Problem 40 - 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? \(y=2+m(x+3) \quad\) for \(m=0, \pm 0.5, \pm 1, \pm 2, \pm 6\)

Step-by-Step Solution

Answer

All lines intersect at the point \((-3, 2)\).

Step 1: Identify the Equation Form

The given family of equations is in the form of \( y = 2 + m(x+3) \), where \( m \) represents different slopes. This is an equation of a line with points transformed horizontally and vertically.

Step 2: Determine the Effect of Parameters

The variable \( m \) changes the slope of the line. As \( m \) varies across the values 0, \( \pm 0.5, \pm 1, \pm 2, \pm 6 \), the angle at which each line tilts from the horizontal axis changes.

Step 3: Graph Each Line

Using a graphing device, input and plot each line for the given values of \( m \). The lines will be plotted for slopes of 0 (horizontal line), \( \pm 0.5, \pm 1, \pm 2, \pm 6 \) showing different inclinations.

Step 4: Analyze the Graph

Observe the graph to identify common features of all the lines. Notice that each line intersects at the point \((-3, 2)\), which is the y-intercept transformed by the equation.

Step 5: Conclude Shared Characteristics

From the observation, the shared characteristic is that all lines pass through the fixed point \((-3, 2)\), which means they all intersect at this common point regardless of the slope \( m \).

Key Concepts

Slope-Intercept Form

Understanding linear equations in the slope-intercept form is crucial for graphing lines. The slope-intercept form of a linear equation is usually given as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept. In the exercise equation \( y = 2 + m(x+3) \), it can be rewritten to resemble the slope-intercept form: \( y = mx + mx + 2 \). Breaking Down the Components:

• Slope \( (m) \): Determines the steepness or inclination of the line. A positive slope means the line ascends from left to right, while a negative slope descends. If \( m \) is zero, the line is horizontal.
• Y-intercept \( (b) \): The point where the line crosses the y-axis, marking a fixed point in the vertical direction. This exercise shows the intercept at the point transformed to \( (-3, 2) \).

Now that you understand the role of slope and intercept, it's easier to visualize and graph each equation from the family of lines by substituting different values of \( m \).

Family of Lines

A family of lines refers to a set of lines that share particular properties. In this case, all lines vary by different values of slopes \( m \), but they intersect at a shared constant point. Exploring Common Characteristics:

• Common Point of Intersection: Despite differing slopes, all lines in this family intersect at the fixed point \((-3, 2)\). This is a hallmark of a family of lines, showing they have a shared lateral and vertical anchoring.
• Variation with Slope \( (m) \): By changing \( m \), each line demonstrates a distinct inclination or orientation without altering their common intersection, illustrating the unique relationship between slopes while maintaining other parameters.

This exercise helps visualize how lines can evolve or rotate around a pivotal point, revealing the dynamic yet consistent nature of line families.

Graph Analysis

Graph analysis involves interpreting the visual representation of equations to understand relationships and characteristics. Through analyzing the graph of the given equations, we identify key patterns and insights.Key Observations:

• Consistent Intersection: All lines converge at the point \((-3, 2)\), which provides an anchor and visual confirmation that they all belong to the same family, regardless of slope variations.
• Line Inclinations: As \( m \) takes values 0, \( \pm 0.5, \pm 1, \pm 2, \pm 6 \), the lines create varying angles with the horizontal, showcasing horizontal to steep inclines. This demonstrates how varying the slope affects line direction.
• Graphing Tool Usage: Utilizing a graphing device simplifies the visualization of multiple lines and their intersections, making it easier to identify shared points and alignments.

Analyzing graphs allows a deeper comprehension of linear relationships and offers a concrete method to observe abstract mathematical ideas unfold visually.