1. Characteristics of Functions

2. x and y intercepts
The x intercept is where the graph crosses the x axis.
The y intercept is where the graph
crosses the y axis.
x intercepts

3. x and y intercepts
The x intercept is a point in the equation where the y-value is zero.
Example:
Find the x intercepts of
Set y =0, solve for x.
x intercepts:

4. x and y intercepts
The y intercept is a point in the equation where the x-value is zero.
Example:
Find the y intercepts of
Set x =0, solve for y.
y intercepts:
(0,-2) (0,2)

5. Increasing, Decreasing, Constant Intervals
A function f is increasing on an interval if as x increases, then f(x) increases.
A function f is decreasing on an interval if as x increases, then f(x) decreases.
A function f is constant on an interval if as x increases, then f(x) remains the same.

6. Increasing, Decreasing, Constant Intervals
A function f is increasing on an interval if as x increases, then f(x) increases.
A function f is decreasing on an interval if as x increases, then f(x) decreases.
f(x) is decreasing
in the interval .
vertex (1.5,-2)
f(x) is increasing
in the interval .

7. Increasing, Decreasing, Constant Intervals
A function f is increasing on an interval if as x increases,
then f(x) increases.
A function f is decreasing on an interval if as x increases, then f(x) decreases.
A function f is constant on an interval if as x increases, then f(x) remains the same.
Find the interval(s) over which the interval is increasing, decreasing and constant?
Answer Now

8. Increasing, Decreasing, Constant Intervals
A function f is increasing on an interval if as x increases,
then f(x) increases.
A function f is decreasing on an interval if as x increases, then f(x) decreases.
A function f is constant on an interval if as x increases, then f(x) remains the same.
f(x) is decreasing
in the interval
(-1,1).
f(x) is increasing
in the intervals

9. Increasing, Decreasing, Constant Intervals
A function f is increasing on an interval if as x increases,
then f(x) increases.
A function f is decreasing on an interval if as x increases, then f(x) decreases.
A function f is constant on an interval if as x increases, then f(x) remains the same.
Find the interval(s) over which the interval is increasing, decreasing and constant?
Answer Now

10. Increasing, Decreasing, Constant Intervals
A function f is increasing on an interval if as x increases,
then f(x) increases.
A function f is decreasing on an interval if as x increases, then f(x) decreases.
A function f is constant on an interval if as x increases, then f(x) remains the same.
f(x) is decreasing
over the intervals
f(x) is increasing
over the interval (3,5).
f(x) is constant
over the interval (-1,3).

11. Minimum/Maximum
A min/max is where there is a "turning point" on a graph . It is a point at which the graph changes from increasing to decreasing (it looks like a "hill"), or from decreasing to increasing (it looks like a "valley").

12. These turning points are referred to as relative (or local) maxima or minima. The designation of "relative" (or "local") tells you that this point may not be the largest (or smallest) value reached by this function. It is only a maximum (or minimum) "relative" to a small interval of the function surrounding this point.
The one, true, largest (or smallest) value reached by the entire function is called the absolute maximum (or minimum), or the global maximum (or minimum).

13. 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.

14. End Behavior of Functions
First determine whether the degree of the polynomial is even or odd.
degree = 2 so it is even
Next determine whether the leading coefficient is positive or negative.
Leading coefficient = 2 so it is positive

15. Leading Coefficient: +
END BEHAVIOR
Degree: Even
Leading Coefficient: +
End Behavior: Up Up
So we say f(x) ∞ , as x ∞ and f(x) ∞ , as x ∞

16. End Behavior: Down Down
Degree: Even
Leading Coefficient:
End Behavior: Down Down
So we say f(x) ∞ , as x ∞ and f(x) ∞ , as x ∞

17. Leading Coefficient: +
END BEHAVIOR
Degree: Odd
Leading Coefficient: +
End Behavior: Down Up
So we say f(x) ∞ , as x ∞ and f(x) ∞ , as x ∞

18. Degree: Odd Leading Coefficient: End Behavior: Up Down END BEHAVIOR
So we say f(x) ∞ , as x ∞ and f(x) ∞ , as x ∞

19. END BEHAVIOR
PRACTICE
Give the End Behavior:

20. END BEHAVIOR PRACTICE Give the End Behavior: Up Down Up Up Down Up
Down Down

21. Symmetry
Some graphs have an axis of symmetry over which the graph becomes a reflection (or mirror image) of itself.

22. Horizontal Symmetry
If a graph possesses horizontal symmetry (such as a reflection over the x-axis), the graph is not a function.
possesses both (x,y) and (x,-y)

23. Vertical Symmetry
A function graph may possess vertical symmetry (such as a reflection over the y-axis).
possesses both (x,y) and (-x,y)

24. About the origin Symmetry
A function graph may possess symmetry about the origin (0,0). This is the same as a rotation of 180º 
possesses both (x,y) and (-x,-y).

25. Rate of Change Recall formula for slope of a line through two points
For any function we could determine the slope for two points on the graph
This is the average rate of change for the function on the interval from x1 to x2

26. Rate of Change
Jasper has invested an amount of money into a savings account. The graph to the right shows the value of his investment over a period of time. What is the rate of change for the interval [1, 3]?

27. Rate of Change
Find the average rate of change of f(x) = 2x2 – 3 when x1= 2 and x2 = 4