Problem 59 - Algebra 2

Question

Graph each set of ordered pairs on a coordinate plane. \(\\{(-1,2),(2,4),(3,-3),(4,-1)\\}\)

Step-by-Step Solution

Answer

Plot points: (-1,2), (2,4), (3,-3), and (4,-1) on the coordinate plane.

Step 1: Understanding the Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points. Each point is defined by an ordered pair \( (x, y) \), where \( x \) is the horizontal coordinate and \( y \) is the vertical coordinate. The origin, where both \( x \) and \( y \) are zero, divides the plane into four quadrants.

Step 2: Plotting the First Point: (-1, 2)

To plot the point \((-1, 2)\) on the coordinate plane, start at the origin (0,0). Move 1 unit to the left on the x-axis to reach -1. Then move 2 units up on the y-axis to reach 2. Place a point at this location.

Step 3: Plotting the Second Point: (2, 4)

For the point \( (2, 4) \), begin at the origin. Move 2 units to the right on the x-axis. From there, move 4 units upwards on the y-axis. Mark this spot with a point.

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

To plot \( (3, -3) \), start at the origin. Move 3 units right along the x-axis, then move 3 units downward on the y-axis. The point should be placed here.

Step 5: Plotting the Fourth Point: (4, -1)

For \( (4, -1) \), from the origin, move 4 units to the right along the x-axis, then move 1 unit downward on the y-axis. Place a point at this location.

Key Concepts

Ordered Pairs

When working with a coordinate plane, everything begins with ordered pairs. An ordered pair is a simple way of identifying a spot on this grid-like surface. It's written as \(x, y\), with 'x' representing the horizontal position and 'y' the vertical. This sequence is crucial because switching the order would point to a completely different location on the plane. Imagine giving directions to a friend: telling them to go three blocks east (x) and two blocks north (y). It’s indispensable to follow the sequence, or your friend will end up elsewhere!
In the exercise, we have several ordered pairs like \((-1,2)\) and \((2,4)\). Each of these directs exactly where to plot points on the plane. Without ordered pairs, mapping out any points on a coordinate grid would be almost impossible.

Plotting Points

Plotting points is akin to a treasure hunt for exact locations on the coordinate plane. You begin at the origin, which is at the center of the plane (0,0). This is your starting point for finding any other spot. From there, follow the x-coordinate, moving left or right on the x-axis just like you would on a street. Next, follow the y-coordinate, moving up or down the y-axis just like climbing a ladder or heading underground.
For example, when plotting \((-1,2)\), start by moving 1 unit left for -1 on the x-axis, then go 2 units up for 2 on the y-axis. Mark a point where you end up. Each point on the plane is plotted this way: moving along the x-axis first and then the y-axis second. It’s like joining dots in exactly defined spaces.

Quadrants

The coordinate plane is divided into four sections known as quadrants. These are numbered using Roman numerals I, II, III, and IV, counterclockwise starting from the upper right corner. Each quadrant has different sign rules for the x and y values in the ordered pairs. It’s a manner of quickly understanding where a particular point lies on the coordinate plane.

• Quadrant I: Both x and y are positive (e.g., \(2,4\))
• Quadrant II: x is negative and y is positive (e.g., \((-1,2)\))
• Quadrant III: Both x and y are negative
• Quadrant IV: x is positive and y is negative (e.g., \(3,-3\))

When plotting our points like \(3,-3\), which is in Quadrant IV, or \((-1,2)\), in Quadrant II, you are essentially learning how to navigate the entire plane just by looking at the signs and values of each component in the ordered pair.

X-axis and Y-axis

Think of the x-axis and y-axis as the main roads in a city grid. The x-axis runs horizontally, dividing the plane into upper and lower halves. Meanwhile, the y-axis runs vertically, splitting the plane into left and right sections. Where these two axes meet is called the origin, \(0,0\), a critical reference point.

• X-axis: Helps determine left-right movements
• Y-axis: Guides up-down movements

Understanding these axes is fundamental; they guide you in plotting and finding points precisely. For instance, for the point \(4,-1\), knowing that 4 units positive takes you to the right on the x-axis and -1 takes you downwards on the y-axis is key. The axes are not just lines; they're the grid system's backbone, offering vital orientation and navigation as you pinpoint locations accurately on the plane.