Graph Of Y 1 3x

Understanding the Graph of y = 1/(3x): A Deep Dive into Rational Functions

At first glance, the equation y = 1/(3x) appears elegantly simple. Yet, beneath its compact notation lies a fascinating and non-intuitive curve that serves as a cornerstone for understanding more complex mathematical relationships. This is not the graph of a straight line, a parabola, or any other elementary function you might first encounter. Instead, it represents a specific type of hyperbola, a curve with two distinct branches that approach, but never touch, invisible boundary lines called asymptotes. Mastering the graph of this function is a critical step in moving from basic algebra to the richer worlds of precalculus and calculus, as it perfectly illustrates the behavior of inverse proportionality and the powerful concept of transformations of parent functions. This article will provide a comprehensive, step-by-step exploration of this graph, ensuring you understand not just how to sketch it, but why it looks the way it does and where such relationships appear in the real world.

Detailed Explanation: Deconstructing the Equation

To understand y = 1/(3x), we must first recognize its lineage. It is a direct transformation of the most fundamental rational function: the parent function y = 1/x. The parent function y = 1/x is the archetypal hyperbola. Its defining characteristics are:

1. Domain: All real numbers except x = 0 (division by zero is undefined).
2. Range: All real numbers except y = 0 (the output can never be zero for any finite x).
3. Asymptotes: It has two asymptotes. The vertical asymptote is the line x = 0 (the y-axis). The horizontal asymptote is the line y = 0 (the x-axis). The curve forever approaches these lines but never intersects them.
4. Symmetry: It is symmetric with respect to the origin (it has rotational symmetry of 180 degrees). If a point (a, b) is on the graph, then (-a, -b) is also on the graph.

Now, let's introduce the transformation in y = 1/(3x). The "3" is a constant multiplier inside the denominator, directly affecting the input variable x. This is a horizontal transformation. Specifically, multiplying x by a factor of 3 before taking the reciprocal results in a horizontal stretch of the parent graph by a factor of 3. To understand this, consider a point on y = 1/x, say (1, 1). For the same y-value of 1 to occur in y = 1/(3x), we solve 1 = 1/(3x), which gives x = 1/3. The point that was at x=1 is now at x=1/3. All x-coordinates are divided by 3, meaning the graph is stretched away from the y-axis. Consequently, the curve appears "flatter" and "wider" than y = 1/x, especially noticeable in the first and third quadrants.

Crucially, this transformation does not change the asymptotes. The vertical asymptote remains x = 0 because the denominator 3x is zero only when x=0. The horizontal asymptote remains y = 0 because as x grows infinitely large (positive or negative), 1/(3x) grows infinitely close to zero. The domain and range also remain identical to the parent function: x ≠ 0 and y ≠ 0. The origin-symmetry is preserved as well.

Step-by-Step Concept Breakdown: Graphing the Function

Graphing y = 1/(3x) systematically ensures accuracy. Follow these logical steps:

Step 1: Identify the Parent Function and Transformation. Acknowledge you are starting with y = 1/x. The equation y = 1/(3x) can be rewritten as y = (1/3) * (1/x). This form reveals it is a vertical compression by 1/3? Careful! While algebraically equivalent, the standard interpretation for y = f(kx) is a horizontal transformation. For y = 1/(3x), the input x is multiplied by 3, so it is a horizontal stretch by a factor of 1/3? Let's clarify the rule: For y = f(kx), if k > 1, it's a horizontal compression by 1/k. If 0 < k < 1, it's a horizontal stretch by 1/k. Here, k=3 (>1), so y = 1/(3x) is a horizontal compression by a factor of 1/3? This is a common point of confusion. The most intuitive method is to use a T-chart based on the inverse relationship.

Step 2: Use an Inverse T-Chart. Because y is inversely proportional to x, it's easier to pick y-values and solve for x.

• Choose simple y-values: