Problem 10 - Precalculus with Limits

Question

In Exercises 7-10, plot the points in the Cartesian plane. \( (1, -\frac{1}{3}) \), \( (\frac{3}{4}, 3) \), \( (-3, 4) \), \( (-\frac{4}{3}, -\frac{3}{2}) \)

Step-by-Step Solution

Answer

The points \( (1, -1/3) \), \( (3/4, 3) \), \( (-3, 4) \) and \( (-4/3, -3/2) \) have been plotted on the Cartesian plane by following their respective x and y coordinates.

Step 1: Understanding Coordinate System

Before starting, you must understand the Cartesian coordinate system, shaped like a cross and divided into 4 quadrants. The intersection point of the horizontal and vertical lines is called the origin, denoted by (0,0). The x-coordinate moves along horizontally, starting from the origin while the y-coordinate moves along vertically. Now, let's begin plotting.

Step 2: Plotting the Point (1, -1/3)

In this pair, the x-coordinate is 1 and the y-coordinate is -1/3. Start from the origin, move 1 unit to the right (as per the positive x-coordinate), then drop down by 1/3 of a unit (as per the negative y-coordinate). This is where the first point is located.

Step 3: Plotting the Point (3/4, 3)

For the second point, move 3/4 units to the right from the origin (as per the positive x-coordinate), then go up by 3 units (as per the positive y-coordinate) to plot the point.

Step 4: Plotting the Point (-3, 4)

In this pair, move 3 units to the left from the origin (as per the negative x-coordinate), then go up by 4 units (as per the positive y-coordinate). This is where this point lies.

Step 5: Plotting the Point (-4/3, -3/2)

For the final point, move 4/3 units to the left from the origin (as per the negative x-coordinate), then drop down by 3/2 unit (as per the negative y-coordinate). This is the location of the last point.

Key Concepts

Plotting Points

Plotting points is like connecting the dots but on a grid. You use two numbers, an x-coordinate and a y-coordinate, to find a point's exact spot on the Cartesian plane. The x-coordinate tells you how far to move right or left from the origin. If it's positive, you move to the right, and if it's negative, you go to the left. The y-coordinate shows how far to move up or down. With positive values, you climb upwards, and negative ones mean you step down.

• Start at the origin, where x is 0 and y is 0.
• Use the x-coordinate to move right or left.
• Then, from that point, use the y-coordinate to move up or down to find your spot.

Remember, careful measurement and plotting lead to accurate results. Practice these steps to become skilled at plotting points.

Quadrants

In the Cartesian coordinate system, the plane is divided into four sections called quadrants. Each quadrant is like a slice of a pizza. They help to identify the direction of a point based on its coordinates.

• Quadrant I: Both x and y are positive. You're moving right and up.
• Quadrant II: x is negative, y is positive. You move left but still go up.
• Quadrant III: Both x and y are negative. You move left and down.
• Quadrant IV: x is positive, y is negative. You go right but then down.

These quadrants are key when figuring out where points sit on the plane. For instance, the point \(1, -\frac{1}{3}\) is in Quadrant IV because x is positive and y is negative.

Origin

The origin is like the home base in the Cartesian coordinate system. It is the starting point for all plotting, marked by \(0,0\). This point is where the x-axis and y-axis intersect, dividing the plane into its four quadrants.

• It serves as the zero point for both axes.
• From here, you can move vertically along the y-axis or horizontally along the x-axis.
• It helps in determining the direction and location of other points.

Understanding the function of the origin is essential for grasping how the coordinate plane works. It is the point from which every other point's location is measured.