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Chapter 2: Problem 19

Complete the table without using the Quotient Rule. $$ y=\frac{x^{2}+2 x}{3} $$

Short Answer

Expert verified
The function can be simplified to \(y=\frac{1}{3}x^{2}+\frac{2}{3}x\). Using this simplified equation, any x term can be substituted into this equation to find the corresponding y-value.

Step by step solution

01

Simplify the Function

To simplify the function, distribute the denominator into the numerator. The new equation will be \(y=\frac{1}{3}x^{2}+\frac{2}{3}x\).
02

Identify x and y values

For a table of x and y values, we would select values for x and substitute them into the simplified equation to find the corresponding y-value.
03

Example Calculation

Suppose we choose 1 as a value for x. Substitute x = 1 into the simplified equation. So, the y-value would be \(y=\frac{1}{3}(1)^{2}+\frac{2}{3}(1) = \frac{1}{3} + \frac{2}{3} = 1\).
04

Complete the table

Repeat Step 3 for various values of x to complete the entire table.

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Key Concepts

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

Polynomial Functions
Polynomial functions are expressions involving variables raised to whole number exponents. They are fundamental in algebra as they form the basis for equations and expressions you encounter. Each term in a polynomial function is a product of a constant (coefficient) and a variable raised to a non-negative integer power. When examining polynomial functions, it's crucial to focus on the degree, which is the highest power of the variable.
  • The standard form of a polynomial is written as: \( ax^n + bx^{n-1} + cx^{n-2} + \ldots + k \), where \( a, b, c, \ldots, k \) are coefficients and \( n \) is a non-negative integer.
  • The degree of a polynomial helps us understand its general shape and behavior. For example, a polynomial of degree 2, known as a quadratic, will have a parabolic graph.
  • Operations on polynomials, such as addition, subtraction, and multiplication, maintain the polynomial structure. Division by a non-zero polynomial yields a rational function, not a polynomial.
Understanding these elements is crucial when breaking down more complex problems like rational functions.
Rational Functions
A rational function is a ratio of two polynomial functions. It can be written as \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are both polynomials, and \( Q(x) eq 0 \). Rational functions exhibit behaviors influenced by the degrees of both the numerator and the denominator.
  • The behavior of a rational function is often characterized by asymptotes, which are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal asymptotes are determined by the degrees of the polynomials involved.
  • To simplify rational functions, like in the exercise, you might need to divide each term in the numerator by the denominator. This practice, as seen in the conversion of \( y=\frac{x^{2}+2x}{3} \) to \( y=\frac{1}{3}x^{2}+\frac{2}{3}x \), facilitates easier computations and graph evaluations.
  • Identifying characteristics of polynomial functions within a rational function helps in graphically representing and solving them.
Through this understanding, you can handle manipulations such as those needed to construct function tables without relying solely on rules like the quotient rule.
Function Tables
Function tables are a valuable tool in visualizing relationships between variables. They allow you to see how changes in one variable affect another, giving concrete insights into how a function behaves across a range of values.
  • To construct a function table, choose a set of input values for \( x \). Substitute these into the function to determine the corresponding output values for \( y \).
  • Creating a table helps in predicting and verifying the function's behavior, as shown by substituting various values in the simplified form \( y=\frac{1}{3}x^{2}+\frac{2}{3}x \).
  • Function tables are particularly useful for sketching graphs. By plotting the \( (x, y) \) pairs derived from the table, you can visually represent the function.
Using function tables provides a straightforward method to observe trends and patterns, crucial for deeper comprehension of both polynomial and rational functions.

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Most popular questions from this chapter

Find the derivative of the function. \(f(t)=\frac{3^{2 t}}{t}\)

Find equations of both tangent lines to the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) that passes through the point (4,0).

Find the derivative of the function. \(f(t)=t^{3 / 2} \log _{2} \sqrt{t+1}\)

In Exercises \(115-118,\) evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. \(h(x)=\frac{1}{9}(3 x+1)^{3}, \quad\left(1, \frac{64}{9}\right)\)

Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. \(g(t)=\tan 2 t, \quad\left(\frac{\pi}{6}, \sqrt{3}\right)\)
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