Select The Correct Answer.Which Graph Is The Graph Of The Function $r(x)=\frac{x^2+6x+8}{2x+8}$?A. B.

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Understanding Rational Functions

Rational functions are a type of function that can be expressed as the ratio of two polynomials. In other words, they are functions of the form $f(x) = \frac{p(x)}{q(x)}$, where p(x)p(x) and q(x)q(x) are polynomials. Rational functions can be used to model a wide range of real-world phenomena, from the motion of objects to the growth of populations.

Graphing Rational Functions

Graphing rational functions can be a bit more challenging than graphing other types of functions, but with the right tools and techniques, it can be done. Here are the steps to graph a rational function:

  1. Factor the numerator and denominator: If possible, factor the numerator and denominator of the rational function. This can help simplify the function and make it easier to graph.
  2. Identify any restrictions: Identify any restrictions on the domain of the function. These are values of xx that would make the denominator equal to zero.
  3. Graph the function: Use a graphing calculator or software to graph the function. You can also use a table of values to help you graph the function.
  4. Identify any asymptotes: Identify any vertical or horizontal asymptotes of the function. These are lines that the graph approaches but never touches.

**Graphing the Function r(x)=x2+6x+82x+8r(x)=\frac{x^2+6x+8}{2x+8}

To graph the function r(x)=x2+6x+82x+8r(x)=\frac{x^2+6x+8}{2x+8}, we can follow the steps outlined above.

Step 1: Factor the numerator and denominator

The numerator of the function can be factored as follows:

x2+6x+8=(x+4)(x+2)x^2+6x+8 = (x+4)(x+2)

The denominator of the function can be factored as follows:

2x+8=2(x+4)2x+8 = 2(x+4)

So, the function can be rewritten as follows:

r(x)=(x+4)(x+2)2(x+4)r(x) = \frac{(x+4)(x+2)}{2(x+4)}

Step 2: Identify any restrictions

The function is restricted when the denominator is equal to zero. This occurs when x+4=0x+4=0, which means that x=−4x=-4.

Step 3: Graph the function

Using a graphing calculator or software, we can graph the function. The graph of the function is shown below:

Step 4: Identify any asymptotes

The function has a vertical asymptote at x=−4x=-4, since the denominator is equal to zero at this point. The function also has a horizontal asymptote at y=1y=1, since the degree of the numerator is equal to the degree of the denominator.

Conclusion

Graphing rational functions can be a bit more challenging than graphing other types of functions, but with the right tools and techniques, it can be done. By following the steps outlined above, we can graph the function r(x)=x2+6x+82x+8r(x)=\frac{x^2+6x+8}{2x+8} and identify any restrictions, asymptotes, and other important features of the function.

Graph A vs. Graph B

Based on the graph of the function, we can determine which of the two graphs is the correct graph of the function.

Graph A

Graph A is a graph of a rational function with a vertical asymptote at x=−4x=-4 and a horizontal asymptote at y=1y=1. The graph also has a hole at x=−4x=-4, since the numerator and denominator are both equal to zero at this point.

Graph B

Graph B is a graph of a rational function with a vertical asymptote at x=−4x=-4 and a horizontal asymptote at y=1y=1. However, the graph does not have a hole at x=−4x=-4, since the numerator and denominator are not both equal to zero at this point.

Conclusion

Based on the graph of the function, we can conclude that Graph A is the correct graph of the function r(x)=x2+6x+82x+8r(x)=\frac{x^2+6x+8}{2x+8}.

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Understanding Rational Functions

Rational functions are a type of function that can be expressed as the ratio of two polynomials. In other words, they are functions of the form $f(x) = \frac{p(x)}{q(x)}$, where p(x)p(x) and q(x)q(x) are polynomials. Rational functions can be used to model a wide range of real-world phenomena, from the motion of objects to the growth of populations.

Graphing Rational Functions

Graphing rational functions can be a bit more challenging than graphing other types of functions, but with the right tools and techniques, it can be done. Here are some common questions and answers about graphing rational functions:

Q: What is the first step in graphing a rational function?

A: The first step in graphing a rational function is to factor the numerator and denominator, if possible. This can help simplify the function and make it easier to graph.

Q: How do I identify any restrictions on the domain of a rational function?

A: To identify any restrictions on the domain of a rational function, you need to find the values of xx that would make the denominator equal to zero. These values are called the restrictions on the domain.

Q: What is a vertical asymptote?

A: A vertical asymptote is a line that the graph of a rational function approaches but never touches. It occurs when the denominator of the function is equal to zero.

Q: What is a horizontal asymptote?

A: A horizontal asymptote is a line that the graph of a rational function approaches as xx approaches infinity or negative infinity. It occurs when the degree of the numerator is equal to the degree of the denominator.

Q: How do I graph a rational function?

A: To graph a rational function, you can use a graphing calculator or software. You can also use a table of values to help you graph the function.

Q: What is a hole in a graph?

A: A hole in a graph is a point where the graph is not defined. It occurs when the numerator and denominator of the function are both equal to zero at the same point.

Common Graphing Mistakes

Here are some common mistakes to avoid when graphing rational functions:

Mistake 1: Not factoring the numerator and denominator

A: Failing to factor the numerator and denominator can make it difficult to graph the function.

Mistake 2: Not identifying any restrictions on the domain

A: Failing to identify any restrictions on the domain can result in a graph that is not accurate.

Mistake 3: Not identifying any asymptotes

A: Failing to identify any asymptotes can result in a graph that is not accurate.

Mistake 4: Not using a graphing calculator or software

A: Failing to use a graphing calculator or software can make it difficult to graph the function accurately.

Conclusion

Graphing rational functions can be a bit more challenging than graphing other types of functions, but with the right tools and techniques, it can be done. By following the steps outlined above and avoiding common mistakes, you can graph rational functions accurately and effectively.

Additional Resources

Here are some additional resources to help you learn more about graphing rational functions:

Online Graphing Tools

A: There are many online graphing tools available, including Desmos, Graphing Calculator, and Mathway.

Graphing Software

A: There are many graphing software programs available, including Mathematica, Maple, and MATLAB.

Textbooks and Online Courses

A: There are many textbooks and online courses available that cover graphing rational functions, including Khan Academy, Coursera, and edX.

Conclusion

Graphing rational functions is an important skill to have in mathematics and science. By following the steps outlined above and using the resources available, you can graph rational functions accurately and effectively.