Problem 40 - Algebra for College Students

Question

How do we know that the graph of \(y=-3 x\) is a straight line that contains the origin?

Step-by-Step Solution

Answer

It is a straight line because it has a constant slope, and it includes the origin since the y-intercept is 0.

Step 1: Understanding the Equation Form

The equation given is in the form of \[y = mx + c\]where \(m\) is the slope, and \(c\) is the y-intercept. The equation \(y = -3x\) is already in this form with \(m = -3\) and \(c = 0\).

Step 2: Analyzing the Slope

The slope, \(m = -3\), indicates that for each unit increase in \(x\), the value of \(y\) decreases by 3 units. This "constant rate of change" confirms that the graph will be a straight line, as a constant slope suggests a linear relationship.

Step 3: Identifying the Y-Intercept

In the equation \(y = -3x\), the y-intercept \(c = 0\) implies that the line crosses the y-axis at the origin (0,0). The value of \(c\) represents where the line meets the y-axis.

Step 4: Drawing a Conclusion

Since the line has a constant slope and passes through the y-intercept at the origin, the graph of \(y = -3x\) is a straight line that contains the origin.

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is an incredibly useful mathematical tool. It is written as \( y = mx + c \), where \( m \) represents the slope of the line, and \( c \) is the y-intercept. Understanding this form helps to quickly identify key characteristics of a line.

In the equation \( y = -3x \), it's already neatly arranged in this form. Here, the slope \( m \) is \(-3\) and the y-intercept \( c \) is \(0\). This means that the equation directly tells us important details without the need for extra manipulation.

When you see an equation in slope-intercept form, think of:

• Slope (\( m \)): It tells you how steep the line is and whether it slopes up or down.
• Y-intercept (\( c \)): This is where the line meets the y-axis.

Knowing this format helps in quickly graphing the line without needing to plot multiple points.

straight line graph

Graphs of linear equations like \( y = -3x \) produce straight lines. This is because the relationship between \( x \) and \( y \) is linear, which means the change between them is consistent throughout.

The slope, or \( m \) in slope-intercept form, is what defines the angle of this line. Our slope of \(-3\) indicates that the line will descend from left to right. For every increase of 1 in \( x \), \( y \) goes down by 3. This uniform change across the graph produces a straight line.

Key insights into a straight line graph include:

• A consistent slope value ensures straightness.
• The slope determines if the line is ascending or descending.

Thus, with a constant rate of change reflected by the slope, you're always guaranteed a straight line in linear equations.

y-intercept

The y-intercept is a fundamental concept in understanding where a line penetrates the y-axis. In the slope-intercept form \( y = mx + c \), \( c \) denotes this point on the graph.

Analyzing our equation \( y = -3x \), we note that \( c = 0 \). This means the line touches the y-axis at the origin, the point (0,0). Since the y-intercept is zero, this line neatly passes through the origin without any vertical offset.

Some points to note about the y-intercept:

• It indicates the starting point of a graph on the y-axis.
• A zero y-intercept suggests the line goes through the origin.

Understanding the y-intercept allows for a straightforward starting point when drawing or analyzing the graph of a linear equation.