1. Unit 2 Day 6: Characteristics of Functions
Essential Questions: What is the maximum and minimum of a graph? How do the degree and the leading coefficient determine end behavior?

2. Vocabulary
Minimum: the lowest output (y-value) in a function. (lowest point on a graph)
Maximum: the highest output (y-value) in a function. (highest point on a graph)
End Behavior: the behavior of the graph as x approaches positive infinity or negative infinity
Degree: the highest exponent.
Leading Coefficient: the number in
front of the variable with the
highest exponent.
Asymptote: A line that a graphed function approaches as the value of x gets very large or very small.
Ex:
Degree: x7 7
2x3 + 2x6 6
-3x23 23

3. Finding a Maximum and Minimum
The maximum is the highest output (y-value) that a function produces. The input (x-value) doesn’t matter. 10
The maximum in this case is the highest the shot put went, which is about 9 ft.
-10 10
-10

4. Finding a Maximum and Minimum
The minimum is the lowest output (y-value) that a function produces. The input (x-value) doesn’t matter. 10
The minimum in this case is when the shot put hit the ground, which is at 0 ft.
-10 10
-10

5. For some functions there is no minimum or maximum output value.
(not a max or min)

6. End Behavior
Remember, end behavior is the behavior of the graph as x approaches positive infinity or negative infinity.
By knowing the degree and leading coefficient we can identify the end behavior of graphs.

7. Example 1 f(x) = x2 Degree: Leading Coefficient: End Behavior: Even
Positive
Up Up

8. Example 2 f(x) = -x2 Degree: Leading Coefficient: End Behavior: Even
Negative
Down Down

9. Example 3 f(x) = x3 Degree: Leading Coefficient: End Behavior: Odd
Positive
Down Up

10. Example 4 f(x) = -x3 Degree: Leading Coefficient: End Behavior: Odd
Negative
Up Down

11. Example 5 Give the end behavior: a. f(x) = -2x3 + 5x - 9
b. f(x) = 4x4 - 2x2 + 6x - 3
c. f(x) = 4x5 - 3x2 + 2x
d. f(x) = -3x4 + 2x3 - x2 + 3x - 4
Up Down
Up Up
Down Up
Down Down

12. Exponential Growth and Decay
Growth: function increases rapidly as x increases
Decay: function decreases rapidly as x increases

13. Asymptotes
f(x) = 2x
Line will get closer and closer to the x-axis but never reaches it because 2x cannot be zero.
Imagine a kitten that is 5 feet away from a box. Every second the kitten moves halfway closer to the box. The kitten would never reach the box because each time it is going halfway but it would continue to get closer.

14. Review Game
behavior-game

15. Summary
Essential Questions: What is the maximum and minimum of a graph? How do the degree and the leading coefficient determine end behavior?
Take 1 minute to write 2 sentences answering the essential questions.