Problem 43 - 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? $$ y=m(x-3) \quad \text { for } m=0, \pm 0.25, \pm 0.75, \pm 1.5 $$

Step-by-Step Solution

Answer

All lines intersect the x-axis at x = 3.

Step 1: Understand the equation form

The family of lines is given by the equation \(y = m(x-3)\), where \(m\) represents different slopes. The x-term is adjusted by subtracting 3, indicating a horizontal shift of the graph.

Step 2: Identify values for m

We will graph the equation for the following values of \(m\): 0, \(\pm 0.25\), \(\pm 0.75\), \(\pm 1.5\). These represent a horizontal line, gentle slopes, and steep slopes.

Step 3: Graph each line

For each value of \(m\), plot the line on the same graph using a graphing device or software. For example, for \(m = 0.25\), the line would be \(y = 0.25(x-3)\). Repeat for each \(m\) value.

Step 4: Analyze the lines

After plotting all the lines, observe their positions and how they interact on the graph. Notice they all intersect the x-axis at \(x = 3\). This x-intercept is the same for all lines.

Key Concepts

Equation of a Line

The equation of a line provides a way to represent a straight line graphically. A common form used is the slope-intercept form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. However, there are other forms, like the point-slope form used in this exercise: \( y = m(x - 3) \). Here, instead of focusing on the y-intercept, we draw attention to a specific horizontal shift.
In this equation, \( m \) still represents the slope, but the subtraction of 3 from \( x \) indicates a shift of the graph to the right on the x-axis. We can analyze how changing the slope \( m \) will affect the steepness of the line. Understanding the equation form is crucial as it serves as a foundation for graphing and analyzing linear equations effectively.

Slope

The slope of a line, denoted by \( m \) in the equation \( y = m(x - 3) \), is a measure of how steep the line is. It indicates the rate of change of \( y \) with respect to \( x \).
When \( m = 0 \), the line is horizontal, meaning it has no steepness. This was one of the cases explored in the exercise. A horizontal line has consistent \( y \) values across all \( x \) values.

• Positive slopes like 0.25 or 1.5 mean the line rises as it moves from left to right.
• Negative slopes like -0.25 or -1.5 mean the line falls as it moves from left to right.

The steeper the line, the greater the absolute value of the slope. A slope (or "m") that is far from zero indicates a very steep line, whether positive or negative. Observing the changes in slope helps understand how the graph's steepness changes across different values.

X-intercept

In the context of graphing linear equations, the x-intercept is the point where the line crosses the x-axis. It occurs when \( y = 0 \). In the equation \( y = m(x - 3) \), you set \( y \) to 0 and solve for \( x \):
\[0 = m(x - 3)\]Solving this gives the x-intercept:

• If \( m ≠ 0 \), \( x = 3 \).
• If \( m = 0 \), the equation becomes a horizontal line with no defined x-intercept.

Throughout the step-by-step solution, it was noted that regardless of the slope (as long as \( m ≠ 0 \)), the x-intercept remains at \( x = 3 \). This is due to the constant horizontal shift embedded in the equation. The x-intercept provides a consistent point of reference for assessing the position of different lines relative to one another.

Horizontal Shift

Horizontal shift in the equation \( y = m(x - 3) \) indicates that all lines are shifted to the right by 3 units on the x-axis compared to \( y = mx \). In graphs, such a shift will move the position of the line consistently, based on the equation's formula.
This is visible because the expression \((x-3)\) in the equation results in an x-intercept at \( x=3 \), as explained before. A horizontal shift does not affect the steepness or vertical intercept of the line but affects where it crosses the x-axis. For students graphing such lines, knowing that all shifts to the x-values will maintain a common x-intercept is key to understanding the overall graph setup. You can confidently predict the base position even when the slope changes. Shifting helps visualize how different line orientations interact on a common graph plane.