Graphing Polynomial FunctionsWe will now use the strategies from the previous sections to

sketch the graphs of higher order polynomials.

Graphing Polynomial FunctionsWe will now use the strategies from the previous sections to

sketch the graphs of higher order polynomials.

Follow these steps and the process is quite easy…

Graphing Polynomial FunctionsWe will now use the strategies from the previous sections to

sketch the graphs of higher order polynomials.

Follow these steps and the process is quite easy…

1. Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function.

Graphing Polynomial FunctionsWe will now use the strategies from the previous sections to

sketch the graphs of higher order polynomials.

Follow these steps and the process is quite easy…

1. Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function.

2. Plot the roots on the graph

Graphing Polynomial FunctionsWe will now use the strategies from the previous sections to

sketch the graphs of higher order polynomials.

Follow these steps and the process is quite easy…

1. Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function.

2. Plot the roots on the graph3. Create a test table using the roots of the function

Graphing Polynomial FunctionsWe will now use the strategies from the previous sections to

sketch the graphs of higher order polynomials.

Follow these steps and the process is quite easy…

1. Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function.

2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.

Graphing Polynomial FunctionsWe will now use the strategies from the previous sections to

sketch the graphs of higher order polynomials.

Follow these steps and the process is quite easy…

1. Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function.

2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

Then, using synthetic division…

1 1 6 3 - 10

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

Then, using synthetic division…

1 1 6 3 - 10

+1

1

7

7

10

10

0

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

Then, using synthetic division…

1 1 6 3 - 10

+1

1

7

7

10

10

0

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

The roots of the function are the x – values that create a zero…

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

- The intervals are set by the roots

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

- The test value will be some “x” within the interval or close to the root…

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

- The test value will be some “x” within the interval or close to the root…

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

- Now plug each test value into the function…

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

- The point is off the graph, no need to plot extreme values…we will use arrows to show the “ends” of the graph

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Plot the point…

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions1. Factor completely

2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Plot the point…

Graphing Polynomial Functions1. Factor completely

2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions1. Factor completely

2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

- The point is off the graph, no need to plot extreme values…we will use arrows to show the “ends” of the graph

Graphing Polynomial Functions1. Factor completely

2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Now sketch in the rest of the graph using smooth curves…

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Then use synthetic division…

1 -31 -1

3

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Then use synthetic division…

1 -31 -1

3

+

1

1

-2

-2

-3

-3

0

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Then use synthetic division…

1 -31 -1

3

+

1

1

-2

-2

-3

-3

0

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

The roots of the function are the x – values that create a zero…

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

The roots of the function are the x – values that create a zero…

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

The roots of the function are the x – values that create a zero…

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

- The intervals are set by the roots

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

- The test value will be some “x” within the interval or close to the root…

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

- The test value will be some “x” within the interval or close to the root…

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

- Now plug each test value into the function…

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

- The point is off the graph, no need to plot extreme values…we will use arrows to show the “ends” of the graph

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Plot the point…

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Plot the point…

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

- The point is off the graph, no need to plot extreme values…we will use arrows to show the “ends” of the graph

Graphing Polynomial Functions

1. Factor completely2. Plot the roots on the graph3. Create a test table using the roots of the function4. Test value(s) in each interval and plot the points.5. Sketch the function

Now sketch in the rest of the graph using smooth curves…