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Chapter 3: Problem 78

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\).

Short Answer

Expert verified
The graph of the function \(f(x)\) surges upward towards positive infinity as \(x\) approaches -2 from the left, and plunges downward towards negative infinity as \(x\) approaches -2 from the right, creating a stark 'split' or 'divide' in the graph at \(x = -2\) and forming a vertical asymptote.

Step by step solution

01

Understand the function behavior on the left of the asymptote

On observing, we can see that as the value of \(x\) comes close to -2 from the left side (e.g., -2.1, -2.01, -2.001, etc.), the value of \(f(x)\) increases significantly towards positive infinity. This behavior can be seen as climbing a mountain with its peak pointing towards the sky (positive infinity).
02

Understand the function behavior on the right of the asymptote

Simultaneously, as \(x\) approaches -2 from the right side (e.g., -1.9, -1.99, -1.999, etc.), the value of \(f(x)\) decreases sharply towards negative infinity. It's like digging a hole that gets deeper and deeper towards the bottom (negative infinity).
03

Visualizing a vertical asymptote

When seen collectively, the function near the asymptote at \(x = -2\), from left it shoots upwards towards the sky while from right it plunges downwards towards the bottom creating a gap that cannot be reached or crossed, mounting a vertical wall at \(x = -2\). This vertical wall is the vertical asymptote.

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Most popular questions from this chapter

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-2 x^{4}+2 x^{3}$$

In Exercises \(74-77\), use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-x^{5}+5 x^{4}-6 x^{3}+2 x+20$$

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$g(x)=7 x^{5}-\pi x^{3}+\frac{1}{5} x$$

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}+5 x^{2}-9 x-45$$

If you are given the equation of a rational function, explain how to find the horizontal asymptote, if any, of the functions graph.
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