Problem 34 - A Graphical Approach to Precalculus with Limits

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 \((0,0) .\) 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

The line passes through the points (0,0) and (1,-2), forming a straight line through these points.

Step 1: Identify the Equation

The equation of the line is given as \( y = -2x \). This is in the form \( y = kx \), where \( k = -2 \), which indicates the slope of the line.

Step 2: Determine the Origin Point

Since the equation is in the form \( y = kx \), we know the line passes through the origin. The first point on the graph is \( (0,0) \).

Step 3: Find a Second Point

To graph the line, we need another point besides the origin. Choose a value for \( x \), for example, let \( x = 1 \). Substitute \( x = 1 \) into the equation: \( y = -2(1) = -2 \). This gives us the second point \( (1, -2) \).

Step 4: Draw the Line

Plot the points \( (0,0) \) and \( (1,-2) \) on a coordinate plane. Draw a straight line through these points to graph the line \( y = -2x \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is perhaps the most common way to express a line. It is written as \( y = mx + b \), where:

• \( m \) is the slope of the line
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis

For the equation \( y = -2x \), we note that it fits into a special case of the slope-intercept form where \( b = 0 \). This means the line crosses the y-axis exactly at the origin (0, 0).
Since \( m = -2 \), the slope indicates that for every one unit the line moves to the right (increasing \( x \)), it moves two units down (decreasing \( y \)). This negative slope gives the line a downward tilt from left to right.
Understanding the slope is important as it tells us how steep the line is and in which direction it tilts. This basic yet powerful form of the equation allows us to quickly grasp both the line’s steepness and where it starts if it had an intercept other than zero.

Coordinate Plane

The coordinate plane is where all the graphing magic happens. It is a two-dimensional surface formed by the intersection of two lines:

• The horizontal line, known as the x-axis
• The vertical line, known as the y-axis

These axes divide the plane into four sections called quadrants.
In this environment, each point can be located using an ordered pair \((x, y)\). The equation \( y = -2x \) lives on this plane as a line that moves through certain points, such as the origin \((0,0)\) and another point we found, \((1, -2)\).
Plotting points on the coordinate plane helps visualize how equations translate into pictures, making abstract algebraic concepts more tangible. This visualization aids in comprehending relationships between numbers and better interpreting solutions to linear functions.

Linear Function

A linear function represents a constant rate of change between two variables, typically \( x \) and \( y \). Its graph is always a straight line, hence the name "linear." The general form of a linear function is \( y = kx + b \).
The linear function \( y = -2x \) exemplifies this concept as it depicts a direct, proportional relationship between \( x \) and \( y \) with a fixed rate of change.

• For each unit increase in \( x \), \( y \) decreases by 2 units due to the slope \( k = -2 \).
• Such functions graph as straight lines and capture scenarios where one quantity depends linearly on another.

linear functions are foundational in mathematics for modeling simple relationships, such as speed (distance-time) or cost analysis. Understanding linear functions such as \( y = -2x \) allows students to explore complex real-world problems where variables change at a steady rate.