Problem 34 - A Graphical Approach to College Algebra

Question

A line having an equation of the form \(y=k x,\) where \(k\) is a real number, \(k \neq 0,\) will always pass through the origin. To graph such an equation by hand, we can determine a second point and then join the origin and that second point with a straight line. Use this method to graph each line. $$y=-2 x$$

Step-by-Step Solution

Answer

Graph the line by plotting points (0,0) and (1,-2) and drawing a line through them.

Step 1: Identify the Equation

The given line equation is \(y = -2x\). Here, \(k = -2\). This line is in the form \(y = kx\), where \(k\) is a real number.

Step 2: Determine the First Point

For any line of the form \(y = kx\), one point is always the origin \((0, 0)\), since when \(x = 0\), \(y = 0\cdot k = 0\). This gives the first point as \((0, 0)\).

Step 3: Choose a Value for x to Find a Second Point

Choose a value for \(x\) to find a second point. Let \(x = 1\). Substitute this into the equation to find \(y\).

Step 4: Calculate y for the Chosen x

Substitute \(x = 1\) into the equation: \(y = -2(1)\). Calculate the result to find the second point. Here, \(y = -2\times 1 = -2\). Hence, the second point is \((1, -2)\).

Step 5: Plot Points on Graph

Plot the first point \((0, 0)\) and the second point \((1, -2)\) on a coordinate plane. The point \((0, 0)\) represents the origin, and \((1, -2)\) is one unit to the right and two units down from the origin.

Step 6: Draw the Line

Use a ruler to draw a straight line through the plotted points \((0, 0)\) and \((1, -2)\). This line represents the equation \(y = -2x\).

Key Concepts

Slope-Intercept Form

The Slope-Intercept Form of a linear equation is one of the most common ways to express the equation of a line on a graph. It is given by the equation \(y = mx + b\). In this format, \(m\) represents the slope of the line, and \(b\) indicates the y-intercept, where the line crosses the y-axis. In our specific exercise, the equation \(y = -2x\) is a simplified version of the slope-intercept form with \(b = 0\), signifying that the line passes through the origin.

• When \(b = 0\), the equation simplifies to \(y = mx\), demonstrating that a direct relationship between \(x\) and \(y\) exists without a constant offset.
• The slope \(m\) represents how steep the line is, and, in this case, \(m = -2\) indicates a downward slope as we move from left to right across the graph.

Understanding the Slope-Intercept Form aids in quickly graphing linear equations and recognizing the basic pattern they follow on a graph.

Coordinate Plane

The Coordinate Plane is a two-dimensional surface on which we can plot points, lines, and curves. It consists of two perpendicular number lines, the horizontal x-axis and the vertical y-axis, which intersect at the origin \(0, 0\).

• The x-axis runs left and right, allowing us to determine horizontal positions.
• The y-axis runs up and down, indicating vertical positions.
• Each point on the coordinate plane is determined by an ordered pair \(x, y\), where \(x\) denotes placement along the x-axis and \(y\) along the y-axis.

In graphing the equation \(y = -2x\), the coordinate plane is our canvas. We begin at the origin and use the slope of the equation to guide us in plotting the necessary points. This system allows us to visualize the relationship defined by linear equations effectively.

Slope of a Line

The Slope of a Line is a measure of its steepness and direction, typically described by the rise over run between two points on the line. It is commonly denoted by \(m\) in the slope-intercept form equation \(y = mx + b\).

• The slope \(m\) is calculated as \(\frac{\text{change in } y}{\text{change in } x}\), or \(\frac{y_2 - y_1}{x_2 - x_1}\).
• A positive slope means the line ascends from left to right, while a negative slope indicates it descends.
• The steepness of the line increases with the absolute value of the slope.

In the example of \(y = -2x\), the slope \(-2\) signifies that for every unit we move horizontally to the right, the line moves 2 units downwards. This negative value clearly shows the line slopes downward on the coordinate plane, providing a visual representation of the mathematical relationship defined by the equation.