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Chapter 3: Problem 69

Create a function whose graph has the given characteristics. Vertical asymptote: \(x=5\) Slant asymptote: \(y=3 x+2\)

Short Answer

Expert verified
A function whose graph meets the characteristics is \(f(x) = \frac{(3x^2 +10x - 5)}{(x-5)}\).

Step by step solution

01

Identifying Vertical Asymptote

A function will have a vertical asymptote at \(x=a\) when the denominator of the function equals zero at \(x=a\). Therefore, a possible denominator of the function could be \(x-5\) giving a vertical asymptote at \(x=5\).
02

Establishing Slant Asymptote

A rational function will have a slant asymptote \(y = mx + b\) when the degree of the numerator is one more than the degree of the denominator. Here, we're given the slant asymptote \(y=3x+2\). A possible numerator for the function that could give this slant asymptote is \(3x^2 +10x - 5\). When divided by the denominator \(x-5\), it would result in the slant asymptote \(y=3x+2\).
03

Creation of Function

With the considerations from Step 1 and Step 2, a potential function to meet all the characteristics is \(f(x) = \frac{(3x^2 +10x - 5)}{(x-5)}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Asymptotes
Vertical asymptotes occur in a rational function when the denominator becomes zero, making the function undefined at that specific point. In simple terms, they indicate values of x where the function goes to infinity or negative infinity. For example, if we're told that there is a vertical asymptote at \(x=5\), the denominator of the function must be zero when \(x=5\). This is because the denominator causes the function to spike up indefinitely or drop off to negative infinity at this point.

Here's how it works:
  • Identify the given vertical asymptote, which is a line \(x=a\).
  • Set the denominator of your rational function equal to zero using this \(x=a\) value. Thus, for \(x=5\), we have \(x-5=0\).
  • This means that \(x-5\) is a factor of the denominator, ensuring a vertical asymptote occurs at \(x=5\).
This method allows us to easily structure the denominator of the function so we can predict and control where vertical asymptotes appear.
Slant Asymptotes
Slant (or oblique) asymptotes typically occur in rational functions when the degree of the numerator is exactly one more than the degree of the denominator. Instead of aligning horizontally or vertically, slant asymptotes are diagonal lines, described by equations of the form \(y = mx + b\).

To have a slant asymptote, follow these steps:
  • Since we need the degree of the numerator to be one higher than the degree of the denominator, choose a polynomial degree accordingly (e.g., a quadratic numerator and a linear denominator).
  • For the given slant asymptote \(y=3x+2\), perform polynomial long division on the function's terms.
  • The quotient obtained from dividing the numerator by the denominator gives the slant asymptote, \(y=3x+2\), ensuring it matches the given condition.
These slant asymptotes give the graph its overall trajectory, indicating where it heads as \(x\) moves towards positive or negative infinity.
Rational Functions
Rational functions are of the form \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\). These functions are interesting because they can have various asymptotic behaviors, leading to different graph shapes and characteristics.

Key characteristics include:
  • Vertical asymptotes occur at values of \(x\) where the denominator \(Q(x)\) equals zero, given that the numerator \(P(x)\) is not zero at that point.
  • Slant asymptotes appear when the degree of \(P(x)\) is exactly one more than \(Q(x)\), as shown in the slant asymptote \(y=3x+2\).
  • Horizontal asymptotes can also occur, typically when the degrees of \(P(x)\) and \(Q(x)\) are equal or when \(Q(x)\) has a higher degree.
Understanding these key properties gives us the ability to sketch or understand the graph's behavior and anticipate the type and position of asymptotes.

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Most popular questions from this chapter

Engine Efficiency The efficiency of an internal combustion engine is Efficiency \((\%)=100\left[1-\frac{1}{\left(v_{1} / v_{2}\right)^{c}}\right]\) where \(v_{1} / v_{2}\) is the ratio of the uncompressed gas to the compressed gas and \(c\) is a positive constant dependent on the engine design. Find the limit of the efficiency as the compression ratio approaches infinity.

Use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{x^{2}}{x^{2}-1} $$

Consider \(\lim _{x \rightarrow-\infty} \frac{3 x}{\sqrt{x^{2}+3}}\). Use the definition of limits at infinity to find values of \(N\) that correspond to (a) \(\varepsilon=0.5\) and (b) \(\varepsilon=0.1\).

Use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function may indicate that there should be one.$$ h(x)=\frac{6-2 x}{3-x} $$

Numerical, Graphical, and Analytic Analysis The cross sections of an irrigation canal are isosceles trapezoids of which three sides are 8 feet long (see figure). Determine the angle of elevation \(\theta\) of the sides such that the area of the cross section is a maximum by completing the following. (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) \begin{tabular}{|c|c|c|c|} \hline Base 1 & Base 2 & Altitude & Area \\ \hline 8 & \(8+16 \cos 10^{\circ}\) & \(8 \sin 10^{\circ}\) & \(\infty 22.1\) \\ \hline 8 & \(8+16 \cos 20^{\circ}\) & \(8 \sin 20^{\circ}\) & \(\approx 42.5\) \\ \hline \end{tabular} (b) Use a graphing utility to generate additional rows of the table and estimate the maximum cross-sectional area. (Hint: Use the table feature of the graphing utility.) (c) Write the cross-sectional area \(A\) as a function of \(\theta\). (d) Use calculus to find the critical number of the function in part (c) and find the angle that will yield the maximum cross-sectional area. (e) Use a graphing utility to graph the function in part (c) and verify the maximum cross-sectional area.
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