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## Digital SAT Math

### Course: 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 $f\left(x\right)={x}^{3}+2{x}^{2}-5x-6$. 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, $f\left(x\right)=\frac{1}{x}$ is a rational function.
In this lesson, we'll:
1. Relate the factors of polynomial functions to the $x$-intercepts of polynomial graphs
2. Apply the polynomial remainder theorem
3. Understand the behavior of basic rational functions
This lesson builds upon the following skills:
• 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

Zeros of polynomials introductionSee video transcript

### 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, $f\left(x\right)={x}^{3}+2{x}^{2}-5x-6$, would be written as $f\left(x\right)=\left(x+3\right)\left(x+1\right)\left(x-2\right)$.
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 $y=\left(x-a\right)\left(x-b\right)\left(x-c\right)$, the $x$-intercepts of graph are located at $\left(a,0\right)$, $\left(b,0\right)$, and $\left(c,0\right)$.
According to the graph of $y=\left(x+3\right)\left(x+1\right)\left(x-2\right)$ above, the graph intercepts the $x$-axis at $-3$, $-1$, and $2$, which means their corresponding factors are equal to $0$ when $x$ is equal to $-3$, $-1$, and $2$:
• $-3+3=0$
• $-1+1=0$
• $2-2=0$
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:
1. Set each factor equal to $0$.
2. 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 $y=\left(2x-1\right)\left(x-3\right)\left(x+5\right)$ ?

To write a polynomial function when its zeros are provided:
1. For each given zero, write a linear expression for which, when the zero is substituted into the expression, the value of the expression is $0$.
2. Each linear expression from Step 1 is a factor of the polynomial function.
3. The polynomial function must include all of the factors without any additional unique binomial factors.

Example: The real roots of the polynomial function $p\left(x\right)$ are $-1$, $3$, and $8$. Write a function that could be $p\left(x\right)$.

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

When a polynomial function is written in standard form, e.g., $y={x}^{3}+2{x}^{2}-5x-6$, 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 $x=0$, and so is equal to the constant term of the polynomial expression. So, the $y$-intercept for $y={x}^{3}+2{x}^{2}-5x-6$ is $-6$.
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 $a{x}^{n}$:
• If $a>0$, then $y$ ultimately approaches positive infinity as $x$ increases.
• If $a<0$, then $y$ ultimately approaches negative infinity as $x$ increases.
• If $n$ is even, then the ends of the graph point in the same direction.
• If $n$ is odd, then the ends of the graph point in different directions.
The highest power term in $y={x}^{3}+2{x}^{2}-5x-6$ is $1{x}^{3}$:
• Since $1>0$, $y$ approaches positive infinity as $x$ increases.
• Since $3$ is odd, the other end of the graph points in the opposite direction as $x$ and ${x}^{3}$ become more and more negative: negative infinity.

### Try it!

Try: determine the factors of a polynomial function based on its graph
The polynomial graph shown above has
unique zeros, which means it has the same number of unique factors.
Select all of the unique factors of the polynomial function representing the graph above.

Try: determine the end behaviors of polynomial functions
The highest power term in the polynomial function $f\left(x\right)=-2{x}^{4}-7{x}^{3}+8{x}^{2}-10x-1$ is $-2{x}^{4}$. Since the coefficient in $-2{x}^{4}$ is
and the exponent in $-2{x}^{4}$ is
:
• As $x$ increases, $y$ ultimately approaches
.
• As $x$ decreases, $y$ ultimately approaches
.

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

### Intro to the polynomial remainder theorem

Intro to the Polynomial Remainder TheoremSee video transcript

### The polynomial remainder theorem

The polynomial remainder theorem states that when a polynomial function $p\left(x\right)$ is divided by $x-a$, the remainder of the division is equal to $p\left(a\right)$.
The polynomial remainder theorem lets us calculate the remainder without doing polynomial long division. It also tells us whether an expression $x-a$ is a factor of an unknown polynomial function as long as we know the value of $p\left(a\right)$:
• If $p\left(a\right)=0$, then $\left(a,0\right)$ is an $x$-intercept, and $x-a$ is a factor of $p\left(x\right)$.
• If $p\left(a\right)\ne 0$, then $x-a$ is not a factor of $p\left(x\right)$.

