Which of the following rational functions is graphed below 5 – In this exploration, we delve into the realm of rational functions, embarking on a journey to identify the function that corresponds to a given graph lying below the y-axis at a value of 5. Along the way, we will uncover the intricacies of rational functions, unraveling their characteristics, graphing techniques, and practical applications.

As we navigate through this mathematical landscape, we will dissect the key features of rational functions, examining their vertical and horizontal asymptotes, intercepts, and domain and range. By comparing different rational functions, we will gain insights into their similarities and differences, ultimately enabling us to pinpoint the function that aligns with the provided graph.

Rational Functions: Which Of The Following Rational Functions Is Graphed Below 5

Rational functions are functions that can be expressed as the quotient of two polynomials. They have the general form f(x) = p(x)/q(x), where p(x)and q(x)are polynomials and q(x) ≠ 0.

Rational functions can be classified based on the degrees of their numerator and denominator polynomials. A rational function is said to be proper if the degree of its numerator is less than the degree of its denominator, and improper if the degree of its numerator is greater than or equal to the degree of its denominator.

Types of Rational Functions

• Linear Rational Functions:These functions have a numerator of degree 1 and a denominator of degree 1 or greater. Their graphs are straight lines with slopes and y-intercepts determined by the coefficients of the polynomials.
• Quadratic Rational Functions:These functions have a numerator of degree 2 and a denominator of degree 2 or greater. Their graphs are parabolas that may have vertical or horizontal asymptotes.
• Higher Degree Rational Functions:These functions have numerators and denominators of degree greater than 2. Their graphs can exhibit more complex shapes, such as multiple asymptotes, inflection points, and local extrema.

Graphing Rational Functions

To graph a rational function, follow these steps:

1. Find the vertical asymptotes by setting the denominator equal to zero and solving for x.
2. Find the horizontal asymptote by dividing the numerator by the denominator using long division or synthetic division.
3. Find the x- and y-intercepts by setting y = 0and x = 0, respectively.
4. Plot the asymptotes, intercepts, and any additional points to sketch the graph.

Analyzing the Graph, Which of the following rational functions is graphed below 5

Once the graph of a rational function is obtained, its key features can be analyzed:

• Domain:The set of all x-values for which the function is defined.
• Range:The set of all y-values that the function can take on.
• X-intercepts:The points where the graph crosses the x-axis.
• Y-intercept:The point where the graph crosses the y-axis.
• Vertical Asymptotes:Vertical lines that the graph approaches but does not touch.
• Horizontal Asymptote:A horizontal line that the graph approaches as xapproaches infinity or negative infinity.

Comparison of Rational Functions

Comparing the graphs of different rational functions can reveal similarities and differences in their key features.

• Functions with the same vertical asymptotes but different horizontal asymptotes will have different ranges.
• Functions with the same horizontal asymptote but different vertical asymptotes will have different domains.
• Functions with the same degree numerator and denominator will have the same horizontal asymptote.

Applications of Rational Functions

Rational functions are used in various real-world applications, including:

• Modeling growth and decay:Exponential functions can be used to model exponential growth or decay processes, such as population growth or radioactive decay.
• Inverse relationships:Rational functions can be used to model inverse relationships, such as the relationship between the price and demand of a product.
• Physics and engineering:Rational functions are used to solve problems in physics and engineering, such as modeling the trajectory of a projectile or the frequency of a vibrating system.

Commonly Asked Questions

What are the key characteristics of rational functions?

Rational functions are defined as functions that can be expressed as the quotient of two polynomials, where the denominator is not zero. They exhibit vertical asymptotes at the zeros of the denominator and horizontal asymptotes at the ratio of the leading coefficients of the numerator and denominator.

How do you graph rational functions?

To graph rational functions, find the vertical and horizontal asymptotes first. Then, plot the intercepts and use the asymptotes as guides to sketch the graph. The behavior of the graph near the asymptotes can be determined by examining the degrees of the numerator and denominator.

What are some real-world applications of rational functions?

Rational functions have numerous applications in fields such as physics, engineering, and economics. They can be used to model population growth, radioactive decay, and the trajectory of projectiles, among other phenomena.