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Digital SAT Math
Course: Digital SAT Math > Unit 12
Lesson 13: Polynomial and other nonlinear graphs: advancedPolynomial 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 . This function is a third degree polynomial; its highest exponent of is , and it has three -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, is a rational function.
In this lesson, we'll:
- Relate the factors of polynomial functions to the
-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 " -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, , would be written as .
This is because the factors tell us the -intercepts of the graph. At each -intercept, the value of is zero. By the zero product property, if any of the factors is equal to , then the entire polynomial expression is equal to . Therefore, for , the -intercepts of graph are located at , , and .
According to the graph of above, the graph intercepts the -axis at , , and , which means their corresponding factors are equal to when is equal to , , and :
Higher order polynomials behave similarly. For any polynomial graph, the number of distinct -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
. - 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 ?
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
. - 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 are , , and . Write a function that could be .
Standard form, -intercept, and end behavior
When a polynomial function is written in standard form, e.g., , we can't identify the zeros as easily, but we can determine the -intercept and end behavior of the graph.
The -intercept happens when , and so is equal to the constant term of the polynomial expression. So, the -intercept for is .
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 as becomes very large in both the positive and negative directions. For the highest power term :
- If
, then ultimately approaches positive infinity as increases. - If
, then ultimately approaches negative infinity as increases. - If
is even, then the ends of the graph point in the same direction. - If
is odd, then the ends of the graph point in different directions.
The highest power term in is :
- Since
, approaches positive infinity as increases. - Since
is odd, the other end of the graph points in the opposite direction as and 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 is divided by , the remainder of the division is equal to .
The polynomial remainder theorem lets us calculate the remainder without doing polynomial long division. It also tells us whether an expression is a factor of an unknown polynomial function as long as we know the value of :
- If
, then is an -intercept, and is a factor of . - If
, then is not a factor of .
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 and less like .
Graphing rational functions 1
Undefined and vertical asymptotes
As with other functions, we can find points on the graphs of rational functions by plugging -values into the function to get pairs. What makes rational functions different from linear, quadratic, exponential, and polynomial functions is that for certain values of , the function can be undefined.
In math, division by is impossible: any attempt to divide by 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 .
For example, for , is undefined because when , the denominator is .
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 as approaches ?
- For
, as approaches , the value of the denominator remains negative and gets closer and closer to . When is divided by a very small negative number, the quotient is a very large negative number. In other words, as approaches from the left, approaches negative infinity. - For
, as approaches , the value of the denominator remains positive and gets closer and closer to . When is divided by a very small positive number, the quotient is a very large positive number. In other words, as approaches from the right, approaches positive infinity.
The graph of and its vertical asymptote at are shown below.
Try it!
Your turn!
Things to remember
For a polynomial function in standard form, the constant term is equal to the -intercept.
For the highest power term in the standard form of a polynomial function:
- If
, then ultimately approaches positive infinity as increases. - If
, then ultimately approaches negative infinity as increases. - If
is even, then the ends of the graph point in the same direction. - If
is odd, then the ends of the graph point in different directions.
The polynomial remainder theorem states that when a polynomial function is divided by , the remainder of the division is equal to .
A rational function is undefined when division by 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(81 votes)
- I guess you're right as I too faced the same glitch.(12 votes)
- theres a mistake, one of the try its the answer is truly -4 but the system takes 4 as correct(53 votes)
- 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(0 votes)
- This was so hard!(36 votes)
- god damn i hate this LOL(25 votes)
- 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)(24 votes)
- 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?(18 votes)
- You are right.(1 vote)
- There is a problem in the "Try it" question(TRY: DETERMINE HOW A RATIONAL FUNCTION BEHAVES).(18 votes)
- Brain is not braining:`((17 votes)
- Does rational Functions really come on the SAT?(11 votes)
- there is some kind of mistake in the last try it question? the explanation says that the answer is -4 but when I choose the answer -4 it tells me wrong ?(11 votes)
- 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
besides that you can always try put 4 in the f(x) and you'll get 0 but if you try out -4 you'll get 56(0 votes)