1. Graphing Polynomial Functions
Sophia Zhang
Graphing Polynomial Functions
Section 7-2

2. What You Will Learn
Graph polynomial functions and locate their real zeros
Find the maxima and minima of polynomial functions

3. How To Graph Polynomial Functions
Make a table of values to find points and connect them into a smooth curve.
Knowing the end behavior* will help you sketch the general shape of the graph.
*Remember ? If the degree of the polynomial is even, then both ends will point the same direction, and if the degree is odd, then the ends will point opposite directions !

4. Graph a Polynomial Function
Example 1
Graph by making a table of values. x
f(x) -2
-1.5
-1.3 -1 1 -6
1.5
-6.6
This is an even-degree polynomial with a positive coefficient, so you know that the graph will face up with both ends going the same direction.
NOTICE that the graph intersects the x axis at 4 points, which means there are 4 REAL zeros of this function!

5. How do you find Zeros? By using the Location Principle :
The Location Principle states that there will be a zero, or x-intercept, for every time pairs of x values with corresponding y values change signs ( from + to – or vice versa) x
f(x)
For Example... -2 6
Change in Signs
Thus, there will be a zero between x=-2 and x=-1 and x=0 and x=1 -1 -2 -2
Change in Signs 1 7

6. Locate Zeros of a Function
Example 2
Locate Zeros of a Function
Determine consecutive values of x between which each real zero of the function is located. x
f(x)
1) Make a table of values -1 -7
Change in Signs
(Since this is a third-degree polynomial function, it will have either 1, 2, or 3 real zeros). 2 1 1
Change in Signs 2 -4
2 ) Look at the values of f(x) to locate the zeros. 3 -7 4 -2
Change in Signs 5 17
The changes in sign show that there are zeros between x = -1 and x = 0, between x = 1 and x = 2, and between x = 4 and x = 5.

7. Maximum and Minimum Points
Relative Maximum: A point on the graph where no other surrounding points have a greater y-coordinate.
Relative Minimum : A point on the graph where no other surrounding points have a lesser y-coordinate.
* Called the “Turning Points”
Relative Maximum
Relative Minimum
A function with degree n has at MOST n-1 turning points.
(Plurals of Maximum and Minimum are
Maxima Minima)
&

8. Maximum and Minimum Points
Example 3
Maximum and Minimum Points
Graph Estimate the x-coordinates at which the relative maxima and relative minima occur.
Make a table of values & graph. x
f(x) -2
-15
Zero between x = -2 and x = -1. -1 1
5 is a Relative Maximum because it’s surrounded by 1 and 3 which BOTH have a lesser value than it. 5 1 3 2 1
1 is a Relative Minimum because it’s surrounded by 3 and 5 which BOTH have a greater value than it. 3 5

9. (Continued)
Example 3
Looking at the table and the graph, you can conclude that…
The values of f(x) change signs between x = -2 and x = -1, indicating a zero of the function.
The value of f(x) at x = 0 is greater than the surrounding points, so it is a relative maximum.
The value of f(x) at x = 2 is less than the surrounding points, so it is a relative minimum.

10. Polynomial Functions can reveal trends in real-world data.
For Example…
The Consumer Price Index (CPI) gives the relative price for a fixed set of goods and services. The CPI from September, 2000 to July, 2001 is shown in the graph.
179
178
177
176
175
174
173
Describe the turning points on the graph.
The Maximum is the 9th month and the Minimum is the 3rd month.
Consumer Price Index
If the graph were modeled by a polynomial equation, what is the least degree the equation could have?
(1 more than turning points!)
Months Since September, 2000

11. Homework
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