Graph Y 2 3x 2

Understanding and Graphing the Linear Equation y = (2/3)x + 2

Introduction

In the vast landscape of algebra, few concepts are as foundational and universally applicable as the ability to graph a linear equation. The equation y = (2/3)x + 2 is a perfect exemplar of this power—a simple, elegant expression that describes a straight line on the Cartesian plane. Mastering its graph provides a gateway to understanding more complex functions and real-world relationships, from calculating costs to modeling physical phenomena. This article will serve as your complete guide, demystifying every component of this equation and walking you through the precise, logical steps to plot it accurately. By the end, you will not only have a clear picture of this specific line but also possess a transferable skill for graphing any equation in slope-intercept form.

Detailed Explanation: Decoding the Slope-Intercept Form

The equation y = (2/3)x + 2 is written in slope-intercept form, which is universally represented as y = mx + b. This format is exceptionally useful because it explicitly reveals two critical characteristics of the line it describes: its slope and its y-intercept.

Let's dissect our equation. The coefficient of x is 2/3. This fractional value is the slope (m) of the line. In geometric terms, slope is the measure of a line's steepness and direction. It is defined as the "rise over run"—the change in the vertical (y) direction divided by the change in the horizontal (x) direction between any two points on the line. A slope of 2/3 means that for every 3 units you move to the right along the x-axis (the "run"), the line rises by 2 units (the "rise"). Because this slope is positive, the line will ascend from left to right.

The constant term, +2, is the y-intercept (b). This is the point where the line crosses the vertical y-axis. At this exact point, the x-coordinate is always zero. Therefore, the y-intercept for our equation is the coordinate point (0, 2). This gives us a guaranteed starting point for our graph—a fact that simplifies the plotting process immensely. The entire equation is a concise instruction: "Start at (0, 2) on the y-axis, and from there, for every 3 steps right, take 2 steps up."

Step-by-Step Breakdown: Plotting the Line with Precision

Graphing this equation is a systematic process that combines the insights from the slope-intercept form with basic plotting skills. Following these steps will ensure accuracy every time.

Step 1: Identify and Plot the Y-Intercept. First, locate the y-intercept b = 2 from the equation. On your coordinate plane, find the point on the y-axis where y = 2. This is the point (0, 2). Place a clear dot or mark at this location. This is your foundational anchor point.

Step 2: Interpret and Apply the Slope. The slope m = 2/3 is a ratio. The numerator (2) represents the rise (vertical change), and the denominator (3) represents the run (horizontal change). From your y-intercept point (0, 2), you will use this ratio to find a second point.

• Since the slope is positive, you will move up (positive rise) and right (positive run).
• From (0, 2), move up 2 units. This brings your imaginary pencil to y = 4.
• From there, move right 3 units. This brings your imaginary pencil to x = 3.
• You have now arrived at the second point: (3, 4). Plot this point clearly on the grid.

Step 3: Draw the Line. Using a ruler, draw a perfectly straight line that passes through both plotted points: (0, 2) and (3, 4). Extend this line in both directions, adding arrowheads to indicate it continues infinitely. This line is the graphical representation of all (x, y) pairs that satisfy the equation y = (2/3)x + 2.

Alternative Method: Using a Table of Values. For those who prefer a more methodical approach or need to verify points, creating a table of values is excellent. Choose several convenient x-values (like -3, 0, 3, 6), substitute each into the equation, and solve for the corresponding y-value.

• If x = -3, y = (2/3)(-3) + 2 = -2 + 2 = 0. Point: (-3, 0).
• If x = 0, y = 2. Point: (0, 2).
• If x = 3, y = 4. Point: (3, 4).
• If x = 6, y = (2/3)(6) + 2 = 4 + 2 = 6. Point: (6, 6). Plotting these points will also yield the same line, providing multiple checks for accuracy.

Real-World Examples: Why This Line Matters

The abstract line on a graph has concrete meaning in countless scenarios. Any situation involving a constant rate of change with a fixed starting value can be modeled by an equation like y = (2/3)x + 2.

• Example 1: A Taxi Fare. Imagine a taxi service that charges a flat base fee of $2.00 (the y-intercept) plus a per-mile rate of $1.33... Wait, that doesn't match. Let's adjust. Suppose the rate is $0.66... per 1/3 mile? To make it clean, consider a service charging $2.00 to start and then $2.00 for every 3 miles traveled. The cost per mile is 2/3 dollars. If x is miles traveled and y is total cost, the equation is y = (2/3)x + 2. After 3 miles (x=3), your cost is (2/3)*3 + 2 = 2 + 2 = $4.00.
• Example 2: Plant Growth. A seedling is