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Chapter 4: Problem 16

Find the \(x\) -and \(y\) -intercepts of the rational function. $$ r(x)=\frac{x^{3}+8}{x^{2}+4} $$

Short Answer

Expert verified
The x-intercept is at \((-2, 0)\) and the y-intercept is at \((0, 2)\).

Step by step solution

01

Find the x-intercept

To find the x-intercept(s), set the numerator of the function equal to zero, since the x-intercept of a fraction occurs where the numerator is zero (and the denominator is not zero). The numerator of the function is \(x^3 + 8\). Set it to zero:\[x^3 + 8 = 0\]Solve for \(x\):\[x^3 = -8\]\[x = -2\]The x-intercept is at \((x = -2, y = 0)\).
02

Find the y-intercept

To find the y-intercept, evaluate the function where \(x = 0\) since the y-intercept happens when \(x = 0\). Substitute \(x=0\) into the function: \[r(0) = \frac{0^3 + 8}{0^2 + 4} = \frac{8}{4} = 2\]The y-intercept is at \((x = 0, y = 2)\).
03

Verify the Denominator is Non-Zero at x-intercept

Double-check that the denominator is not zero at the x-intercept found in Step 1. At \(x = -2\), the denominator is: \[(-2)^2 + 4 = 4 + 4 = 8 \]Since 8 is not zero, the original x-intercept \((x = -2, y = 0)\) is valid.

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Key Concepts

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

X-Intercepts
Understanding the concept of x-intercepts is essential in graphing and analyzing rational functions. An x-intercept of a function is a point where the graph crosses the x-axis. At this point, the value of the function is zero (i.e., the output or y-coordinate is zero). To find the x-intercept of a rational function like \( r(x)=\frac{x^{3}+8}{x^{2}+4} \), follow these steps:

  • Focus on the numerator. This is because the output of the function equals zero when the numerator is zero (and the denominator is non-zero).
  • Write the equation of the numerator equal to zero. For \( r(x) \), this means setting \( x^3 + 8 = 0 \).
  • Solve the equation. Here, \( x^3 = -8 \), leading to \( x = -2 \).
Thus, the x-intercept is at the point \((x = -2, y = 0)\). It鈥檚 crucial to ensure that the denominator is not zero at this intercept so that we have a valid solution. This means we avoid division by zero, keeping the function defined.
Y-Intercepts
The y-intercept provides important insights into a rational function鈥檚 graph. It tells where the function crosses the y-axis, which occurs when the input, \( x \), equals zero. To find the y-intercept for the function \( r(x)=\frac{x^{3}+8}{x^{2}+4} \), use the following approach:

  • Substitute \( x = 0 \) directly into the function.
  • Calculate the resulting expression. For this function, substituting gives \( r(0) = \frac{0^3 + 8}{0^2 + 4} = \frac{8}{4} = 2 \).
This means the y-intercept is at \((x = 0, y = 2)\). The y-intercept is straightforward to find as you simply replace the input with zero and compute the result. This point represents where the function鈥檚 graph intersects the y-axis, providing a starting point for graphing or sketching out the rest of the function behavior.
Numerator and Denominator
The structure of a rational function, like \( r(x)=\frac{x^{3}+8}{x^{2}+4} \), is hinged on its numerator and denominator. Comprehending how these components affect the overall behavior of the function is vital:

  • Numerator: The numerator, \( x^3 + 8 \), determines the x-intercepts. Setting it to zero gives the values of \( x \) where the function's output is zero, facilitating the finding of x-intercepts.
  • Denominator: The denominator, \( x^2 + 4 \), plays a critical role in defining where the function is undefined. The function is not valid when this part equals zero, as it would imply division by zero.
  • Double-checking at critical points like x-intercepts ensures the denominator is never zero ensuring those points are real and valid on the graph.
Together, the numerator and denominator dictate not only the intercepts but also highlight potential asymptotes and undefined regions, key for plotting and fully understanding rational functions.

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Most popular questions from this chapter

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ 2 x^{4}-5 x^{3}-14 x^{2}+5 x+12=0 ; \quad[-2,5] \text { by }[-40,40] $$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=x^{4}-2 x^{3}-9 x^{2}+2 x+8 ; \quad a=-3, b=5 $$

Constructing a Rational Function from Its Asymptotes Give an example of a rational function that has vertical asymptote \(x=3\) . Now give an example of one that has vertical asymptote \(x=3\) and horizontal asymptote \(y=2\) Now give an example of a rational function with vertical asymptotes \(x=1\) and \(x=-1,\) horizontal asymptote \(y=0,\) and \(x\) -intercept \(4 .\)

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=8 x^{3}+10 x^{2}-39 x+9 ; \quad a=-3, b=2 $$

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\) (b) Sketch the graph of \(P\) . $$ P(x)=2 x^{3}-7 x^{2}+4 x+4 $$
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