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# Topic 13 - Polynomial Functions

Symmetry
• A polynomial function with all even exponents is an even function and symmetric about the y-axis.
• 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 .

x-Intercepts

The roots of a polynomial function correspond to the x- intercepts of its graph.
• If a root has odd multiplicity , the graph crosses the x-axis at the corresponding x-intercept.
• If a root has even multiplicity , the graph touches but does not cross the x-axis at the corresponding x-intercept.

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 x-intercepts.
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 x-intercepts are (−3, 0), (1, 0), and (2, 0).
The y-intercept is (0, 6).

3. The graph crosses the x-axis at each x-intercept.

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 x-intercepts are (−3, 0), (0, 0), and (3, 0).
The y-intercept is (0, 0).

3. The graph crosses the x-axis at each x-intercept.

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 y-axis.

The x-intercepts are
The y-intercept is (0, 0).

3. The graph crosses the x-axis at
The graph touches but does not cross the x-axis 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.

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