Y 1 2x 4 On A Graph

How to Graph the Rational Function y = 1/(2x - 4)

Understanding how to graph a rational function like y = 1/(2x - 4) is a fundamental skill in algebra that unlocks a deeper comprehension of how equations translate into visual shapes. This function, while appearing simple, beautifully illustrates core concepts such as asymptotes, intercepts, and the dramatic behavior of curves near undefined points. Mastering its graph provides a template for analyzing a vast family of similar functions. This guide will walk you through every step, from initial analysis to sketching the final curve, ensuring you grasp not just the "how" but the crucial "why" behind each point.

Step 1: Analyzing the Function's Structure

Before drawing a single point, we must understand the function's DNA. The equation y = 1/(2x - 4) is a rational function, meaning it is a ratio of two polynomials. Here, the numerator is the constant 1 (a polynomial of degree 0), and the denominator is the linear polynomial (2x - 4). This structure immediately tells us two critical things:

1. The function is undefined where the denominator equals zero. This is because division by zero is mathematically impossible.
2. The degrees of the numerator and denominator dictate the long-term behavior of the graph, which we identify through horizontal or slant asymptotes.

Let's find where the function is undefined. Set the denominator equal to zero: 2x - 4 = 0 2x = 4 x = 2

This single value, x = 2, is the most important feature of this graph. It is the location of our vertical asymptote.

Step 2: Identifying the Asymptotes

Asymptotes are invisible lines that the graph of the function approaches infinitely closely but never actually touches or crosses. They act as gravitational anchors for the curve.

• Vertical Asymptote: As determined above, the line x = 2 is the vertical asymptote. The graph will exist on either side of this line—to the left (x < 2) and to the right (x > 2)—but will never have a point where x = 2. As x-values get closer and closer to 2 from either side, the y-values will shoot off towards positive or negative infinity.
• Horizontal Asymptote: To find this, we compare the degrees of the numerator and denominator.
  • Degree of numerator (1): 0
  • Degree of denominator (2x - 4): 1 Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the line y = 0, which is the x-axis. This means as x becomes very large positive (→ ∞) or very large negative (→ -∞), the y-value of the function will get closer and closer to zero.

Step 3: Finding Intercepts

Intercepts are the points where the graph crosses the coordinate axes.

• Y-intercept: This occurs where x = 0. Substitute x = 0 into the equation: y = 1/(2(0) - 4) = 1/(-4) = -1/4 or -0.25 So, the graph crosses the y-axis at the point (0, -0.25).
• X-intercept: This occurs where y = 0. Set the equation equal to zero: 0 = 1/(2x - 4) A fraction equals zero only when its numerator is zero. Here, the numerator is the constant 1, which is never zero. Therefore, there is no x-intercept. The graph never crosses the x-axis.

Step 4: Determining Behavior and Plotting Key Points

With asymptotes and intercepts marked, we now need to understand the curve's shape in each region. The vertical asymptote at x = 2 divides the graph into two separate branches: one on the left (Region I: x < 2) and one on the right (Region II: x > 2). We will test points in each region to see if the function yields positive or negative y-values.

Region I: x < 2 (Left of the asymptote) Choose a value less than 2, say x = 1. y = 1/(2(1) - 4) = 1/(2 - 4) = 1/(-2) = -0.5 The point is (1, -0.5). Since the y-value is negative, this entire left branch will be below the x-axis. Choose another point further left, x = 0 (we already have this y-intercept: -0.25). Choose x = -1: y = 1/(2(-1) - 4) = 1/(-2 - 4) = 1/(-6) ≈ -0.167 As x moves left from the asymptote (from x=1.9 to x=1 to x=0 to x=-1), y increases from large negative values towards 0 (the horizontal asymptote). The curve in this region is increasing and lies in Quadrant III.

Region II: x > 2 (Right of the asymptote) Choose a value greater than 2, say x = 3. y = 1/(2(3) - 4) = 1/(6 - 4) = 1/2 = 0.5 The point is (3, 0.5). The y-value is positive, so this entire right branch will be above the x-axis. Choose x = 4: y = 1/(2(4) - 4) = 1/(8 - 4) = 1/4 = 0.25 Choose a value very close to the asymptote from the right, x = 2.1: y = 1/(2(2.1) - 4) = 1/(4.2 - 4) = 1/0.2 = 5 As x approaches 2 from the right (x=2.1, x=2.01, x=2.001), y becomes very large positive (→ +∞). As x moves further right (x=3, x=4, x=10), y decreases towards 0. The curve in this region is decreasing and lies in Quadrant I.

Table of Values for Plotting