Algebra II, Unit 1
Activity
Algebra II: A Functions Approach
Unit I: Graphical Analysis of Functions
Increasing and Decreasing Functions
Increasing and Decreasing
We have learned that the turning points of a function correspond to local
maxima and local minima. At these turning points, functions change from
increasing to decreasing or from decreasing to increasing. In order to
understand the concept of increasing and decreasing, let’s look at the graph of a
function.
If we start looking at the graph from the left and move to the right, we see a local
minimum at (-1, -2). Notice that as we approach (-1, -2) from the left, the yvalues are getting smaller and smaller. Then, as we leave (-1, -2), the y-values are
no longer getting smaller. So on the interval    x  1 the function is
decreasing.
Similarly, if we move from (-1, -2) to the local maximum, (0, -1), we see that the yvalues of the function are increasing. Thus we can say that on the interval
1  x  0 the function is increasing.
Can you find another interval over which this function is decreasing? If so,
identify it.
Can you find another interval over which the function is increasing? If so,
identify it.
HCPSS Algebra II
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Algebra II, Unit 1
Activity
It is possible for a function to have an interval over which the function is neither
increasing nor decreasing. In the graph below, the function is neither increasing
nor decreasing over the interval    x  1.
Can you find another interval over which this function is decreasing? If so,
identify it.
Can you find another interval over which the function is increasing? If so,
identify it.
Practice Exercises
For each of the graphs below:
 Identify any maximum and/or minimum values of the function. Indicate
whether each is absolute or local.
 Identify the interval(s) over which the function is increasing and the interval(s)
over which the function is decreasing
1.
2.
Maximum:
Minimum:
Increasing:
Decreasing:
3.
Maximum:
Minimum:
Increasing:
Decreasing:
4.
Maximum:
Minimum:
Increasing:
Decreasing:
HCPSS Algebra II
Maximum:
Minimum:
Increasing:
Decreasing:
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Algebra II, Unit 1
Activity
Practice Exercises Part 2


Using a graphing calculator, graph the function. Sketch the graph on the
grid provided. Label the maximum and minimum.
Indicate the range, minimum and maximum values, and the interval(s)
over which the function is increasing and/or decreasing.
1.
f ( x)  x 2  4 x  5
Range:
Maximum:
Minimum:
Increasing:
Decreasing:
D: [0, 4]
2.
f ( x)  x 2  x  3
Range:
Maximum:
Minimum:
Increasing:
Decreasing:
D: (-5, 5)
3.
f ( x)  x 3  4 x 2
Range:
Maximum:
Minimum:
Increasing:
Decreasing:
D: [-1, 3]
4.
f ( x)  16  x 2
Range:
Maximum:
Minimum:
Increasing:
Decreasing:
D: (-4, 4)
HCPSS Algebra II
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Algebra II, Unit 1
Activity
Alcohol Concentration Problem
The function f ( x)  .015x3  1.058x gives the approximate alcohol concentration
(in tenths of a percent) in an average person’s bloodstream x hours after drinking
8 ounces of 100 proof whiskey. (Advanced Algebra, Scott Foresman 1993) Using
a graphing calculator and an appropriate window, sketch the complete graph
below and label as y1 .
WINDOW:
Xmin=
Xmax=
Xscl=
Ymin=
Ymax=
Yscl=
1.
For what values of x does the graph make sense for this situation?
2.
After drinking 8 ounces of 100 proof whiskey, how long does it take for
the average person’s blood alcohol concentration to return to zero?
3.
At what time was the concentration of alcohol in the blood the greatest?
4.
A person is considered to be legally drunk when the blood alcohol is 8
hundredths (0.08). Let y2 = 0.08 and restrict the domain to represent the
problem situation. Sketch the graphs of y1 and y2 .
WINDOW:
Xmin=
Xmax=
Xscl=
Ymin=
Ymax=
Yscl=
5.
How long is the average person legally drunk after drinking 8 ounces of
100 proof whiskey?
HCPSS Algebra II
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