Graph The Line With Slope 3/2 And Y-intercept 3.

Graphing the Line with Slope 3/2 and Y-intercept 3: A Comprehensive Guide

Understanding how to graph a line given its slope and y-intercept is a fundamental skill in algebra. This guide will walk you through the process of graphing the line with a slope of 3/2 and a y-intercept of 3, explaining the concepts involved in a clear and accessible way. We’ll delve into the theoretical underpinnings, provide step-by-step instructions, and address frequently asked questions, ensuring a thorough understanding of this essential mathematical concept.

Introduction: Understanding Slope and Y-intercept

Before we begin graphing, let's define the key terms:

• Slope (m): The slope of a line represents its steepness. It's the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. A positive slope indicates an upward trend from left to right, while a negative slope indicates a downward trend. A slope of 0 represents a horizontal line. Undefined slope describes a vertical line. In our case, the slope (m) is 3/2. This means for every 2 units we move to the right (run), the line rises 3 units (rise).

• Y-intercept (b): The y-intercept is the point where the line intersects the y-axis. It represents the value of y when x is 0. In our example, the y-intercept (b) is 3. This means the line passes through the point (0, 3).

The equation of a line can be represented in slope-intercept form: y = mx + b, where 'm' is the slope and 'b' is the y-intercept. For our line, the equation is y = (3/2)x + 3.

Step-by-Step Graphing Process

Now let's graph the line y = (3/2)x + 3 using two primary methods:

Method 1: Using the Slope and Y-intercept

1. Plot the y-intercept: Begin by plotting the y-intercept on the coordinate plane. Since the y-intercept is 3, plot the point (0, 3).

2. Use the slope to find another point: The slope is 3/2. This means the rise is 3 and the run is 2. Starting from the y-intercept (0, 3):

   • Move 2 units to the right (positive x-direction).
   • Move 3 units up (positive y-direction). This brings us to the point (2, 6).

3. Plot the second point: Plot the point (2, 6) on the coordinate plane.

4. Draw the line: Draw a straight line that passes through both points (0, 3) and (2, 6). This line represents the equation y = (3/2)x + 3. Extend the line beyond these two points to show that it continues infinitely in both directions.

Method 2: Creating a Table of Values

This method involves creating a table of x and y values that satisfy the equation y = (3/2)x + 3.

1. Choose x-values: Select several x-values. It's generally useful to choose both positive and negative values, including 0. For simplicity, let's choose x = -2, 0, 2, and 4.

2. Calculate corresponding y-values: Substitute each x-value into the equation y = (3/2)x + 3 to calculate the corresponding y-value:

   • When x = -2: y = (3/2)(-2) + 3 = -3 + 3 = 0. Point: (-2, 0)
   • When x = 0: y = (3/2)(0) + 3 = 3. Point: (0, 3)
   • When x = 2: y = (3/2)(2) + 3 = 3 + 3 = 6. Point: (2, 6)
   • When x = 4: y = (3/2)(4) + 3 = 6 + 3 = 9. Point: (4, 9)

3. Plot the points: Plot the points (-2, 0), (0, 3), (2, 6), and (4, 9) on the coordinate plane.

4. Draw the line: Draw a straight line that passes through all the plotted points. This line, again, represents the equation y = (3/2)x + 3.

Both methods will yield the same line. Choose the method that you find more intuitive and comfortable.

The Mathematical Explanation: Linear Equations

The equation y = (3/2)x + 3 is a linear equation because it represents a straight line. Linear equations are of the form y = mx + b, where:

• y and x are variables representing the coordinates of points on the line.
• m is the slope, representing the rate of change of y with respect to x.
• b is the y-intercept, representing the y-coordinate where the line intersects the y-axis (when x = 0).

The slope of 3/2 indicates that for every unit increase in x, y increases by 3/2 units. This constant rate of change is a defining characteristic of linear relationships. The y-intercept of 3 signifies the starting point of the line on the y-axis.

Understanding the Graph: Visualizing the Line

The graph of y = (3/2)x + 3 is a straight line that slopes upwards from left to right. The steepness of the line is determined by the slope (3/2). A larger slope would result in a steeper line, while a smaller slope would result in a less steep line. The y-intercept (3) indicates where the line crosses the y-axis. The line extends infinitely in both directions, representing all possible x and y values that satisfy the equation.

Extending the Understanding: Different Forms of Linear Equations

While the slope-intercept form (y = mx + b) is convenient for graphing, linear equations can also be expressed in other forms:

• Standard Form: Ax + By = C, where A, B, and C are constants.
• Point-Slope Form: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Converting between these forms can be helpful depending on the given information and the desired outcome.

Frequently Asked Questions (FAQs)

• What if the slope is negative? A negative slope means the line will slant downwards from left to right. The rise would be negative (moving down), while the run remains positive (moving right).

• What if the y-intercept is 0? If the y-intercept is 0, the line passes through the origin (0, 0).

• Can I use only one point to draw a line? No, you need at least two points to define a unique line. A single point could be part of infinitely many lines.

• What if the slope is undefined? An undefined slope represents a vertical line. Vertical lines have the equation x = k, where k is a constant.

• How can I check if a point is on the line? Substitute the x and y coordinates of the point into the equation of the line. If the equation holds true, the point lies on the line.

Conclusion: Mastering Linear Equations

Graphing the line with a slope of 3/2 and a y-intercept of 3 is a straightforward process once you understand the concepts of slope and y-intercept. By mastering these concepts and the various methods for graphing linear equations, you'll build a strong foundation for more advanced mathematical topics. Remember to practice regularly to reinforce your understanding and develop confidence in tackling different types of linear equations. The ability to visualize and interpret linear relationships is crucial for success in algebra and beyond. Through understanding the slope-intercept form and the graphical representation, you unlock a deeper understanding of how these equations describe relationships between variables in the real world.