Problem 43 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

All lines intersect the x-axis at \( x = 3 \).

Step 1: Understand the Equation

The given equations are of the form \( y = m(x-3) \). This means each line is a straight line with slope \( m \) and intersecting the x-axis at \( x = 3 \). The family of lines explores various slopes \( m = 0, \pm 0.25, \pm 0.75, \pm 1.5 \).

Step 2: Graph Each Line

Using a graphing device, plot each equation \( y = 0(x-3) \), \( y = 0.25(x-3) \), \( y = -0.25(x-3) \), \( y = 0.75(x-3) \), \( y = -0.75(x-3) \), \( y = 1.5(x-3) \), \( y = -1.5(x-3) \). Each line will pass through the point \( (3,0) \) on the x-axis. Vary the slope according to the value of \( m \).

Step 3: Analyze the Graphs

Observe the plotted lines. Regardless of the slope, each line crosses the x-axis at \( x = 3 \). This is due to the form \( y = m(x-3) \), which ensures all lines are centered on the x-axis at this point.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of writing equations so it's easy to identify both the slope and the intercept of a line. The general form is \( y = mx + b \), where \( m \) represents the slope and \( b \) signifies the y-intercept. The slope \( m \) indicates how steep the line is, showing the rise over run or the change in \( y \) over the change in \( x \). In the equation from our exercise, \( y = m(x - 3) \), we can rewrite this to be closer to slope-intercept form: \( y = mx - 3m \). This form provides insight into how the line changes as \( m \) varies.
In every linear equation, the slope is a crucial component because it defines the angle of tilt of the line from the horizontal. A positive slope means the line ascends from left to right, while a negative slope descends. Utilizing the slope-intercept form can simplify graphing and analyzing linear functions significantly.

Graphing Technology

Graphing technology refers to the use of digital tools like graphing calculators or software to create visual representations of mathematical equations. These tools are incredibly helpful for graphing lines, especially when dealing with multiple equations at once. In our exercise, using graphing technology, you can visualize the family of lines \( y = m(x-3) \) with ease.
By inputting these equations into a graphing tool, you can quickly see how each line is affected by changes in slope \( m \). The technology highlights how the steepness of each line varies while maintaining a common x-axis intercept at \( x = 3 \). Graphing technology enhances understanding by allowing you to zoom in, explore different views, and even animate changes to make learning interactive.

• Graphing calculators and apps can save time, providing instant visual feedback.
• They help in checking calculations and making predictions about line behaviors.
• Facilitating a deeper understanding of algebraic concepts through visual means.

Straight Lines

Straight lines are fundamental in geometry and algebra, representing the simplest form of a linear equation. They have constant slopes and extend infinitely in both directions. In our exercise, all lines are straight, intersecting the x-axis at \( x = 3 \) due to the equation \( y = m(x-3) \).
Each line here is defined by its slope \( m \), and though they share the same intercept at \( x = 3 \), their directions and steepness differ based on the value of \( m \). The point \( (3,0) \) acts as a pivot for these lines. The concept of straight lines is straightforward yet essential as it forms the basis for understanding more complex mathematical graphs.
In essence, understanding the geometry of straight lines lays the groundwork for exploring more advanced topics such as calculus or linear transformations. By examining straight lines in various contexts, especially through exercises like plotting families of lines, students gain valuable insight into how linear behavior is modeled mathematically.