1.4 Modeling aspects of scenarios using polynomial and rational functions

• What is a polynomial function?

  A function in polynomial form
• A polynomial function can be expressed as $f(x) =$$a_{n}x^{n} +$$a_{n - 1}x^{n - 1} +$$... +$$a_{1}x +$$a_{0}$, where $n$ is the degree
• The degree of a polynomial function is the highest exponent of the variable.
• What are the coefficients of a polynomial function?

  The numerical multipliers of each term
• Match the characteristic with its description:

  Degree ↔️ Highest exponent of the variable

  Coefficients ↔️ Numerical multipliers of each term

  Notation ↔️ Written in the form $f(x) =$$a_{n}x^{n} +$$...$
• How does the behavior of a polynomial function depend on its degree and coefficients?

  It can increase, decrease, or oscillate
• Understanding the characteristics of polynomial functions is essential for modeling real-world scenarios.
• A rational function is of the form $f(x) =$$\frac{P(x)}{Q(x)}$, where P(x)</latex> and $Q(x)$ are polynomial
• What is the domain of a rational function?

  All real numbers except $Q(x) =$$0$
• Vertical asymptotes occur where $Q(x) =$$0$ and $P(x) \neq 0$.
• What determines the horizontal asymptotes of a rational function?

  The degrees of $P(x)$ and $Q(x)$
• Oblique asymptotes occur when the degree of $P(x)$ is one greater than the degree of $Q(x)$.oblique
• How are quadratic polynomials used in projectile motion?

  To describe the trajectory
• Polynomial functions can be used to approximate trends in datasets.
• What can polynomials model in business cost functions?

  Fixed and variable expenses
• Match the characteristic of polynomial functions with its description:

  Degree ↔️ Highest exponent of the variable

  Coefficients ↔️ Numerical multipliers of each term

  Notation ↔️ Written in the form $f(x) =$$a_{n}x^{n} +$$...$
• What shapes can polynomial functions exhibit?

  Lines, parabolas, cubics, etc.
• A rational function is expressed as f(x) = \frac{P(x)}{Q(x)}</latex>, where $Q(x)$ cannot equal zero
• What is the highest exponent of the variable in a polynomial function called?

  Degree
• Numerical multipliers of each term in a polynomial function are called coefficients
• The notation for a polynomial function is f(x) = a_{n}x^{n} + a_{n - 1}x^{n - 1} + ... + a_{1}x + a_{0}</latex>
• What types of shapes can polynomial functions exhibit?

  Lines, parabolas, cubics
• In a rational function, Q(x) cannot equal zero.
• What is the domain of a rational function?

  All real numbers except where Q(x) = 0
• Vertical asymptotes occur at values of x where Q(x) = 0 and P(x) ≠ 0
• Horizontal asymptotes are determined by comparing the degrees of P(x) and Q(x).
• When does an oblique asymptote occur in a rational function?

  When the degree of P(x) is one greater than Q(x)
• A polynomial function must have coefficients that are real numbers.
• The degree of a polynomial function is denoted by the variable n
• Match the real-world scenario with its polynomial function model:

  Growth and Decay ↔️ $P(t) =$$P_{0}e^{rt}$

  Projectile Motion ↔️ $y(x) =$$ax^{2} +$$bx +$$c$

  Cost Functions ↔️ $C(x) =$$ax^{3} +$$bx^{2} +$$cx +$$d$

  Data Approximation ↔️ $f(x) =$$a_{n}x^{n} +$$a_{n - 1}x^{n - 1} +$$... +$$a_{1}x +$$a_{0}$
• The polynomial function for growth and decay is P(t) = P_{0}e^{rt}</latex>
• What is the condition for the denominator Q(x) in a rational function?

  Q(x) ≠ 0
• A rational function can be expressed in the form $f(x) =$$\frac{P(x)}{Q(x)}$ where P(x) and $Q(x)$ are polynomial functions.
• Steps to construct polynomial models

  1️⃣ Identify variables and their relationships

  2️⃣ Select the degree that best fits the scenario

  3️⃣ Determine coefficients using given data points or conditions

  4️⃣ Verify that the model aligns with the context
• What is an example of a polynomial model for population growth?

  $P(t) =$$a t^{2} +$$b t +$$c$
• The degree of a polynomial function is the highest exponent of the variable.
• What is the general form of a polynomial model for distance vs. time?

  $D(t) =$$k t^{3} +$$l t^{2} +$$m t +$$n$
• A key term for polynomial functions is their degree
• Coefficients in polynomial functions are numerical values multiplying x terms.
• What is an example of a polynomial model for population growth?

  $P(t) =$$100t^{2} +$$500t +$$1000$