A polynomial function is a function of the form:
\$\$f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\$\$
where \$a_n, a_{n-1}, \ldots, a_1, a_0\$ are constants, \$a_n \neq 0\$, and \$n\$ is a non-negative integer.
The degree of a polynomial is the highest power of \$x\$ with a non-zero coefficient.
Graphing Polynomial Functions
The graph of a polynomial function is a smooth curve with no sharp corners or breaks. The shape of the graph is determined by:
- The degree of the polynomial.
- The leading coefficient (the coefficient of the highest power of \$x\$).
- The roots (or zeros) of the polynomial.
- The multiplicity of the roots.
The multiplicity of a root is the number of times the root appears in the factorization of the polynomial.
Steps to Graph a Polynomial Function
- Find the roots of the polynomial by factoring or using other methods.
- Determine the multiplicity of each root.
- Find the y-intercept by evaluating the polynomial at \$x = 0\$.
- Determine the end behavior of the graph using the degree and leading coefficient.
- Plot the roots and y-intercept on the coordinate plane.
- Sketch the graph using the information from the previous steps.
When sketching the graph, remember that:
- If the multiplicity of a root is odd, the graph crosses the x-axis at that root.
- If the multiplicity of a root is even, the graph touches the x-axis and bounces off at that root.
Example 1: Graphing a Polynomial with Distinct Roots
Graph the polynomial \$f(x) = (x + 1)(x - 1)(x - 3)\$.
- Find the roots: The roots are \$x = -1\$, \$x = 1\$, and \$x = 3\$.
- Determine the multiplicity: Each root has multiplicity 1 (odd).
- Find the y-intercept: \$f(0) = (0 + 1)(0 - 1)(0 - 3) = 3\$. So the y-intercept is \$(0, 3)\$.
- Determine the end behavior: The degree is 3 (odd) and the leading coefficient is 1 (positive), so the graph starts from the bottom left and ends at the top right.
- Plot the points: Plot the roots \$(-1, 0)\$, \$(1, 0)\$, \$(3, 0)\$, and the y-intercept \$(0, 3)\$.
- Sketch the graph: The graph crosses the x-axis at each root and follows the end behavior.
The graph of \$f(x) = (x + 1)(x - 1)(x - 3)\$ is shown below.
Example 2: Graphing a Polynomial with Double Roots
Graph the polynomial \$f(x) = (x - 2)^2(x + 3)\$.
- Find the roots: The roots are \$x = 2\$ and \$x = -3\$.
- Determine the multiplicity: The root \$x = 2\$ has multiplicity 2 (even), and the root \$x = -3\$ has multiplicity 1 (odd).
- Find the y-intercept: \$f(0) = (0 - 2)^2(0 + 3) = 12\$. So the y-intercept is \$(0, 12)\$.
- Determine the end behavior: The degree is 3 (odd) and the leading coefficient is 1 (positive), so the graph starts from the bottom left and ends at the top right.
- Plot the points: Plot the roots \$(2, 0)\$, \$(-3, 0)\$, and the y-intercept \$(0, 12)\$.
- Sketch the graph: The graph crosses the x-axis at \$x = -3\$ and bounces off the x-axis at \$x = 2\$.
The graph of \$f(x) = (x - 2)^2(x + 3)\$ is shown below.
