Problem 32 - College Algebra

Question

For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line). $$(-3,0)(-3,4)(-3,-3)$$

Step-by-Step Solution

Answer

Yes, the points are collinear; they lie on a vertical line with x-coordinate -3.

Step 1: Understand the Problem

We need to plot the three points \((-3, 0), (-3, 4), (-3, -3)\) on a coordinate plane. After plotting, we need to determine if they are collinear, meaning they all lie on the same straight line.

Step 2: Identify the Points

The points given are \((-3, 0)\), \((-3, 4)\), and \((-3, -3)\). Notice that each point has the same x-coordinate, which is \(-3\). This indicates they might be on a vertical line.

Step 3: Plotting the Points

Plot each point on the coordinate plane:- Point A: \((-3, 0)\) is plotted by moving 3 units left on the x-axis from the origin and staying at y-coordinate 0.- Point B: \((-3, 4)\) is plotted by moving 3 units left on the x-axis and 4 units up.- Point C: \((-3, -3)\) is plotted by moving 3 units left on the x-axis and 3 units down.

Step 4: Determine Collinearity

Since all points have the x-coordinate \(-3\), they form a vertical line. Thus, the points \((-3, 0), (-3, 4), (-3, -3)\) are collinear because they lie on the same vertical line.

Key Concepts

coordinate plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It consists of two perpendicular axes: the horizontal axis is called the x-axis, and the vertical axis is the y-axis. The point where these axes intersect is known as the origin, with coordinates \(0,0\).
Understanding the coordinate plane helps us visualize mathematical concepts and solve problems involving geometry. Each point on this plane is defined by a pair of numbers \(x,y\). Here, 'x' represents the horizontal position, while 'y' conveys the vertical position.
- The x-axis divides the plane into two halves: the upper part, where y-values are positive, and the lower part, with negative y-values. - Similarly, the y-axis also splits it into left and right halves. To the left of the y-axis, x-values are negative, and to the right, they are positive. - Every position on this grid is unique, allowing us to represent spatial relationships accurately.

plotting points

Plotting points on a coordinate plane involves placing a marker at a specific location defined by an \(x,y\) coordinate pair. This act is crucial in many geometry problems, helping us visualize patterns and relationships among different points.
Here's a step-by-step on how to plot points:

• Identify the coordinates that you want to plot. Let's consider \((-3,0)\) as an example.
• Start at the origin \(0,0\). Since our x-coordinate is \(-3\), move 3 units left along the x-axis.
• Since our y-coordinate is 0, we don't move up or down from this spot, keeping the point right on the x-axis.
• Place your point at \((-3,0)\) and repeat for each coordinate you have.

Plotting points helps in laying the groundwork for determining relationships, like collinearity, which may emerge from the positions of these points.

vertical line

A vertical line is a straight up-and-down line on the coordinate plane, where all the points on it have the same x-coordinate. As a result, these lines run parallel to the y-axis and can be represented by an equation in the form of \(x = a\), where 'a' is a constant.
In our exercise, all plotted points \((-3,0)\), \((-3,4)\), and \((-3,-3)\) share the same x-coordinate, \(-3\). This means they fall on the same vertical line at x = -3. Here’s why vertical lines are important:

• Vertical lines demonstrate the concept of collinearity. When multiple points lie on a vertical line, they are considered collinear.
• They serve as foundational elements in many geometric shapes and graphs, assisting with more complex graphical interpretations.
• Understanding these lines promotes better comprehension of how shifts in coordinates affect positions on the plane.

One crucial note is that vertical lines do not represent a function, as they violate the basic function rule where each input should have only one output. Thus, while vertical lines help with visual understanding, they also illustrate boundaries within mathematical concepts.