Problem 44 - 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? $$ y=2+m(x+3) \text { for } m=0, \pm 0.5, \pm 1, \pm 2, \pm 6 $$

Step-by-Step Solution

Answer

The lines all pass through the point (-3, 2).

Step 1: Identify the Family of Lines

The given equation is \( y = 2 + m(x+3) \). Here, the variable \( m \) is the slope of the line, and the different values of \( m \) represent different lines. The lines are given for \( m = 0, \pm 0.5, \pm 1, \pm 2, \pm 6 \). We need to graph each line with these slopes.

Step 2: Simplify the Equation

Let's simplify each line's equation by expanding \( y = 2 + m(x + 3) \). \[ y = 2 + mx + 3m \] Which can be rewritten as:\[ y = mx + 3m + 2 \]This gives us the equation of the line in slope-intercept form \( y = mx + b \), where \( b = 3m + 2 \).

Step 3: Graph Each Line

Use a graphing device or software to graph each line.- For \( m = 0 \); \( y = 2 \)- For \( m = \pm 0.5 \); \( y = 0.5x + 3.5 \) and \( y = -0.5x + 0.5 \)- For \( m = \pm 1 \); \( y = x + 5 \) and \( y = -x - 1 \)- For \( m = \pm 2 \); \( y = 2x + 8 \) and \( y = -2x - 4 \)- For \( m = \pm 6 \); \( y = 6x + 20 \) and \( y = -6x - 16 \)

Step 4: Analyze the Graph

By observing the graph, you can see that all the lines pass through the same point. Simplify the original equation for each slope to find this common point. This happens because each equation transforms as:For \( m eq 0 \), solve \( y = mx + 3m + 2 \) and \( x+3=0 \). The solution for \( x \) is \( x=-3 \). Substitute \( x = -3 \) into the unmodified equation:\[ y = 2 + m(-3 + 3) \]\( y = 2 \)Thus, all lines intersect at \((-3, 2)\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a linear equation of the form \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) represents the y-intercept, which is where the line crosses the y-axis. It provides a clear structure to understand the linear relationship between variables. The slope \( m \) tells us how steep the line is and in which direction it moves. When \( m \) is positive, the line slopes upwards as we move from left to right. Conversely, when \( m \) is negative, the line slopes downwards.
The y-intercept \( b \) indicates the starting point of the line on the y-axis when \( x = 0 \).
If you want to graph a line using the slope-intercept form, follow these steps:

• Start by plotting the y-intercept \( b \) on the y-axis.
• Use the slope \( m \) to determine the direction and steepness of the line. For example, if the slope is \( \frac{2}{3} \), move up 2 units and right 3 units from the y-intercept.
• Draw a straight line through these points to complete the graph.

This form is extremely handy for quickly sketching the line and understanding how changes in the equation affect its graph.

Family of Lines

A "family of lines" refers to a set of lines that are described by the same general equation but differ based on a parameter that can vary. In our exercise, the parameter is \( m \), which is the slope in the equation \( y = 2 + m(x + 3) \).
By changing the parameter \( m \), we get different lines within the same family. This means:

• For each value of \( m \), the line appears differently on the graph, each with its unique slope.
• The given values for \( m \) could include positive, negative, and zero slopes, providing a variety of line orientations (flat, rising, or falling).

These lines are related by their common mathematical form but differ in their slopes. Despite the slope changes, certain properties may remain consistent across the entire family. For instance, in our case, no matter the slope, all lines intersect through a common point. Understanding families of lines is integral for visualizing how mathematical parameters affect the behavior and position of lines on a graph.

Intersection Point

The intersection point is where two or more lines on a graph cross each other, or in this case, where all the lines in our family intersect at a common point. This point is significant because it reveals particular solutions where the equations are equal.
For the given exercise, after graphing all the lines using their respective slopes, we found that they all intersect at the point \((-3, 2)\).
This happens because, when the equation is simplified to its slope-intercept form and assessed for an intersection, plugging in \( x = -3 \) yields the same \( y \) value across all lines. This consistent result occurs irrespective of the slope \( m \), indicating a universal intersection point.
To determine an intersection, especially in families of lines, set the \( x \)-values equal and solve for \( y \). The result will offer the common point shared by the lines, demonstrating their relationship beyond merely being parallel or perpendicular. Understanding the intersection provides insights into solutions for equations that would otherwise seem unconnected.