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Chapter 4: Problem 15

In each part, make a rough sketch of the graph using asymptotes and appropriate limits but no derivatives. Compare your graph to that generated with a graphing utility. $$\text { (a) } y=\frac{3 x^{2}-8}{x^{2}-4}$$ $$\text { (b) } y=\frac{x^{2}+2 x}{x^{2}-1}$$

Short Answer

Expert verified
For (a), vertical asymptotes are at \( x = \pm 2 \) and horizontal at \( y = 3 \). For (b), vertical asymptotes are at \( x = \pm 1 \) and horizontal at \( y = 1 \).

Step by step solution

01

Identify Vertical Asymptotes for Part (a)

The function given is \( y = \frac{3x^2 - 8}{x^2 - 4} \). Vertical asymptotes occur when the denominator equals zero. Set \( x^2 - 4 = 0 \) and solve for \( x \). This gives \( x = \pm 2 \). Therefore, there are vertical asymptotes at \( x = 2 \) and \( x = -2 \).
02

Identify Horizontal Asymptote for Part (a)

For horizontal asymptotes, compare the degrees of the numerator and the denominator. Both have degree 2. The horizontal asymptote is found by dividing the leading coefficients, which are 3 for the numerator and 1 for the denominator. Therefore, the horizontal asymptote is \( y = 3 \).
03

Identify Limits for Part (a)

Examine the behavior of the function as \( x \to \pm \infty \). Since the horizontal asymptote is \( y = 3 \), the function approaches 3 as \( x \to \pm \infty \). Additionally, check the behavior near the vertical asymptotes at \( x = 2 \) and \( x = -2 \). Around these points, the value of the function tends towards \( \pm \infty \).
04

Sketch Graph for Part (a)

Draw the vertical lines \( x = -2 \) and \( x = 2 \) for the vertical asymptotes. Draw a horizontal line at \( y = 3 \) for the horizontal asymptote. Indicate that the curve approaches 3 at extremes and rises or falls near the vertical lines, based on the signs of \( (x^2) - 4 \) in the numerator.
05

Identify Vertical Asymptotes for Part (b)

The function given is \( y = \frac{x^2 + 2x}{x^2 - 1} \). Vertical asymptotes occur where \( x^2 - 1 = 0 \), so solve \( x^2 - 1 = 0 \), yielding \( x = \pm 1 \). Vertical asymptotes occur at these points.
06

Identify Horizontal Asymptote for Part (b)

Again compare the degrees, both are 2. So, the horizontal asymptote is found by the leading coefficients as before. The numerator coefficient is 1, and the denominator coefficient is 1. Thus, the horizontal asymptote is \( y = 1 \).
07

Identify Limits for Part (b)

Consider behavior as \( x \to \pm \infty \). The function approaches \( y = 1 \) following the horizontal asymptote. Investigate near \( x = 1 \) and \( x = -1 \), where the function will rise to \( \pm \infty \) depending on the sign changes in the asymptotic behavior.
08

Sketch Graph for Part (b)

Draw vertical asymptotes at \( x = 1 \) and \( x = -1 \), with a horizontal asymptote at \( y = 1 \). Illustrate the curve approaching \( y = 1 \) as \( x \to \pm \infty \), while rising or falling sharply through the vertical asymptote positions.

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Key Concepts

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

Graph Sketching
When sketching a graph, understanding asymptotes is crucial. They guide the general shape of the graph without needing to calculate every single detail manually. Here鈥檚 a step-by-step look into what you need for graph sketching:
  • First, identify vertical asymptotes, where the denominator of the function equals zero. For instance, in the function \[y = \frac{3x^2 - 8}{x^2 - 4}\], vertical asymptotes occur at the roots of \(x^2 - 4\), which are \(x = 2\) and \(x = -2\).
  • Then, discover the horizontal asymptote by comparing the degrees of the numerator and the denominator. In part (a), both degrees are 2, so the horizontal asymptote is found by dividing the leading coefficients: 3 by 1, giving \(y = 3\).
  • Once you've noted these asymptotes, sketch the vertical and horizontal lines corresponding to them. Observe the behavior of the function near these asymptotes. Approaching a vertical asymptote, the function typically moves towards infinity or negative infinity. Meanwhile, the function approaches the horizontal asymptote as \(x\) approaches infinity.
Apply this method to every function you encounter to visualize a rough but accurate graph quickly. Once the asymptotes are plotted, observe how the function behaves relative to these lines to complete the sketch.
Limits in Calculus
Limits help us understand the behavior of functions at specific points or as they approach infinity. Grasping limits is key for identifying asymptotic behavior. Let鈥檚 see how they work in graph sketching:
  • As \(x\) tends to infinite values (positive or negative), the limit gives insights into how the function behaves on the edges of the graph. For example, in \[y = \frac{3x^2 - 8}{x^2 - 4}\], the function approaches the horizontal asymptote \(y = 3\) as \(x\) approaches \(\pm\infty\).
  • Analyze the function鈥檚 behavior near vertical asymptotes, like \(x = 2\) and \(x = -2\). As \(x\) nears these values, you鈥檒l often find that the function rises or falls drastically, heading toward positive or negative infinity, respectively.
  • The concept of limits in calculus extends beyond mere approaches to infinity or specific points; it measures the "tendency" or "approach" of a function near these critical points, helping graph its climb or dip more accurately.
Understanding limits significantly eases the task of sketching a graph, especially for complex functions, as it showcases the real behavior at unobservable points like infinities or discontinuous breaks.
Vertical and Horizontal Asymptotes
Asymptotes are lines that a graph approaches but never touches. They play a critical role in understanding the shape and direction of the graph. Here is a deeper dive into identifying them:
  • Vertical Asymptotes: These occur when the function is undefined, typically where the denominator equals zero. For a given function like \[y = \frac{3x^2 - 8}{x^2 - 4}\], solve \(x^2 - 4 = 0\) to find vertical asymptotes at \(x = 2\) and \(x = -2\).
  • Horizontal Asymptotes: Examine the degrees of the polynomial terms in the numerator and denominator. If they are equal, the horizontal asymptote is a line determined by the fraction of their leading coefficients. Hence, for this function, the asymptote is \(y = 3\) since the leading coefficients are 3 and 1.
  • While vertical asymptotes indicate undefined points and steep increases or decreases, horizontal asymptotes highlight limit behavior as \(x\) extends towards infinity, describing stable outcomes of the function far from the origin.
Grasping the concept of asymptotes is fundamental to mastering graph sketching and truly understanding function behaviors in calculus.

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Most popular questions from this chapter

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