Try our Free Online Math Solver!

Topic 13  Polynomial Functions
Symmetry
• A polynomial function with all even exponents is an even function and
symmetric about the yaxis.
• A polynomial function with all odd exponents is an odd function and symmetric
about the origin.
• A quadratic function is symmetric about its axis of symmetry x = −b/2a .
xIntercepts
The roots of a polynomial function correspond to the
x intercepts of its graph.
• If a root has odd multiplicity , the graph crosses the xaxis at the
corresponding xintercept.
• If a root has even multiplicity , the graph touches but does not cross the
xaxis at the corresponding xintercept.
Consider the polynomial
• For n odd and
• For n odd and
• For n odd and
• For n odd and
Strategy for Graphing Polynomial Functions
1. Check for symmetry.
2. Find the x intercepts and the y intercept.
3. Determine the behavior of the graph at the xintercepts.
4. Determine the behavior of the graph as and
as
5. Calculate ordered pairs if necessary.
6. Draw a smooth curve connecting the points.
Example 1
Graph the polynomial function f(x) = x^3 − 7x + 6.
1. This is not an even, odd, or quadratic function . The
graph has no symmetry.
The xintercepts are (−3, 0), (1, 0), and (2, 0).
The yintercept is (0, 6).
3. The graph crosses the xaxis at each xintercept.
5. f(−2) = (−2)^3 − 7(−2) + 6 = 12
The ordered pair (−2, 12) is a point on the graph.
Example 2
Graph the polynomial function g (x) = −x^3 + 9x.
1. This is an odd function. The graph is symmetric about the origin.
The xintercepts are (−3, 0), (0, 0), and (3, 0).
The yintercept is (0, 0).
3. The graph crosses the xaxis at each xintercept.
The ordered pairs (1, 8) and (−1,−8) are points on the graph.
Example 3
Graph the polynomial function h (x) = −2x^4 + 6x^2.
1. This is an even function. The graph is symmetric about the yaxis.
The xintercepts are
The yintercept is (0, 0).
3. The graph crosses the xaxis at
The graph touches but does not cross the xaxis at (0, 0).
5. h(1) = −2 + 6 = 4
h(−1) = h(1) = 4
The ordered pairs (1, 4) and (−1, 4) are points on the graph.
Prev  Next 