Problem 21 - Algebra 1: Concepts and Skills

Question

Without plotting the point, tell whether it is in Quadrant I, Quadrant II, Quadrant III, or Quadrant IV. $$ (6,17) $$

Step-by-Step Solution

Answer

The point (6,17) is in Quadrant I.

Step 1 Identify the coordinates

The point given in the problem is (6,17). This means the x-coordinate is 6 and the y-coordinate is 17.

Step 2 Understand quadrants and their rules

In the Cartesian coordinate system, the plane is divided into four quadrants. In Quadrant I, both x and y coordinates are positive. In Quadrant II, x is negative while y is positive. In Quadrant III, both x and y are negative. In Quadrant IV, x is positive and y is negative.

Step 3 Determine the quadrant of the given point

Given the point (6,17), since both x-coordinate and y-coordinate are positive, it belongs to Quadrant I according to the rules we have established.

Key Concepts

Cartesian coordinate system

The Cartesian coordinate system is a fundamental framework used for plotting points, lines, and shapes on a plane. It's like a map for mathematics where every location has a unique address defined by numbers. René Descartes, a French mathematician and philosopher, developed this system, which is why it's named after him.

This system consists of two perpendicular lines or axes. The horizontal line is known as the x-axis, and the vertical line is the y-axis. The point where they intersect is called the origin, marked as (0,0). The axes divide the plane into four sections, known as quadrants, which are numbered counterclockwise starting from the upper right as Quadrant I, upper left as Quadrant II, lower left as Quadrant III, and lower right as Quadrant IV.

The coordinates of a point in this system are written as \( (x, y) \), where \( x \) corresponds to the position along the x-axis, and \( y \) corresponds to the position along the y-axis. Positive or negative values of \( x \) and \( y \) reveal not only the location but also the quadrant that the point belongs to.

Plotting points

Plotting points is a skill that allows us to visualize and better understand the relationships between numbers in a spatial context. When you're given a set of coordinates, such as \( (6, 17) \), you can 'plot' this point on the Cartesian coordinate system by starting from the origin.

Firstly, move horizontally from the origin to the right if your x-coordinate is positive, or to the left if it's negative. The number of units you move corresponds to the value of the x-coordinate. In our example, move 6 units to the right because the x-coordinate is 6. Secondly, move vertically from that point, up if the y-coordinate is positive, or down if it's negative. In this case, move 17 units up because the y-coordinate is 17. The spot where you end up is the location of your point on the graph.

Coordinate geometry

Coordinate geometry, also known as analytic geometry, is where algebra meets geometry. This field of mathematics allows us to analyze geometric shapes, such as lines and circles, using algebraic equations.

Using the Cartesian coordinate system, we can describe geometric figures through equations and solve geometric problems by applying algebraic techniques. For example, the equation of a line can be written in the format \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. These equations and others like them give us a powerful tool for exploring the relationships between points and shapes on the plane.

Finding quadrants

Understanding how to find the quadrant a point belongs to is a critical task in coordinate geometry. Each of the four quadrants is characterized by different signs of x and y coordinates. Here's a quick guide to help you identify in which quadrant a point will fall:

• Quadrant I: Both \( x \) and \( y \) are positive.
• Quadrant II: \( x \) is negative, \( y \) is positive.
• Quadrant III: Both \( x \) and \( y \) are negative.
• Quadrant IV: \( x \) is positive, \( y \) is negative.

Considering our example \( (6,17) \), since both the x-coordinate (6) and the y-coordinate (17) are positive, we can conclude without plotting that the point lies in Quadrant I.