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## Digital SAT Math

### Unit 12: Lesson 13

Polynomial and other nonlinear graphs: advanced# Polynomial and other nonlinear graphs | Lesson

A guide to polynomial and other nonlinear graphs on the digital SAT

## What are polynomial and other nonlinear graphs?

In a

**polynomial function**, the output of the function is based on a**polynomial expression**in which the input is raised to the second power or higher.**Quadratic functions**are a type of polynomial function. However, this lesson focuses on polynomial functions raised to the

**third power or higher**.

Consider the function f, left parenthesis, x, right parenthesis, equals, x, cubed, plus, 2, x, squared, minus, 5, x, minus, 6. This function is a

**third degree polynomial**; its highest exponent of x is 3, and it has three x-intercepts: In addition to polynomial functions, we may also encounter other nonlinear functions such as

**rational functions**. The input of a rational function appears in the denominator of an expression. For example, f, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, x, end fraction is a rational function. In this lesson, we'll:

- Relate the factors of polynomial functions to the x-intercepts of polynomial graphs
- Apply the
**polynomial remainder theorem** - Understand the behavior of basic rational functions

This lesson builds upon the following skills:

- Quadratic graphs
- Radical, rational, and absolute value equations

**You can learn anything. Let's do this!**

## How do I identify features of graphs from polynomial functions?

### Zeros of polynomials introduction

### Features of polynomial graphs

#### Factored form and zeros

**Note:**The terms "zeros", "roots", and "x-intercepts" are used interchangeably here and on the test!

On the SAT, polynomial functions are usually shown in

**factored form**. For example, the function above, f, left parenthesis, x, right parenthesis, equals, x, cubed, plus, 2, x, squared, minus, 5, x, minus, 6, would be written as f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 3, right parenthesis, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 2, right parenthesis.This is because the factors tell us the

**x-intercepts**of the graph. At each x-intercept, the value of y is zero. By the**zero product property**, if any of the factors is equal to 0, then the entire polynomial expression is equal to 0. Therefore, for y, equals, left parenthesis, x, minus, start color #7854ab, a, end color #7854ab, right parenthesis, left parenthesis, x, minus, start color #ca337c, b, end color #ca337c, right parenthesis, left parenthesis, x, minus, start color #208170, c, end color #208170, right parenthesis, the x-intercepts of graph are located at left parenthesis, start color #7854ab, a, end color #7854ab, comma, 0, right parenthesis, left parenthesis, start color #ca337c, b, end color #ca337c, comma, 0, right parenthesis, and left parenthesis, start color #208170, c, end color #208170, comma, 0, right parenthesis. According to the graph of y, equals, left parenthesis, x, plus, 3, right parenthesis, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 2, right parenthesis above, the graph intercepts the x-axis at start color #7854ab, minus, 3, end color #7854ab, start color #ca337c, minus, 1, end color #ca337c, and start color #208170, 2, end color #208170, which means their corresponding factors are equal to 0 when x is equal to start color #7854ab, minus, 3, end color #7854ab, start color #ca337c, minus, 1, end color #ca337c, and start color #208170, 2, end color #208170:

- start color #7854ab, minus, 3, end color #7854ab, plus, 3, equals, 0
- start color #ca337c, minus, 1, end color #ca337c, plus, 1, equals, 0
- start color #208170, 2, end color #208170, minus, 2, equals, 0

Higher order polynomials behave similarly. For any polynomial graph, the number of distinct x-intercepts is equal to the number of unique factors.

To determine the zeros of a polynomial function in factored form:

- Set each factor equal to 0.
- Solve the equations from Step 1. The solutions to the linear equations are the zeros of the polynomial function.

**Example:**What are the roots of y, equals, left parenthesis, 2, x, minus, 1, right parenthesis, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 5, right parenthesis ?

To write a polynomial function when its zeros are provided:

- For each given zero, write a linear expression for which, when the zero is substituted into the expression, the value of the expression is 0.
- Each linear expression from Step 1 is a factor of the polynomial function.
- The polynomial function must include all of the factors without any additional unique binomial factors.

**Example:**The real roots of the polynomial function p, left parenthesis, x, right parenthesis are minus, 1, 3, and 8. Write a function that could be p, left parenthesis, x, right parenthesis.

#### Standard form, y-intercept, and end behavior

When a polynomial function is written in

**standard form**, e.g., y, equals, x, cubed, plus, 2, x, squared, minus, 5, x, minus, 6, we can't identify the zeros as easily, but we*can*determine the**y-intercept**and**end behavior**of the graph.The

**y-intercept**happens when x, equals, 0, and so is equal to the**constant term**of the polynomial expression. So, the y-intercept for y, equals, x, cubed, plus, 2, x, squared, minus, 5, x, minus, 6 is minus, 6. The highest power term tells us the end behavior of the graph. End behavior is just another term for what happens to the value of y as x becomes very large in both the positive and negative directions. For the highest power term start color #7854ab, a, end color #7854ab, x, start superscript, start color #ca337c, n, end color #ca337c, end superscript:

- If start color #7854ab, a, end color #7854ab, is greater than, 0, then y ultimately approaches positive infinity as x increases.
- If start color #7854ab, a, end color #7854ab, is less than, 0, then y ultimately approaches negative infinity as x increases.
- If start color #ca337c, n, end color #ca337c is
**even**, then the ends of the graph point in the same direction. - If start color #ca337c, n, end color #ca337c is
**odd**, then the ends of the graph point in different directions.

