1. Function Characteristics – Increasing/Decreasing Intervals
AII.7 - The student will investigate and analyze functions algebraically and graphically. Key concepts include a) domain and range, including limited and discontinuous domains and ranges; b) zeros; c) x- and y-intercepts; d) intervals in which a function is increasing or decreasing; e) asymptotes; f) end behavior; g) inverse of a function; and h) composition of multiple functions. Graphing calculators will be used as a tool to assist in investigation of functions.

2. Increasing/Decreasing Intervals
How would you describe the shape of these graphs?
f(x)
g(x)

3. How would you describe the shape of these graphs?
f(x) is a quadratic function
Changes directions once
Shaped like a U (parabola)
DECREASING
INCREASING
f(x)

4. How would you describe the shape of these graphs?
g(x) is a quartic function.
3 directional changes
Shape of a W
DECREASING
INCREASING
g(x)

5. Increasing/Decreasing Intervals
One characteristic we can use to describe a function is the intervals over which the function increases and decreases (Remember, we always ‘read’ a graph from left to right.)
A function is increasing if it rises from left to right, (meaning the function has a positive slope).
Thus a function decreases if it falls from left to right (has a negative slope).

6. Increasing/Decreasing Intervals
Where does f(x) stop decreasing and start increasing?
As we approach that position from the left f(x) is decreasing.
To the right, f(x) is increasing.
DECREASING
INCREASING

7. Increasing/Decreasing Intervals
Notice, we referenced the positions the left and right of our change in direction. Thus we base our intervals on domain, or x, values.
The function changes directions at x = 2.
It decreases to the left of x = 2, so we say the function decreases over the interval x < 2.
DECREASING
INCREASING
x < 2
x = 2

8. Increasing/Decreasing Intervals
f(x) increases to the right of x = 2, so we say the function increases over the interval x > 2.
Recap:
f(x) decreases over the interval x < 2 and
f(x) increases over the interval x > 2
DECREASING
INCREASING
x < 2
x > 2
x = 2

9. Increasing/Decreasing Intervals
Where does g(x) change directions?
x = -4, 0, 4
g(x) is decreasing to the left of x = -4 and between x = 0 and x = 4.
The intervals over which g(x) is decreasing are: x < -4, 0 < x < 4.
DECREASING
DECREASING

10. Increasing/Decreasing Intervals
g(x) is increasing between x = -4 and x = 0 and to the right of x = 4.
The intervals over which g(x) is increasing are: < x < 0, x > 4
Recap:
g(x) decreases over the interval x < -4, 0 < x < 4
g(x) increases over the interval -4 < x < 0, x > 4
DECREASING
INCREASING
DECREASING
INCREASING

11. Determine the interval(s) over which each function increases/decreases.
h(x)
j(x)
Increases: x < -1, x > 3
Decreases: -1 < x < 3
Increases: 𝑥∈ℛ (though the slope changes, this function is always increasing)