Polynomial Functions

class: center, middle, inverse, title-slide

.title[
# Polynomial Functions
]
.subtitle[
## Session 06
]
.author[
### Alejandro Ucan
]
.date[
### 2023-10-04
]

---

<div>
<style type="text/css">.xaringan-extra-logo {
width: 110px;
height: 128px;
z-index: 0;
background-image: url(https://github.com/alxcn/TecLogoEIC/blob/9562a53875418e749a296c85808a19c85fc4be74/IngenieriaCiencias_Horizontal_RGB.png);
background-size: contain;
background-repeat: no-repeat;
position: absolute;
top:2em;right:2em;
}
</style>
<script>(function () {
  let tries = 0
  function addLogo () {
    if (typeof slideshow === 'undefined') {
      tries += 1
      if (tries < 10) {
        setTimeout(addLogo, 100)
      }
    } else {
      document.querySelectorAll('.remark-slide-content:not(.title-slide):not(.inverse):not(.hide_logo)')
        .forEach(function (slide) {
          const logo = document.createElement('div')
          logo.classList = 'xaringan-extra-logo'
          logo.href = null
          slide.appendChild(logo)
        })
    }
  }
  document.addEventListener('DOMContentLoaded', addLogo)
})()</script>
</div>
# Goals

* Describe the general form a the Polynomial function model. <br/><br/>
* Describe the general behavior of a polynomial function. <br/><br/>
* Apply the polynomial function to model business problems. <br/><br/>

---
# Polynomial Functions

> __Definition:__ A polynomial function is a function of the form `$$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$` where `$a_n\neq 0$` and `$n$` is a nonnegative integer. The domain of a polynomial function is `$(-\infty,\infty)$`.

* `$a_n$` are known as coefficientes. <br/><br/>
* `$n$` is called the degree of the polynomial. <br/><br/>
* `$a_0$` is called the constant term. <br/><br/>

---
# Criterium

> Assume that we have a large collection of data (that depends on one variable). This data could fit a polynomial function if there are multiple sign change (in the mean changes of the data).

<br/><br/>
* The degree of the polynomial is the number of sign changes minus one. <br/><br/>
* The polynomial functions take values in `$(-\infty, \infty).$` <br/><br/>
* The numbers of intercepts of the polynomial function with the `$x$`-axis is lower than its degree. <br/><br/>

---
#### Example

Consider the following polynomial function `$f(x)=x^3-3x^2-4x+12.$`

```r
x <- seq(-5, 5, by = 0.1)
y <- x^3 - 3*x^2 - 4*x + 12
plot(x, y, type = "l", col = "blue", lwd = 2)
abline(h = 0, col = "red")
```
---

#### Example

Consider the following polynomial function `$f(x)=x^4 + 2 x^3 - 19 x^2 - 8 x + 60.$`

```r
x <- seq(-6, 6, by = 0.1)
y <- x^4 + 2*x^3 - 19*x^2 - 8*x + 60
plot(x, y, type = "l", col = "blue", lwd = 2)
abline(h = 0, col = "red")
```
---
# What happens when we work with real life data?

> Sometimes collected data does not fit a linear regression model.

---
## How to solve that problem?

> There is a technique called __polynomial regression__ that allows us to fit a polynomial function to a set of data.

## How to decide if my data fits a polynomial function?

1. Plot the data in a scatter plot to see some tendencies. <br/><br/>
2. Use the __criterium__ to decide if the data fits a polynomial function. <br/><br/>
3. Use the __polynomial regression__ to fit a polynomial function to the data. <br/><br/>