Problem 37 - 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? \(y=-2 x+b \quad\) for \(b=0, \pm 1, \pm 3, \pm 6\)

Step-by-Step Solution

Answer

The lines are parallel with the same slope of -2, but different y-intercepts.

Step 1: Understanding the family of lines

According to the equation \( y = -2x + b \), the value of \( b \) represents the y-intercept where the line crosses the y-axis. Different values of \( b \) change the position of the line vertically but maintain the slope.

Step 2: Substitute different values of b

Substitute each specified value of \( b \) into the equation, constructing various equations of the form \( y = -2x + b \): \( y = -2x \), \( y = -2x + 1 \), \( y = -2x - 1 \), \( y = -2x + 3 \), \( y = -2x - 3 \), and \( y = -2x + 6 \), \( y = -2x - 6 \).

Step 3: Graph each line using a graphing device

Using a graphing device, such as Desmos or a graphing calculator, plot each line described in Step 2 on the same graph. Ensure that the viewing window is large enough to encompass all y-intercepts, especially since they range from -6 to 6.

Step 4: Analyze the graphed lines

On examining the graph, observe that all the lines are parallel to each other. This is because each line has the same slope of \(-2\), despite having different y-intercepts. Parallel lines never intersect and maintain a constant distance between each other.

Key Concepts

Understanding the Y-intercept

The y-intercept is a fundamental part of linear equations. It shows where a line crosses the y-axis in a graph. This is particularly important in the equation form \( y = mx + b \), where \( b \) stands for the y-intercept.
In our specific exercise, the equation is \( y = -2x + b \). This line will intersect the y-axis at different points depending on the value of \( b \). Let's break it down further:

• If \( b = 0 \), the line crosses the y-axis at the origin (0,0).
• If \( b = 1 \), the line crosses at (0,1).
• For \( b = -1 \), the crossing is at (0,-1).

By changing \( b \) to values like \( \pm 3 \) or \( \pm 6 \), it's evident that altering \( b \) doesn't affect the steepness of the line, only its vertical position. Knowing where a line will cross the y-axis helps in quickly sketching and understanding its overall positioning on the graph.

Grasping the Slope of a Line

The slope of a line is an essential concept because it measures a line's steepness and direction. Denoted by \( m \) in the standard equation \( y = mx + b \), it shows how much \( y \) changes for a unit change in \( x \).
In our case, the slope for the equation \( y = -2x + b \) is \(-2\). This negative value indicates the line slopes downwards from left to right. Here's what that means:

• A slope of \(-2\) means that for every 1 unit increase in \( x \), \( y \) decreases by 2 units.
• The greater the slope in magnitude, the steeper the line.

Understanding the slope helps predict how the line will look without needing to plot each point. This concept is crucial when comparing the angles formed by different lines in a graph.

Exploring the Concept of Parallel Lines

Parallel lines are lines in a plane that never meet. They run side by side at an equal distance apart. In the context of our exercise, whether they have different y-intercepts or not, they share the same slope \(-2\), making them parallel.
What keeps them parallel?

• Same slope: As they all share \( m = -2 \), these lines will always have the same angle relative to the x-axis.
• Constant distance: Since they never intersect, they maintain equal spacing.

By understanding parallel lines, learners comprehend why these lines, despite being drawn from different starting points on the y-axis, will never touch or cross each other. This ensures a visual consistency across a graph.