1.4 Modeling aspects of scenarios using polynomial and rational functions

• The x-intercepts of a polynomial are also called its roots
• The x-intercepts of a rational function are the zeros of its numerator
• Match the function type with its key characteristic:

  Polynomial ↔️ Degree influences end behavior

  Rational ↔️ Vertical asymptotes where denominator = 0

  Both ↔️ Intercepts are key features
• Polynomial models are suitable for relationships involving division or inverse proportionality.

  False
• Steps to build a polynomial function from scenario data:

  1️⃣ Identify the independent and dependent variables

  2️⃣ Determine the appropriate degree of the polynomial

  3️⃣ Fit the polynomial equation to the data
• Polynomial functions are ideal for relationships that are continuous and smooth
• Rational functions are often used to model scenarios involving inverse relationships or rate problems. They represent quantities such as concentration, efficiency, or cost per unit
• Steps to build polynomial functions from scenario data

  1️⃣ Identify the independent and dependent variables

  2️⃣ Determine the appropriate degree of the polynomial

  3️⃣ Fit the polynomial equation to the data points
• The volume of a cube with side length x can be modeled using a cubic polynomial of the form $V =$$ax^{3}$
• Horizontal or slant asymptotes in rational functions depend on the degrees of the numerator and denominator.

  True
• Which metrics can be used to assess the accuracy of a model?

  RMSE or R-squared
• The end behavior of a polynomial depends on its degree and the sign of its leading coefficient.


  True
• Polynomial models are appropriate when there are sharp changes in the relationship being modeled.

  False
• The location of a horizontal asymptote in a rational function depends on the degrees of the numerator and denominator.
• What is a key feature of rational functions that represents values where the function is undefined?

  Vertical asymptotes
• What determines the appropriate degree of a polynomial when fitting it to data?

  Relationship between variables
• What type of relationships are rational functions best suited for modeling?

  Inverse relationships
• The average speed over a distance can be modeled using a rational function because speed equals distance divided by time.


  True
• The degree of a polynomial influences its shape and end behavior
• Vertical asymptotes in rational functions occur where the denominator is equal to zero.

  True
• What is the general form of a rational function after fitting it to data?

  $f(x) =$ \frac{a_{n} x^{n} +$a_{n - 1} x^{n - 1} +$$... +$ a_{0}}{b_{m} x^{m} +$b_{m - 1} x^{m - 1} +$$... +$ b_{0}}
• Polynomial and rational models can accurately capture exponential behavior in data.

  False
• What does the degree of a polynomial influence?

  End behavior
• What are the three types of asymptotes in rational functions?

  Vertical, horizontal, slant
• What type of scenarios are polynomial functions frequently used to model?

  Volume, cost, growth
• What kind of effect must the independent variable have for a polynomial model to be appropriate?

  Consistent
• Concentration is inversely proportional to volume
• The degree of a polynomial influences its end behavior and shape.

  True
• Vertical asymptotes in rational functions represent values where the function is undefined.


  True
• The cost per unit of production can be modeled using a rational function because cost per unit decreases as production volume increases.
• What is the general form of a polynomial function after fitting it to data?

  $f(x) =$$ax^{n} +$$bx^{n - 1} +$$... +$$c$
• Vertical asymptotes in rational functions occur where the denominator is equal to zero
• The relationship between concentration and volume can be modeled using a rational function of the form C = \frac{a}{V}</latex>
• One strategy to improve model accuracy is to divide the domain into segments and use piecewise functions