Problem 1 - Calculus with Concepts in Calculus

Question

Draw a set of coordinate axes and plot the following points. a. \((2,1)\) b. \((-1,3)\) c. \((4,0)\) d. \(\left(0,-\frac{3}{2}\right)\) e. \((1,-1)\) f. \((-2,-2)\) g. \((0,0)\) h. \(\left(3,-\frac{1}{3}\right)\)

Step-by-Step Solution

Answer

Plot all points on the coordinate axes following steps 2 to 9.

Step 1: Draw the Coordinate Axes

Start by drawing two perpendicular lines on a sheet of paper or graphing interface, labeling the horizontal line as the x-axis and the vertical line as the y-axis. The point where these two lines intersect is the origin, denoted by (0, 0).

Step 2: Plot the Point (2,1)

Locate 2 on the x-axis and move vertically upwards to 1 on the y-axis. Mark this point, which is the position of (2,1).

Step 3: Plot the Point (-1,3)

Locate -1 on the x-axis and move vertically upwards to 3 on the y-axis. Mark this point, which is the position of (-1,3).

Step 4: Plot the Point (4,0)

Locate 4 on the x-axis. Since the y-coordinate is 0, no movement is needed on the y-axis. Mark this point at (4,0).

Step 5: Plot the Point (0, -3/2)

The x-coordinate is 0, which means the point is directly on the y-axis. Move downwards to \(-\frac{3}{2}\) on the y-axis. Mark this point, (0, -3/2).

Step 6: Plot the Point (1,-1)

Locate 1 on the x-axis and move downwards to -1 on the y-axis. Mark this point, which is the position of (1,-1).

Step 7: Plot the Point (-2,-2)

Locate -2 on the x-axis and move downwards to -2 on the y-axis. Mark this point, which is the position of (-2,-2).

Step 8: Plot the Point (0,0)

This is the origin point, already present at the intersection of the x-axis and y-axis. Mark the point if not already marked.

Step 9: Plot the Point (3, -1/3)

Locate 3 on the x-axis and move slightly downwards to \(-\frac{1}{3}\) on the y-axis. Mark this point, which is the position of (3, -1/3).

Key Concepts

Plotting Points

Plotting points is about finding a specific location on a graph, based on given coordinates. Each point is represented by an ordered pair, such as \((2, 1)\). The first number in the pair is the x-coordinate, which tells you how far to move horizontally. The second number is the y-coordinate, indicating how far to move vertically. The movement always starts from the origin, labeled (0, 0).

• For a positive x-coordinate, move to the right of the origin.
• For a negative x-coordinate, move to the left.
• For a positive y-coordinate, move upwards from the x-axis.
• For a negative y-coordinate, move downwards.

By understanding these steps, you can accurately plot any point on the coordinate plane.

Coordinate Axes

Coordinate axes are essential for plotting points. They include the x-axis and y-axis, which divide the plane into four quadrants. The x-axis is the horizontal line, while the y-axis is vertical. These lines intersect at the origin (0, 0), serving as the reference point for plotting.

• Every point is positioned with respect to the origin.
• The axes are labeled, with positive numbers extending to the right of the x-axis and upwards on the y-axis.
• Negative numbers extend to the left of the x-axis and downwards on the y-axis.

These axes are used to measure distances and determine locations on the graph.

Graphing

Graphing involves drawing the coordinate axes and plotting multiple points on the graph. It's a way to visually represent data and relationships between variables. When you graph, ensure the axes are appropriately labeled and spaced to accurately reflect the points' positions.
The process of graphing is not just about placing points but also understanding how these points relate to each other.

• Graphing helps visualize where points are located relative to one another.
• It allows for analyzing patterns, trends, and potential connections.

By mastering the art of graphing, you can make sense of complex data visually.

Ordered Pairs

Ordered pairs are crucial in coordinate geometry. An ordered pair consists of two numbers, \((x, y)\), representing a specific point on the plane. The order matters: \((x, y)\) is not the same as \((y, x)\).

• The first number, the x-coordinate, shows the point's horizontal position.
• The second number, the y-coordinate, indicates the vertical position.
• Understanding ordered pairs is key to accurately locating and relating points.

By carefully considering each pair in its specific order, you can avoid mistakes and confidently work within the coordinate plane.