The coordinate plane is defined by two perpendicular number lines that intersect at a point called the origin. These number lines are:
The plane is divided into four sections called quadrants, each identified by the signs of the x and y coordinates:
Plotting a point on the coordinate plane is a simple yet crucial skill. A point is defined by an ordered pair of numbers (x, y), where the first number represents the x-coordinate (horizontal position), and the second number represents the y-coordinate (vertical position). Here’s how to plot a point step-by-step:
Example: Plotting a Point on the Coordinate Plane
To plot a point like (3, 4) on the coordinate plane:
Step 1: Start at the origin (0,0).
Step 2: Move 3 units to the right along the x-axis.
Step 3: Move 4 units up along the y-axis.
Answer: The point (3, 4) is located in the first quadrant of the coordinate plane.
The coordinate plane is divided into four regions, known as quadrants, by the x-axis and y-axis:
In the first quadrant, both x and y coordinates are positive. For example, the point (3, 2) lies in the first quadrant.
In the second quadrant, x coordinates are negative, and y coordinates are positive. For example, the point (-3, 2) lies in the second quadrant.
In the third quadrant, both x and y coordinates are negative. For example, the point (-3, -2) lies in the third quadrant.
In the fourth quadrant, x coordinates are positive, and y coordinates are negative. For example, the point (3, -2) lies in the fourth quadrant.
Example: Identifying the Quadrant of a Point
To identify the quadrant of the point (-5, 6):
Step 1: Observe the signs of the coordinates.
Step 2: The x-coordinate is negative, and the y-coordinate is positive.
Answer: The point (-5, 6) lies in the second quadrant (Quadrant II).
Graphing linear equations on the coordinate plane is a key skill in algebra. A linear equation, such as y = 2x + 3, represents a straight line on the coordinate plane. To graph a linear equation, you need to find at least two points that satisfy the equation, plot them, and draw a line through them.
Example: Graphing y = 2x + 3
To graph the equation y = 2x + 3:
Step 1: Choose two values for x, such as 0 and 2.
Step 2: Calculate the corresponding y values using the equation.
Step 3: For x = 0, y = 2(0) + 3 = 3, so plot the point (0, 3).
Step 4: For x = 2, y = 2(2) + 3 = 7, so plot the point (2, 7).
Answer: Draw a line through the points (0, 3) and (2, 7) to complete the graph of the equation y = 2x + 3.
The slope of a line is a measure of its steepness and direction. It's calculated as the ratio of the change in the y-coordinates to the change in the x-coordinates between two points on the line. The slope formula is given by:
Slope (m) = (y2 - y1) / (x2 - x1)
Example: Calculating the Slope of a Line
To find the slope of the line passing through the points (1, 2) and (3, 6):
Step 1: Use the slope formula: m = (6 - 2) / (3 - 1) = 4 / 2 = 2.
Answer: The slope of the line is 2, indicating that for every unit increase in x, y increases by 2 units.
The x-intercept is the point where the line crosses the x-axis, and the y-intercept is where the line crosses the y-axis. These intercepts are useful for graphing and understanding the behavior of linear equations.
Example: Finding the X-Intercept and Y-Intercept
For the equation y = 3x - 9:
Step 1: To find the y-intercept, set x = 0: y = 3(0) - 9 = -9. The y-intercept is (0, -9).
Step 2: To find the x-intercept, set y = 0: 0 = 3x - 9, so x = 3. The x-intercept is (3, 0).
Answer: The line crosses the y-axis at (0, -9) and the x-axis at (3, 0).
The distance formula allows you to calculate the distance between two points on the coordinate plane. The formula is derived from the Pythagorean theorem and is given by:
Distance = √[(x2 - x1)2 + (y2 - y1)2]
Example: Calculating the Distance Between Two Points
To find the distance between the points (1, 2) and (4, 6):
Step 1: Use the distance formula: Distance = √[(4 - 1)2 + (6 - 2)2].
Step 2: Calculate: Distance = √[9 + 16] = √25 = 5.
Answer: The distance between the points (1, 2) and (4, 6) is 5 units.
The midpoint formula is used to find the exact middle point between two points on the coordinate plane. The formula is:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Example: Finding the Midpoint
To find the midpoint between the points (1, 2) and (5, 6):
Step 1: Use the midpoint formula: Midpoint = ((1 + 5) / 2, (2 + 6) / 2) = (6 / 2, 8 / 2).
Answer: The midpoint is (3, 4).
Understanding the coordinate plane is essential for success in algebra and geometry. By mastering the concepts of plotting points, graphing linear equations, and calculating distances and midpoints, you'll be well-equipped to tackle more advanced mathematical challenges.
