English | Español

Try our Free Online Math Solver!

Online Math Solver

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


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.

Leading Coefficients

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.

Prev Next