Polynomial and other nonlinear graphs | Lesson

What are polynomial and other nonlinear graphs?

1. Relate the factors of polynomial functions to the $x$x-intercepts of polynomial graphs
2. Apply the polynomial remainder theorem
3. Understand the behavior of basic rational functions

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

How do I identify features of graphs from polynomial functions?

Zeros of polynomials introduction

Features of polynomial graphs

Factored form and zeros

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

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$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.

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

• If $\purpleD{a}>0$start color #7854ab, a, end color #7854ab, is greater than, 0, then $y$y ultimately approaches positive infinity as $x$x increases.
• If $\purpleD{a}<0$start color #7854ab, a, end color #7854ab, is less than, 0, then $y$y ultimately approaches negative infinity as $x$x increases.
• If $\maroonD{n}$start color #ca337c, n, end color #ca337c is even, then the ends of the graph point in the same direction.
• If $\maroonD{n}$start color #ca337c, n, end color #ca337c is odd, then the ends of the graph point in different directions.

• Since $\purpleD{1}>0$start color #7854ab, 1, end color #7854ab, is greater than, 0, $y$y approaches positive infinity as $x$x increases.
• Since $\maroonD{3}$start color #ca337c, 3, end color #ca337c is odd, the other end of the graph points in the opposite direction as $x$x and $x^3$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

• If $p(a)=0$p, left parenthesis, a, right parenthesis, equals, 0, then $(a,0)$left parenthesis, a, comma, 0, right parenthesis is an $x$x-intercept, and $x-a$x, minus, a is a factor of $p(x)$p, left parenthesis, x, right parenthesis.
• If $p(a)\neq0$p, left parenthesis, a, right parenthesis, does not equal, 0, then $x-a$x, minus, a is not a factor of $p(x)$p, left parenthesis, x, right parenthesis.

Try it!

What are the features of simple rational functions?

Graphing rational functions 1

Undefined and vertical asymptotes

• For $x<2$x, is less than, 2, as $x$x approaches $2$2, the value of the denominator remains negative and gets closer and closer to $0$0. When $1$1 is divided by a very small negative number, the quotient is a very large negative number. In other words, as $x$x approaches $2$2 from the left, $y$y approaches negative infinity.
• For $x>2$x, is greater than, 2, as $x$x approaches $2$2, the value of the denominator remains positive and gets closer and closer to $0$0. When $1$1 is divided by a very small positive number, the quotient is a very large positive number. In other words, as $x$x approaches $2$2 from the right, $y$y approaches positive infinity.

Try it!

Your turn!

Things to remember

• If $\purpleD{a}>0$start color #7854ab, a, end color #7854ab, is greater than, 0, then $y$y ultimately approaches positive infinity as $x$x increases.
• If $\purpleD{a}<0$start color #7854ab, a, end color #7854ab, is less than, 0, then $y$y ultimately approaches negative infinity as $x$x increases.
• If $\maroonD{n}$start color #ca337c, n, end color #ca337c is even, then the ends of the graph point in the same direction.
• If $\maroonD{n}$start color #ca337c, n, end color #ca337c is odd, then the ends of the graph point in different directions.