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Chapter 4: Problem 32

Sketch the graph of each rational function. Specify the intercepts and the asymptotes. $$y=\left(4 x^{2}-x-3\right) /\left(2 x^{2}-3 x-5\right)$$

Short Answer

Expert verified
Intercepts: roots of \(4x^2-x-3=0\). Vertical asymptotes: roots of \(2x^2-3x-5=0\). Horizontal asymptote: \(y=2\).

Step by step solution

01

Identify the Intercepts

To find the **x-intercepts**, set the numerator equal to zero and solve for \(x\): \(4x^2 - x - 3 = 0\). Factor or use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a=4\), \(b=-1\), \(c=-3\). The solutions to this are **x-intercepts**. For the **y-intercept**, set \(x = 0\) and calculate \(y\).
02

Determine Vertical Asymptotes

Vertical asymptotes occur at \(x\) values that make the denominator zero (and the numerator isn't zero at that point). Solve \(2x^2 - 3x - 5 = 0\) for \(x\). Factor or use the quadratic formula to find these \(x\) values.
03

Find Horizontal Asymptotes

For horizontal asymptotes, compare the degrees of the numerator and denominator polynomials. Here, both have degree 2. The horizontal asymptote is \(y = \frac{4}{2} = 2\), because the leading coefficients of the terms are 4 and 2.
04

Sketch the Graph

With intercepts and asymptotes identified, sketch the graph. Plot the intercepts on the coordinate axes and draw the vertical asymptotes as dashed vertical lines. The horizontal asymptote is a dashed horizontal line at \(y = 2\). Use test points or the long division of polynomials to refine the curve of the graph.

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Key Concepts

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

Understanding Intercepts
Intercepts are the points where a graph crosses the axes, and they are crucial in graphing rational functions. Let's take a closer look at how to find them:
  • x-intercepts: These occur where the graph crosses the x-axis. To find them, set the numerator of the rational function equal to zero and solve for \(x\). In our example, you solve \(4x^2 - x - 3 = 0\) using the quadratic formula, yielding the x-values where the function intersects the x-axis.
  • y-intercept: The y-intercept is where the graph crosses the y-axis, and it is obtained by substituting \(x = 0\) into the function. Calculate the resulting \(y\) value to find this intercept. In our example function \(y = \frac{(4(0)^2) - (0) - 3}{(2(0)^2) - (3)(0) - 5} = \frac{-3}{-5} = \frac{3}{5}\).
Being precise about intercepts helps in building an accurate sketch of the rational function.
Vertical Asymptotes Explained
In a rational function, vertical asymptotes are the imaginary lines where the graph tends towards positive or negative infinity. They occur where the denominator is zero, but the numerator is not zero. Finding these is a crucial step in graphing.
You can determine the vertical asymptotes by solving the equation formed by setting the denominator equal to zero. For example, in the function \(2x^2 - 3x - 5\), use the quadratic formula, or attempt factoring, to find \(x\) values which make the denominator zero. This will give you the positions of the vertical asymptotes on your graph. Plot these as dashed vertical lines, and remember that the function will approach but not cross these lines. Incorporating vertical asymptotes into your graph helps define its general shape and prevents inaccuracies.
Describing Horizontal Asymptotes
Horizontal asymptotes in a rational function provide insight into the behavior of the graph as \(x\) tends toward infinity. They're determined by comparing the degrees of the numerator and the denominator.
If both the numerator and denominator have the same degree, the horizontal asymptote is \(y = \frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients. In our function example, both terms have degree 2, allowing us to conclude the horizontal asymptote is at \(y = \frac{4}{2} = 2\). This line is represented as a dashed horizontal line on the graph.
Remember, horizontal asymptotes indicate the end-behavior of the function and are particularly useful when x becomes very large or very negative.

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Most popular questions from this chapter

(a) Factor the expression \(4 x^{2}-x^{4}\). Then use the techniques explained in this section to graph the function defined by \(y=4 x^{2}-x^{4}\). (b) Find the coordinates of the turning points. Hint: As in previous sections, use the substitution \(x^{2}=t\).

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