The highest power term in y, equals, x, cubed, plus, 2, x, squared, minus, 5, x, minus, 6 is start color #7854ab, 1, end color #7854ab, x, start superscript, start color #ca337c, 3, end color #ca337c, end superscript:

- Since start color #7854ab, 1, end color #7854ab, is greater than, 0, y approaches positive infinity as x increases.
- Since start color #ca337c, 3, end color #ca337c is odd, the other end of the graph points in the
*opposite*direction as x and x, cubed become more and more negative:*negative*infinity.

### Try it!

## What is the polynomial remainder theorem, and how do I apply it?

### Intro to the polynomial remainder theorem

### The polynomial remainder theorem

The

**polynomial remainder theorem**states that when a polynomial function p, left parenthesis, x, right parenthesis is divided by x, minus, a, the remainder of the division is equal to p, left parenthesis, a, right parenthesis.The polynomial remainder theorem lets us calculate the remainder without doing polynomial long division. It also tells us whether an expression x, minus, a is a factor of an unknown polynomial function as long as we know the value of p, left parenthesis, a, right parenthesis:

- If p, left parenthesis, a, right parenthesis, equals, 0, then left parenthesis, a, comma, 0, right parenthesis is an x-intercept, and x, minus, a is a factor of p, left parenthesis, x, right parenthesis.
- If p, left parenthesis, a, right parenthesis, does not equal, 0, then x, minus, a is
*not*a factor of p, left parenthesis, x, right parenthesis.

### Try it!

## What are the features of simple rational functions?

**Note:**Rational functions can get quite complex, but the SAT tends to focus on simple rational functions! Think rational functions more like f, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, x, minus, 2, end fraction and less like f, left parenthesis, x, right parenthesis, equals, start fraction, 2, x, squared, plus, 7, x, plus, 3, divided by, 2, x, plus, 3, end fraction.

### Graphing rational functions 1

### Undefined and vertical asymptotes

As with other functions, we can find points on the graphs of rational functions by plugging x-values into the function to get left parenthesis, x, comma, y, right parenthesis pairs. What makes rational functions

*different*from linear, quadratic, exponential, and polynomial functions is that for certain values of x, the function can be**undefined**.In math, division by 0 is impossible: any attempt to divide by 0 results in a quotient that is considered "undefined". Therefore, a rational function is undefined when its input results in an expression that asks us to divide by 0.

For example, for f, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, x, minus, 2, end fraction, f, left parenthesis, 2, right parenthesis is undefined because when x, equals, 2, the denominator is 2, minus, 2, equals, 0.

In the graph of a simple rational function, a f, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, x, minus, 2, end fraction as x approaches 2?

exists where the function is undefined. Let's go back to our example: what happens to the value of - For x, is less than, 2, as x approaches 2, the value of the denominator remains negative and gets closer and closer to 0. When 1 is divided by a
*very small*negative number, the quotient is a*very large*negative number. In other words, as x approaches 2 from the left, y approaches*negative infinity*. - For x, is greater than, 2, as x approaches 2, the value of the denominator remains positive and gets closer and closer to 0. When 1 is divided by a
*very small*positive number, the quotient is a*very large*positive number. In other words, as x approaches 2 from the right, y approaches*positive infinity*.

The graph of f, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, x, minus, 2, end fraction and its vertical asymptote at x, equals, 2 are shown below.

### Try it!

## Your turn!

## Things to remember

For a polynomial function in standard form, the constant term is equal to the y-intercept.

For the highest power term start color #7854ab, a, end color #7854ab, x, start superscript, start color #ca337c, n, end color #ca337c, end superscript in the standard form of a polynomial function:

- If start color #7854ab, a, end color #7854ab, is greater than, 0, then y ultimately approaches positive infinity as x increases.
- If start color #7854ab, a, end color #7854ab, is less than, 0, then y ultimately approaches negative infinity as x increases.
- If start color #ca337c, n, end color #ca337c is
**even**, then the ends of the graph point in the same direction. - If start color #ca337c, n, end color #ca337c is
**odd**, then the ends of the graph point in different directions.

The

**polynomial remainder theorem**states that when a polynomial function p, left parenthesis, x, right parenthesis is divided by x, minus, a, the remainder of the division is equal to p, left parenthesis, a, right parenthesis.A rational function is undefined when division by 0 occurs.

## Want to join the conversation?

- Not the biggest deal but the something is up with the f(x)= -1/(3x+12) question . The answer in the explanation is correct but when you select the correct answer in the dropdown menu , it signs incorrect . just a minor thing(15 votes)