Unit 2 Day 6: Characteristics of

FunctionsEssential Questions: What is the maximum and minimum of a graph? How do the degree and

the leading coefficient determine end behavior?

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

Finding a Maximum and Minimum

The maximum in this case is the highest the shot put went,

which is about 9 ft.

The maximum is the highest output (y-value) that a function produces. The input (x-value)

doesn’t matter.

10

10

-10

-10

The minimum in this case is when the shot

put hit the ground, which is at 0 ft.

The minimum is the lowest output (y-value) that a function produces. The input (x-value)

doesn’t matter.

Finding a Maximum and Minimum

10

10

-10

-10

(not a max or min)

For some functions there is no minimum or maximum output value.

No Minimum or Maximum

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.

Positive

Example 1f(x) = x2

Degree:

Leading Coefficient:

End Behavior:Up Up

Even

Example 2f(x) = -x2

Degree:

Leading Coefficient:

End Behavior:

EvenNegative

Down Down

Example 3f(x) = x3

Degree:

Leading Coefficient:

End Behavior:

Odd

Positive

Down Up

Example 4f(x) = -x3

Degree:

Leading Coefficient:

End Behavior:

OddNegative

Up Down

Up Down

Up Up

Down Up

Down Down

Example 5Give 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

Exponential Growth and Decay

• Growth: function increases rapidly as x increases

• Decay: function decreases rapidly as x increases

Asymptotes

• 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.

f(x) = 2x

Review Game• http://www.slideshare.net/RebeckaPeterson/e

nd-behavior-game-11721871

SummaryEssential 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.