CHARACTERISTICS OF FUNCTIONS

x-intercept and y-interceptMaximum and Minimum of a FunctionOdd and Even FunctionsEnd Behavior of Functions

WHAT DOES IT MEAN TO INTERCEPT A PASS IN FOOTBALL?

The path of the defender crosses the path of the thrown football.

In algebra, what are x- and y-intercepts?

THE X- AND Y-INTERCEPTS

The x-intercept is where the graph crosses the x-axis.

The y-coordinate is always 0.

The y-intercept is where the graph crosses the y-axis.

The x-coordinate is always 0.

(2, 0)

(0, 6)

George dropped a rock out of his flying machine and onto a bouncy trampoline. The rock’s height as a function of time, y(t), is plotted bellow.What is the significance of the t-intercept of this graph?

George dropped a rock out of his flying machine and onto a bouncy trampoline. The rock’s height as a function of time, y(t), is plotted bellow.What is the significance of the y-intercept of this graph?

Zidane wants to see how long his bike can keep moving after he stops pedaling. His velocity (in meters per second) as a function of time (in seconds), V(t) is shown below. What is the significance of the V- intercept?

If we examine a typical graph the function y = f(x), we can observe that for an interval throughout which the function is defined, that the function might be increasing, decreasing or neither.

Increasing/Decreasing Patterns:Graphs are always “read” left to right.

If the graph is going up, it is increasing. If the graph is going down, it is decreasing.

Up Up UpDown Down DownIncreasing/decreasing/increasing decreasing/increasing/decreasing

A relative extreme point ( relative maximum point or relative minimum point) of a function is a point at which its graph changes from increasing to decreasing or vice versa.

A relative maximum point is a point at which the graph changes from increasing

to decreasing.

A relative minimum point is a point at which the graph changes from

decreasing to increasing.

A

B

C

D

A

B

C

D

END BEHAVIOR OF FUNCTIONS

The end behavior of a graph describes the far left and the far right portions of the graph.

Using the leading coefficient and the degree of the polynomial, we can determine the end behaviors of the graph. This is often called the Leading Coefficient Test.

END BEHAVIOR OF FUNCTIONS

First determine whether the degree of the polynomial is even or odd.

Next determine whether the leading coefficient is positive or negative.

532)( 2 xxxf degree = 2 so it is even

532)( 2 xxxf Leading coefficient = 2 so it is positive

END BEHAVIOR

Degree: Even

Leading Coefficient: +

End Behavior: Up Up

2)( xxf

END BEHAVIOR

Degree: Even

End Behavior: Down Down

2)( xxf

Leading Coefficient:

END BEHAVIOR

Degree: Odd

Leading Coefficient: +

End Behavior: Down Up

3)( xxf

END BEHAVIOR

Degree: Odd

End Behavior: Up Down

3)( xxf

Leading Coefficient:

A

B

C

D

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

So for an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.

Even functions have y-axis Symmetry

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph.

Odd functions have origin Symmetry

A

B

C

D

A

B

C

D

A

B

C

D

Khan Academy

Comparing and interpreting functions