Graphing Polynomials

Step One: Determine End Behavior Using Lead

Coefficient TestGiven

1. If is odd: 2. If is even:a. , the graph a. , the graph falls to the left and rises to the left and rises to the right rises to the right

b. , the graph b. , the graph rises to the left and falls to the left and falls to the right falls to the right

Summary of End Behavior

+Lead Coefficient -Lead Coefficient

Even Degree

Odd Degree

Notation for End Behavior of Graphs

As As

As x gets bigger or smaller, what happens to the function value?

End Behavior of Graphs

As As

End Behavior of Graphs

As As

Step 2: Determine the Number of Turning Points

A polynomial function of degree n has at most n-1 turning points.

Find the maximum number of turns for each function.

Step 3: Find x-intercepts

1. If a root occurs an even number of times, the graph touches the x-axis there and turns around.

2. If a root occurs an odd number of times, the graph crosses the x-axis there.

3. If a root occurs more than once, the graph flattens out near that point.

Roughly sketch the graph

Even, negative lead coefficient, zeros -3, 2, and 4(x2)

Roughly sketch the graph

Odd degree, zeros -3, 0(x3), 5

Roughly sketch the graph

Odd, zeros -5, -3, -1(x2), 0(x5), 4

Roughly sketch the graph

Even, zeros -3, 0, 1(x4), 4, 5(x3)

Step 4: Find the y-intercept

Find the y-intercept by computing

Step 5: Consider the Symmetry

If the function is even, it has x-axis symmetry.If the function is odd, it has origin symmetry.If the function is neither even or odd, it has no symmetry.

Even or Odd? Symmetry?

Sketch the graph

1. End behavior As As

2. Max number of Turns_______

3. X-intercepts ______________

4. Y-intercept ___________

5. Symmetry _____________

xxxf 2)( 2

Sketch the Graph 1. End Behavior As As 2. Max number of turns ________

3. X-intercepts _____________

4. Y-intercept____________

5. Symmetry_____________

Sketch the Graph 1. End Behavior

As As 2. Max number of turns ________

3. X-intercepts _____________

4. Y-intercept____________

5. Symmetry_____________

Sketch the Graph

1. End Behavior As As 2. Max number of turns ________

3. X-intercepts _____________

4. Y-intercept____________

5. Symmetry_____________

Sketch the Graph

1. End Behavior As As 2. Max number of turns ________

3. X-intercepts _____________

4. Y-intercept____________

5. Symmetry_____________

Using a Calculator to Graph• Graph and give the required information D:________R: ________Zeros: _______Inc: _______Dec: _______as as

More Calculator Graphing

D:________R: ________Zeros: _______Inc: _______Dec: _______as as

More Calculator Graphing

D:________R: ________Zeros: _______Inc: _______Dec: _______as as