Problem 62
Question
Use a graphing utility to graph the rational function. Determine the domain of the function and identify any asymptotes. $$y=\frac{x^{2}+5 x+8}{x+3}$$
Step-by-Step Solution
Verified Answer
The domain of the function is all real numbers except -3. The function has a vertical asymptote at \(x=-3\) and a horizontal asymptote at \(y=1\).
1Step 1: Identify the Rational Function
The given function is a rational function, \(y=\frac{x^{2}+5 x+8}{x+3}\). The numerator of the function is a quadratic polynomial, \(x^{2}+5 x+8\), and the denominator is a linear polynomial, \(x+3\).
2Step 2: Determine the Domain
The domain of a rational function includes all real numbers except those that make the denominator zero. Set the denominator equal to zero and solve for x to find these restrictions. So, \(x+3=0\) implies \(x=-3\). This means all real numbers except -3 are in the domain of the function.
3Step 3: Identify Asymptotes
A rational function has a vertical asymptote at the x-values that make the denominator zero, which we have already found to be \(x=-3\). For horizontal asymptotes, compare the degree of the numerator and denominator. Here, the degree of both is equal (2), so the horizontal asymptote is the ratio of the leading coefficients, which in this case is \(y=1\).
4Step 4: Graph the Function
Using a graphing utility, plot the function \(y=\frac{x^{2}+5 x+8}{x+3}\), taking note of the asymptotes at \(x=-3\) (vertical) and \(y=1\) (horizontal). The graph confirms all calculations.
Key Concepts
Asymptotes of Rational FunctionsDomain of Rational FunctionsGraphing Utility in Algebra
Asymptotes of Rational Functions
Understanding the behavior of rational functions near their asymptotes is essential to graphing them correctly. An asymptote is a line that the graph of a function approaches but never actually touches. Rational functions tend to have two types of asymptotes: vertical and horizontal (or sometimes oblique).
Vertical asymptotes occur at values of x that cause the denominator of the rational function to be zero. In the given function, \[\begin{equation}y = \frac{x^2 + 5x + 8}{x + 3},\end{equation}\]we set the denominator equal to zero and solve for x, which gives us \[\begin{equation}x = -3.\end{equation}\]Therefore, there's a vertical asymptote at \[\begin{equation}x = -3.\end{equation}\]The graph approaches this line, but will not cross it.
Horizontal asymptotes depend on the degrees of the numerator and denominator polynomials. If the degrees are equal, the horizontal asymptote will be at \[\begin{equation}y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}.\end{equation}\]In our function, both the numerator and denominator are second degree polynomials and the leading coefficient for both is 1, thus the horizontal asymptote is at \[\begin{equation}y = 1.\end{equation}\]
Vertical asymptotes occur at values of x that cause the denominator of the rational function to be zero. In the given function, \[\begin{equation}y = \frac{x^2 + 5x + 8}{x + 3},\end{equation}\]we set the denominator equal to zero and solve for x, which gives us \[\begin{equation}x = -3.\end{equation}\]Therefore, there's a vertical asymptote at \[\begin{equation}x = -3.\end{equation}\]The graph approaches this line, but will not cross it.
Horizontal asymptotes depend on the degrees of the numerator and denominator polynomials. If the degrees are equal, the horizontal asymptote will be at \[\begin{equation}y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}.\end{equation}\]In our function, both the numerator and denominator are second degree polynomials and the leading coefficient for both is 1, thus the horizontal asymptote is at \[\begin{equation}y = 1.\end{equation}\]
- The graph will approach but not cross the horizontal asymptote as x approaches positive or negative infinity.
Domain of Rational Functions
The domain of a function is the set of all possible inputs (x-values) it can accept. For rational functions, the domain generally includes all real numbers except those that would make the denominator zero, since division by zero is undefined.
To find the domain of the given function, \[\begin{equation}y = \frac{x^2 + 5x + 8}{x + 3},\end{equation}\]we locate the x-value where the denominator is zero (\[\begin{equation}x = -3\end{equation}\]) and exclude it from the domain. Consequently, the domain of our function is all real numbers except \[\begin{equation}x = -3.\end{equation}\]For practical purposes, when graphing this function or solving equations that include it, you would exclude any \[\begin{equation}x\end{equation}\]-values that fall outside the function's domain.
To find the domain of the given function, \[\begin{equation}y = \frac{x^2 + 5x + 8}{x + 3},\end{equation}\]we locate the x-value where the denominator is zero (\[\begin{equation}x = -3\end{equation}\]) and exclude it from the domain. Consequently, the domain of our function is all real numbers except \[\begin{equation}x = -3.\end{equation}\]For practical purposes, when graphing this function or solving equations that include it, you would exclude any \[\begin{equation}x\end{equation}\]-values that fall outside the function's domain.
Graphing Utility in Algebra
A graphing utility, whether software, an online tool, or a function-capable calculator, can be a powerful ally in algebra. It assists in visualizing the behavior of functions, especially complex ones like rational functions, which might be challenging to plot accurately by hand.
When using a graphing utility to plot the function \[\begin{equation}y = \frac{x^2 + 5x + 8}{x + 3},\end{equation}\]you should first consider the function's domain and asymptotes to set appropriate viewing windows and expect where the graph will tend to infinity. Inputting our function into the graphing utility, we make sure to recognize the vertical asymptote at \[\begin{equation}x = -3\end{equation}\]and the horizontal asymptote at \[\begin{equation}y = 1.\end{equation}\]The graph lines might get closer and closer to these asymptotes as you plot further along the x-axis. This visual representation can significantly enhance understanding of how rational functions behave, leading to better problem-solving skills and mastery of algebraic concepts.
When using a graphing utility to plot the function \[\begin{equation}y = \frac{x^2 + 5x + 8}{x + 3},\end{equation}\]you should first consider the function's domain and asymptotes to set appropriate viewing windows and expect where the graph will tend to infinity. Inputting our function into the graphing utility, we make sure to recognize the vertical asymptote at \[\begin{equation}x = -3\end{equation}\]and the horizontal asymptote at \[\begin{equation}y = 1.\end{equation}\]The graph lines might get closer and closer to these asymptotes as you plot further along the x-axis. This visual representation can significantly enhance understanding of how rational functions behave, leading to better problem-solving skills and mastery of algebraic concepts.
- Remember to use graphing utilities as a supplement to algebraic understanding, not a replacement.
Other exercises in this chapter
Problem 61
Use a graphing utility to graph the function and approximate (accurate to three decimal places) any real zeros and relative extrema. \(f(x)=x^{5}+3 x^{3}-x+6\)
View solution Problem 62
Divide using long division. $$\left(x^{2}-10 x+15\right) \div(x-3)$$
View solution Problem 62
Use the Rational Zero Test to list all possible rational zeros of \(f .\) Then find the rational zeros. $$f(x)=4 x^{5}-8 x^{4}-5 x^{3}+10 x^{2}+x-2$$
View solution Problem 62
Use the given zero to find all the zeros of the function. Function $$g(x)=4 x^{3}+23 x^{2}+34 x-10$$ Zero $$-3+i$$
View solution