### Try it!

Try: find factors and remainders from a table
$x$$g\left(x\right)$
$-2$$3$
$-1$$-6$
$0$$-5$
$1$$0$
$2$$27$
The table above shows the values of polynomial function $g$ at several different values of $x$.
Because $g\left(0\right)=-5$, $x$
a factor of $g\left(x\right)$.
Because $g\left(1\right)=0$, $x-1$
a factor of $g\left(x\right)$.
When $g\left(x\right)$ is divided by $x-2$, the remainder is
.

## 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 $f\left(x\right)=\frac{1}{x-2}$ and less like $f\left(x\right)=\frac{2{x}^{2}+7x+3}{2x+3}$.

### Graphing rational functions 1

Graphing rational functions 1See video transcript

### 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 $\left(x,y\right)$ 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 $f\left(x\right)=\frac{1}{x-2}$, $f\left(2\right)$ is undefined because when $x=2$, the denominator is $2-2=0$.
In the graph of a simple rational function, a
exists where the function is undefined. Let's go back to our example: what happens to the value of $f\left(x\right)=\frac{1}{x-2}$ as $x$ approaches $2$?
• For $x<2$, 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 $x>2$, 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 $f\left(x\right)=\frac{1}{x-2}$ and its vertical asymptote at $x=2$ are shown below.

### Try it!

try: determine how a rational function behaves
$f\left(x\right)=-\frac{1}{3x+12}$
The function above is undefined when the value of $x$ is
, and the graph of the function has a vertical asymptote at that value.
When the function is graphed in the $xy$-plane, as $x$ approaches the vertical asymptote from the negative direction, the value of $y$ approaches
infinity. As $x$ approaches the vertical asymptote from the positive direction, the value of $y$ approaches
infinity.

Practice: select a graph based on the number of zeros
If the function $f$ has four distinct zeros, which of the following could represent the complete graph of $f$ in the $xy$-plane?

Practice: write a function using roots
In the $xy$-plane, the graph of function $g$ has $x$-intercepts at $-2$, $2$, and $7$. Which of the following could define $f$ ?

Practice: Apply the remainder theorem
For a polynomial $h\left(t\right)$, the value of $h\left(-1\right)$ is $-7$. Which of the following must be true about $h\left(t\right)$ ?

practice: identify where a rational function is undefined
If $f\left(x\right)=\frac{1}{{x}^{2}-7x+12}$ is undefined, what is a possible value of $x$ ?

## Things to remember

For a polynomial function in standard form, the constant term is equal to the $y$-intercept.
For the highest power term $a{x}^{n}$ in the standard form of a polynomial function:
• If $a>0$, then $y$ ultimately approaches positive infinity as $x$ increases.
• If $a<0$, then $y$ ultimately approaches negative infinity as $x$ increases.
• If $n$ is even, then the ends of the graph point in the same direction.
• If $n$ is odd, then the ends of the graph point in different directions.
The polynomial remainder theorem states that when a polynomial function $p\left(x\right)$ is divided by $x-a$, the remainder of the division is equal to $p\left(a\right)$.
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
• I guess you're right as I too faced the same glitch.
• theres a mistake, one of the try its the answer is truly -4 but the system takes 4 as correct
• no the answer is 4 . the -4 that u are talking about is in the factored form: f(x)=(x-4)(x-3)
f(x)=0 if
x-4=0 or x-3=0
x=4 or x=3
• This was so hard!
• god damn i hate this LOL
• i know, T_T
i couldnt especially understand the y approaching +ve/-ve infinity thing T_T
(1 vote)
• f(x)=−1/3x+12 the X value that makes the answer 'undefined' is supposed to be -4 not 4 (I think there was a mistake in the TRY IT question)
• I think they have made a mistake in the question in the Try It section. The system gets 4 correct, while -4 is correct in the explanation. Am I right or there is a problem I dunno?
• You are right.
(1 vote)
• There is a problem in the "Try it" question(TRY: DETERMINE HOW A RATIONAL FUNCTION BEHAVES).
• Brain is not braining:`(