Which Graph Represents Y 1 2x 2

Which Graph Represents y = 1/(2x + 2)? Unlocking the Secrets of Rational Functions

Understanding how to represent mathematical functions graphically is a crucial skill in mathematics and its applications. This article delves into the process of identifying the correct graph for the rational function y = 1/(2x + 2). We will explore the key characteristics of this function, including its domain, range, asymptotes, intercepts, and overall behavior, to ultimately determine which graph accurately portrays it. This exploration will equip you with the knowledge to confidently analyze and visualize similar rational functions.

Understanding Rational Functions

Before we dive into the specifics of y = 1/(2x + 2), let's establish a foundation in rational functions. A rational function is simply a function that can be expressed as the ratio of two polynomials, f(x) = p(x)/q(x), where p(x) and q(x) are polynomials, and q(x) is not the zero polynomial. Rational functions often exhibit unique characteristics, including asymptotes and discontinuities, that distinguish them from simpler polynomial functions.

The function we're analyzing, y = 1/(2x + 2), is a prime example of a rational function. The numerator is a constant polynomial (1), and the denominator is a linear polynomial (2x + 2).

Identifying Key Features: Domain and Range

The domain of a function refers to all possible input values (x-values) for which the function is defined. In the case of rational functions, the function is undefined whenever the denominator is equal to zero. Therefore, to find the domain of y = 1/(2x + 2), we set the denominator equal to zero and solve for x:

2x + 2 = 0 2x = -2 x = -1

This means the function is undefined when x = -1. Therefore, the domain of y = 1/(2x + 2) is all real numbers except x = -1. We can express this using interval notation as (-∞, -1) U (-1, ∞).

The range of a function represents all possible output values (y-values). For this particular rational function, observe that as x approaches -1 from the left, y approaches negative infinity. As x approaches -1 from the right, y approaches positive infinity. Furthermore, y can never equal zero because the numerator is a constant (1). Therefore, the range of y = 1/(2x + 2) is all real numbers except y = 0. In interval notation, this is (-∞, 0) U (0, ∞).

Asymptotes: Vertical and Horizontal

Asymptotes are lines that the graph of a function approaches but never touches. There are two main types of asymptotes we need to consider for rational functions: vertical and horizontal.

• Vertical Asymptote: A vertical asymptote occurs at the x-value(s) where the denominator of the rational function is equal to zero and the numerator is non-zero. In our case, we've already determined that the denominator is zero when x = -1. Since the numerator is 1 (non-zero), there is a vertical asymptote at x = -1. This means the graph will approach, but never touch, the vertical line x = -1.

• Horizontal Asymptote: A horizontal asymptote describes the behavior of the function as x approaches positive or negative infinity. To find the horizontal asymptote, we consider the degrees of the numerator and denominator polynomials. Since the degree of the numerator (0) is less than the degree of the denominator (1), the horizontal asymptote is y = 0. This means the graph approaches the x-axis (y = 0) as x becomes very large or very small.

Intercepts: x-intercept and y-intercept

• x-intercept: The x-intercept is the point where the graph intersects the x-axis (where y = 0). However, since the numerator of our function is always 1, there is no x-intercept. The function will never equal zero.

• y-intercept: The y-intercept is the point where the graph intersects the y-axis (where x = 0). To find the y-intercept, substitute x = 0 into the equation:

y = 1/(2(0) + 2) = 1/2

Therefore, the y-intercept is (0, 1/2).

Graphing the Function: Bringing it all Together

Now that we've identified the key features—domain, range, asymptotes, and intercepts—we can sketch the graph.

1. Draw the asymptotes: Draw a vertical line at x = -1 and a horizontal line at y = 0. These lines represent the asymptotes.

2. Plot the y-intercept: Plot the point (0, 1/2).

3. Consider the behavior around the asymptotes: As x approaches -1 from the left, the function approaches negative infinity. As x approaches -1 from the right, the function approaches positive infinity. As x approaches positive or negative infinity, the function approaches 0.

4. Sketch the curve: Using the information from steps 1-3, sketch the curve of the function. The graph should approach, but not touch, the asymptotes. It should pass through the y-intercept (0, 1/2). The graph will consist of two separate branches, one to the left of the vertical asymptote and one to the right.

Differentiating Between Potential Graphs

When presented with multiple graph options, carefully examine each graph for the presence of the vertical asymptote at x = -1 and the horizontal asymptote at y = 0. Verify that the graph does not intersect the x-axis (no x-intercept) and that it passes through the y-intercept (0, 1/2). Any graph lacking these features is incorrect.

Further Analysis: Symmetry and Transformations

While not strictly necessary for identifying the graph, understanding symmetry and transformations can provide deeper insights. This function exhibits neither even nor odd symmetry. However, understanding transformations (shifts, stretches, and reflections) of simpler rational functions can aid in visualizing more complex ones. For instance, consider the graph of y = 1/x; understanding the transformation to obtain y = 1/(2x + 2) can significantly help in visualizing the correct graph.

Frequently Asked Questions (FAQ)

Q: Can the graph ever touch the asymptotes?

A: No, the graph of a rational function will never touch its asymptotes. Asymptotes represent the limiting behavior of the function as it approaches certain values.

Q: What happens if the degree of the numerator is greater than or equal to the degree of the denominator?

A: If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote; instead, there is an oblique (slant) asymptote or a parabolic asymptote. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

Q: How can I use a graphing calculator or software to verify my results?

A: Graphing calculators and software such as Desmos or GeoGebra can be excellent tools for verifying your hand-drawn sketches and ensuring your understanding of the function's behavior. Input the function y = 1/(2x + 2) into the calculator to visualize the graph.

Conclusion

Identifying the correct graph for y = 1/(2x + 2) requires a systematic approach that involves understanding the key characteristics of rational functions. By determining the domain, range, asymptotes, and intercepts, we can accurately sketch the graph and differentiate it from other potential representations. Remember, the presence of a vertical asymptote at x = -1, a horizontal asymptote at y = 0, the absence of an x-intercept, and the y-intercept at (0, 1/2) are all crucial elements in identifying the correct graphical representation of this specific rational function. This process not only helps you solve this specific problem but also builds a strong foundation for analyzing and understanding a wide range of rational functions in the future.