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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:
The polynomial function y=x^3+2x^2-5x-6 is graphed in the xy-plane. The graph intercepts the x-axis at approximately -3, -1, and 2. As x increases, y increases, decreases, and increases. The graph changes direction at the approximate x-values of -2.12 and 0.79.
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.
The rational function y=1/x is graphed in the xy-plane. For x<0, the graph is in Quadrant III. As x decreases, y approaches 0. As x approaches 0, y approaches negative infinity. For x>0, the graph is in Quadrant I. As x increases, y approaches 0. As x approaches 0, y approaches positive infinity.
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:
  • 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

Khan Academy video wrapper
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 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.
The polynomial function y=x^3+2x^2-5x-6 is graphed in the xy-plane. The graph intercepts the x-axis at approximately -3, -1, and 2. The x-intercepts are highlighted.
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:
  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, 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:
  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 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 polynomial function y=x^3+2x^2-5x-6 is graphed in the xy-plane. The y-intercept, (0, -6), is highlighted.
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!

Try: determine the factors of a polynomial function based on its graph
The graph of a polynomial function intercepts the x-axis at -4, 0, 1, and 3.
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.
Choose all answers that apply:
Choose all answers that apply:


Try: determine the end behaviors of polynomial functions
The highest power term in the polynomial function f, left parenthesis, x, right parenthesis, equals, minus, 2, x, start superscript, 4, end superscript, minus, 7, x, cubed, plus, 8, x, squared, minus, 10, x, minus, 1 is minus, 2, x, start superscript, 4, end superscript. Since the coefficient in minus, 2, x, start superscript, 4, end superscript is
and the exponent in minus, 2, x, start superscript, 4, end superscript 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

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Intro to the Polynomial Remainder TheoremSee video transcript

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!

Try: find factors and remainders from a table
xg, left parenthesis, x, right parenthesis
minus, 23
minus, 1minus, 6
0minus, 5
10
227
The table above shows the values of polynomial function g at several different values of x.
Because g, left parenthesis, 0, right parenthesis, equals, minus, 5, x
a factor of g, left parenthesis, x, right parenthesis.
Because g, left parenthesis, 1, right parenthesis, equals, 0, x, minus, 1
a factor of g, left parenthesis, x, right parenthesis.
When g, left parenthesis, x, right parenthesis is divided by x, minus, 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 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

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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 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
vertical asymptote
exists where the function is undefined. Let's go back to our example: what happens to the value of f, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, x, minus, 2, end fraction as x approaches 2?
  • 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.
The rational function y=1/(x-2) is graphed in the xy-plane. For x<2, as x decreases, y approaches 0; as x approaches 2, y approaches negative infinity. For x>2, as x increases, y approaches 0. As x approaches 2, y approaches positive infinity. A vertical dashed line is drawn at x=2.

Try it!

try: determine how a rational function behaves
f, left parenthesis, x, right parenthesis, equals, minus, start fraction, 1, divided by, 3, x, plus, 12, end fraction
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 x, y-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.


Your turn!

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 x, y-plane?
Choose 1 answer:
Choose 1 answer:


Practice: write a function using roots
In the x, y-plane, the graph of function g has x-intercepts at minus, 2, 2, and 7. Which of the following could define f ?
Choose 1 answer:
Choose 1 answer:


Practice: Apply the remainder theorem
For a polynomial h, left parenthesis, t, right parenthesis, the value of h, left parenthesis, minus, 1, right parenthesis is minus, 7. Which of the following must be true about h, left parenthesis, t, right parenthesis ?
Choose 1 answer:
Choose 1 answer:


practice: identify where a rational function is undefined
If f, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, x, squared, minus, 7, x, plus, 12, end fraction is undefined, what is a possible value of x ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


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?

  • blobby green style avatar for user aarohan567
    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)
    Default Khan Academy avatar avatar for user