Problem 58 - Elementary Algebra

Question

How do we know that the graph of \(y=-4 x\) is a straight line that contains the origin?

Step-by-Step Solution

Answer

The graph is a straight line with slope -4, and it contains the origin because the y-intercept is 0.

Step 1: Identify the equation format

The given equation is \(y = -4x\). This is a linear equation in the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Determine the slope and y-intercept

From the equation \(y = -4x\), we identify that the slope \(m = -4\) and the y-intercept \(b = 0\). This means the graph of the equation will be a line with a slope of \(-4\) and it crosses the y-axis at the origin (0,0).

Step 3: Reason why the graph passes through the origin

The y-intercept \(b = 0\) indicates the point where the line crosses the y-axis. Since \(b = 0\), the line passes through the origin (0,0).

Step 4: Verify with a point calculation

To confirm that the line contains the origin, substitute \(x = 0\) into the equation: \(y = -4(0) = 0\). This confirms that the point (0,0) lies on the line, meaning the graph indeed passes through the origin.

Step 5: Visualize the line

A linear equation like \(y = -4x\) always results in a straight line. The equation has no additional powers of \(x\), which confirms that the graph of this equation is a straight line.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to describe a straight line using an easy-to-understand equation. It's written as \(y = mx + b\). Here, \(m\) represents the slope of the line and \(b\) represents the y-intercept. This form helps us quickly recognize key characteristics of a line just by looking at the equation.

• Slope \((m)\): Indicates how steep the line is. A positive slope means the line goes upwards as you move from left to right, while a negative slope, like \(-4\), means the line goes downwards.
• Y-intercept \((b)\): Points to the spot where the line crosses the y-axis. In our exercise, \(b = 0\), indicating the line passes through the origin.

By understanding the slope-intercept form, you can quickly understand how a line will look on a graph. A linear equation like \(y = -4x\) will be easy to graph once you identify \(m\) and \(b\).

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane that satisfy the equation. This process, for the given equation \(y = -4x\), allows us to see the straight line visually. Here's how we can map it out:

• Start with the y-intercept: Since the y-intercept \(b = 0\), the line passes through the origin, (0, 0).
• Use the slope: The slope, \(-4\), shows for each unit increase in \(x\), \(y\) decreases by 4 units. If you move 1 unit to the right on the x-axis, you move down 4 units on the y-axis.
• Plot more points: From the origin, go straight down 4 units while moving right 1 unit, and plot the next point.
• Draw the line: Connect the points you've mapped out, and extend the line to fill the graph.

This methodical approach ensures the graph accurately reflects the linear equation, providing clear visualization of the relationship between \(x\) and \(y\).

Y-Intercept

The y-intercept is an important concept in graphing because it's the point at which a line crosses the y-axis. In slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\). For the equation \(y = -4x\), the y-intercept is 0.

This means the graph crosses the y-axis at the origin, (0, 0). The y-intercept provides a starting point for graphing the line.

• If \(b > 0\), the line would cross the y-axis above the origin.
• If \(b < 0\), the line crosses below the origin.
• When \(b = 0\), as in our example, the line crosses the y-axis exactly at the origin.

Understanding the y-intercept gives a straightforward start to graphing and helps explain how the line aligns with the y-axis. In our equation, the line beautifully illustrates a direct proportionality between \(x\) and \(y\), beginning precisely at \((0, 0)\).