characteristics of functions x-intercept and y-intercept maximum and minimum of a function odd and...
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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
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B
C
D
A
B
C
D
Khan Academy
Comparing and interpreting functions