to the right.
• \( y = 2x \) demonstrates a steeper incline, going two units up for each horizontal unit.
• \( y = -x \) shows a decrease as you move right, descending one unit for each unit on the x-axis.

Linear equations help in visualizing relationships between two quantities, allowing us to predict one based on the other.

Graphing Lines

Graphing lines is a fundamental aspect of understanding coordinate geometry. It provides a visual representation of equations and their slopes. To effectively plot a line, you need two main components: a point and the slope.

In the given exercise, all lines start at the origin \((0,0)\) and extend in the direction the slope dictates:

• For positive slopes, such as \( \frac{1}{2} \) or 3, the line will rise as you move to the right.
• Negative slopes like \(-1\) or \(-\frac{1}{3}\) involve the line dropping below the x-axis as it moves rightward.
• The line with a slope of 0 remains flat along the x-axis.

Understanding how to graph lines allows you to see these mathematical relationships visually, making it easier to grasp terms like slope and intercept. It's a powerful tool for solving and interpreting problems in algebra and beyond.