A polynomial function is a function of the form:
\$\$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\$\$
where \$a_n, a_{n-1}, \ldots, a_1, a_0\$ are real coefficients and \$n\$ is a non-negative integer.
NoteThe degreeof a polynomial function is the highest power of \$x\$ with a non-zero coefficient.
Graphing Polynomial Functions
To graph a polynomial function, we need to:
- Identify the degree and leading coefficient.
- Find the x-intercepts and y-intercept.
- Determine the end behavior.
- Plot additional points if necessary.
- Sketch the graph.
The leading coefficientis the coefficient of the term with the highest degree.
Finding the x-intercepts
The x-intercepts of a polynomial function are the values of \$x\$ for which \$f(x) = 0\$.
- Set the polynomial equal to zero.
- Solve for \$x\$.
If the polynomial is in factored form, the x-interceptsare the rootsof the polynomial.
Finding the y-intercept
The y-intercept of a polynomial function is the value of \$f(x)\$ when \$x = 0\$.
- Substitute \$x = 0\$ into the polynomial.
- Solve for \$f(0)\$.
Determining the End Behavior
The end behavior of a polynomial function is determined by its degree and leading coefficient.
- If the degree is even and the leading coefficient is positive, the graph rises on both ends.
- If the degree is even and the leading coefficient is negative, the graph falls on both ends.
- If the degree is odd and the leading coefficient is positive, the graph falls on the left and rises on the right.
- If the degree is odd and the leading coefficient is negative, the graph rises on the left and falls on the right.
The end behaviorof a polynomial function is determined by its degreeand leading coefficient.
Self review1. Sketch the graph of \$f(x) = (x - 1)(x + 2)^2\$. 2. Identify the x-intercepts, y-intercept, and end behavior. 3. Compare your graph with a graphing calculator.
