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Chapter 2: Problem 3

The key numbers of a rational expression are its________and its______ _______.

Short Answer

Expert verified
The key numbers of a rational expression are its zeroes and its vertical asymptotes.

Step by step solution

01

Rational Expression Overview

A rational function is a fraction where the numerator and denominator are both polynomials. They generally take the form \( \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomial functions.
02

Identifying the Zeroes

The zeroes (or roots) of a rational function are the values where the function equals zero. This is met when the numerator \( p(x) \) is zero, given the denominator isn't zero at these same points to avoid undefined behavior.
03

Identifying the Vertical Asymptotes

The vertical asymptotes of the rational function are the values where the function is undefined, i.e., where the denominator of the function \( q(x) \) equals zero. These represent 'boundaries' that the function will approach but never reach.

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Key Concepts

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

Polynomials
A polynomial is a mathematical expression consisting of variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponents. Think of it like a sum of terms, such as \( x^2 + 3x + 2 \). Here, \( x^2 \), \( 3x \), and \( 2 \) are terms of the polynomial.

The degree of a polynomial is determined by the highest power of the variable in the expression. For example, in \( 4x^3 + 2x^2 + x + 5 \), the degree is 3 because of the term \( 4x^3 \).
  • Constant Polynomial: A polynomial of degree 0, simply a number, like 5.
  • Linear Polynomial: A first-degree polynomial, like \( 3x + 2 \).
  • Quadratic Polynomial: A second-degree polynomial, such as \( x^2 + 4x + 4 \).
  • Cubic Polynomial: Third-degree, for instance, \( x^3 - x^2 + 4x - 6 \).
Polynomials form the building blocks of more complex functions in mathematics, including rational expressions.
Zeroes of a Rational Function
Zeroes of a rational function refer to the values of the variable for which the function equals zero. In technical terms, it's where the polynomial in the numerator equals zero.

Consider the rational function \( \frac{p(x)}{q(x)} \). The zeroes can be found by solving \( p(x) = 0 \) because a rational expression equals zero when its numerator is zero and the denominator is not zero at these points.

These zeroes are crucial in plotting the graph of a rational function since they indicate where the graph will intersect the x-axis. Remember:
  • At the zero, the rational function touches or crosses the x-axis.
  • Solving \( p(x) = 0 \) gives potential zeroes, but verify that they don't cause the denominator to also be zero. If so, these points are not zeroes because the function would be undefined there.
Vertical Asymptotes
Vertical asymptotes occur in a rational function where the expression is undefined, typically where the denominator equals zero.

For a rational function \( \frac{p(x)}{q(x)} \), a vertical asymptote is present where \( q(x) = 0 \), provided that \( p(x) \) isn't zero at these points. These values indicate that the function approaches infinity or negative infinity as it nears these x-values, creating a 'boundary' that the graph will approach but never actually touch.
  • A vertical asymptote tells you where the function behaves wildly, either shooting up to infinity or plummeting downwards.
  • These can be visualized on a graph as a dashed vertical line, indicating the undefined part of the function.
  • Identifying these requires checking if the zeroes of the denominator are not also zeroes of the numerator.
Understanding vertical asymptotes helps in sketching accurate graphs of rational functions and grasping their behavior across various intervals.

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Most popular questions from this chapter

Use a graphing utility to graph the quadratic function. Find the \(x\) -intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when \(f(x)=0\) $$f(x)=-2 x^{2}+10 x$$

Use a graphing utility to graph the quadratic function. Find the \(x\) -intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when \(f(x)=0\) $$f(x)=\frac{7}{10}\left(x^{2}+12 x-45\right)$$

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$h(x)=x^{2}-8 x+16$$

Write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point. $$\text { Vertex: }\left(\frac{5}{2},-\frac{3}{4}\right) ; \text { point: }(-2,4)$$

The total revenue \(R\) earned per day (in dollars) from a pet-sitting service is given by \(R(p)=-12 p^{2}+150 p,\) where \(p\) is the price charged per pet (in dollars). (a) Find the revenues when the prices per pet are \(\$ 4\) \(\$ 6,\) and \(\$ 8\) (b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain your results.
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