Problem 38 - Algebra and Trigonometry

Question

Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? \(y=m x-3\) for \(m=0, \pm 0.25, \pm 0.75, \pm 1.5\)

Step-by-Step Solution

Answer

All lines have the same y-intercept at \(-3\).

Step 1: Understand the Problem

We are given a family of linear equations of the form \( y = mx - 3 \) where \( m \) varies over the values \( 0, \pm 0.25, \pm 0.75, \pm 1.5 \). We need to graph these lines and identify what they have in common.

Step 2: Identify Slopes and Y-Intercept

Each line has a different slope given by \( m \). The common aspect of all these equations is the y-intercept, which is \(-3\) for each line.

Step 3: Graph Each Line

Using a graphing device, input each of the linear equations: \( y = 0x - 3 \), \( y = 0.25x - 3 \), \( y = -0.25x - 3 \), \( y = 0.75x - 3 \), \( y = -0.75x - 3 \), \( y = 1.5x - 3 \), \( y = -1.5x - 3 \). All lines should be graphed in the same coordinate plane.

Step 4: Analyze the Graph

Upon examining the graph, all lines intersect the y-axis at the same point, \(-3\). This shows they all share the same y-intercept.

Step 5: Conclusion

The lines are parallel to one another as their slopes differ. However, they all intersect the y-axis at the same point, indicating that their y-intercept is a common characteristic.

Key Concepts

Slope

In the world of graphing linear equations, the concept of slope is crucial. The slope, denoted as \( m \) in the equation \( y = mx + b \), indicates the steepness and direction of a line on a graph. It tells us how much the value of \( y \) will change for a unit increase in \( x \). For instance:

• A positive slope like \( 0.25 \) or \( 1.5 \) means the line rises as it moves from left to right.


• A negative slope like \( -0.25 \) or \( -1.5 \) indicates the line falls as it goes from left to right.


• A zero slope (\( m = 0 \)) results in a horizontal line.

By understanding slopes, you can easily determine the angle at which a line is tilted, characterized by how steep it is and in which direction it inclines on a graph. In our exercise, highlighting various slopes underscores the diversity of lines even within a single family of equations.

Y-Intercept

The y-intercept is another fundamental aspect of linear equations. It's the point where the line crosses the y-axis, denoted by \( b \) in the formula \( y = mx + b \). For each equation in the given family of lines, the y-intercept is \(-3\). This common y-intercept means that no matter the slope, all lines pass through the point \((0, -3)\).

• Think of the y-intercept as the 'starting point' of a line on the y-axis, when \( x = 0 \).


• In real-life scenarios, the y-intercept might represent a starting fee or a fixed initial charge.

Understanding y-intercepts helps in predicting the positioning of lines on a graph and also assists in comparing different linear equations within a family.

Family of Lines

A family of lines consists of multiple linear equations that share a common characteristic. In this exercise, although each line has a different slope, they all belong to a single family due to their shared y-intercept of \(-3\). Such a set of lines exemplifies the concept of a family of lines, showing how variations in an equation can still result in related graphs.

• The lines have different slopes, meaning they are not parallel, yet they all start from the same point on the y-axis.


• Graphing them on the same plane can help visualize how changing the slope impacts the line's direction and steepness.


• In practical applications, understanding a family of lines can assist in modeling situations where a constant factor, like a fixed expense, is shared.

Recognizing a family of lines enhances your ability to see the big picture in graphing, illustrating both the differences and similarities among various linear equations.