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Section 2 .6: Increasing and decreasing functions.
Chapter 2: Functions, Linear equations, and inequalities
Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant
A function is increasing over an open interval provided the y-coordinates of the points in the interval get
larger, or equivalently the graph gets higher as it moves from left to right over the interval.
A function is decreasing over an open interval provided the ๐ฆ โ ๐๐๐๐๐๐๐๐๐ก๐๐ of the points in the interval
get smaller, or equivalently the graph gets lower as it moves from left to right over the interval.
A point is a local maximum point provided it is higher than any point close to it. (Technically a point is a
local maximum point if the graph changes from increasing to decreasing at that point.)
The local maximum value is the y-coordinate of the local maximum point.
A point is a local minimum point provided it is lower than any point close to it. (Technically a point is a local
minimum point if the graph changes from decreasing to increasing at that point.)
The local minimum value is the y-coordinate of the local minimum point.
Here is a visual representation of what is written
above.
Here is another, example of what is written above.
Notice this graph has a local maximum point, but it does
not have a local minimum.
Section 2 .6: Increasing and decreasing functions.
Chapter 2: Functions, Linear equations, and inequalities
For example: Use the graph of f(x) to determine:
a) interval(s) where the graph is increasing.
b) interval(s) where the graph is decreasing.
c) the coordinates of local maximum point, if any
d) the local maximum value
e) the coordinates of the local minimum point if any
f) the local minimum value
A point is a local maximum point provided it is higher
than any point close to it. (Technically a point is a local
maximum point if the graph changes from increasing to
decreasing at that point.)
The local maximum value is the y-coordinate of the
local maximum point.
d) the local maximum value is ๐ฆ = 16 when x =
A point is a local minimum point provided it is lower
than any point close to it. (Technically a point is a local
minimum point if the graph changes from decreasing to
increasing at that point.)
The local minimum value is the y-coordinate of the
local minimum point.
f) the local minimum value is ๐ฆ = โ 16 ๐คโ๐๐ ๐ฅ = โ 2.
- To determine the intervals where a graph
is increasing and decreasing: break graph
into intervals in terms of ๐ฅ, using only
round parenthesis and determine if the
graph is getting higher or lower in the
interval.
First interval: goes from the left edge of the
graph which has an ๐ฅ โ ๐๐๐๐๐๐๐๐๐ก๐ of ๐ฅ = โโ
to the point (โ 2 , 16 ) which has an ๐ฅ โ
๐๐๐๐๐๐๐๐๐ก๐ of ๐ฅ = โ 2
First interval (โโ, โ 2 )
Second interval goes from the point (โ 2 , 16 )
to the point ( 2 , โ 16 )
Second interval (โ 2 , 3 )
Third interval goes from the point
to the right edge of the graph ๐ฅ โ ๐๐๐๐๐๐๐๐๐ก๐
Third interval ( 2 , โ)
- Determine if the graph is increasing
(getting higher) or decreasing (getting
lower) in each interval.
First interval (โโ, โ 2 ) increasing
Second interval (โ 2 , 1 ) decreasing
Third interval ( 2 , โ) increasing
Answer: a) increasing (โโ, โ 2 ) โช ( 2 , โ)
b) decreasing (โ 2 , 2 )
Section 2 .6: Increasing and decreasing functions.
Chapter 2: Functions, Linear equations, and inequalities
For the table below, select whether the table represents a function that is
increasing or decreasing ,
There are two ways to determine the answer.
- plot the points
- The points are clearly getting higher. The graph is
increasing.
- look at the relative size of the numbers in the g(x)
column.
- Each of the values in the f(x) column is getting
larger. The function is increasing.
Answer: Increasing
Section 2 .6: Increasing and decreasing functions.
Chapter 2: Functions, Linear equations, and inequalities
Use A Graph to Locate the Absolute Maximum and Absolute Minimum
- The absolute maximum is the highest point over the entire domain of a function or relation.
- The absolute maximum value is the ๐ฆ โ ๐๐๐๐๐๐๐๐๐ก๐ of the absolute maximum point.
- The absolute minimum is the lowest point over the entire domain of a function or relation.
- The absolute minimum value is the ๐ฆ โ ๐๐๐๐๐๐๐๐๐ก๐ of the absolute minimum point.
Find the
- Coordinates of the absolute maximum point.
- Value of the absolute maximum
- Coordinates of the absolute minimum point
- Value of the absolute minimum
- Coordinates of the absolute maximum point. ( 2 , 2 )
- Value of the absolute maximum: absolute maximum value is ๐(๐ฅ) = 2 which occurs when ๐ฅ = 2
- Coordinates of the absolute minimum point ( 0 , โ 2 )
- Value of the absolute minimum : absolute minimum value is ๐(๐ฅ) = โ 2 which occurs when ๐ฅ = 0
Section 2 .6: Rates of change, increasing and decreasing functions.
Chapter 2: Functions, Linear equations, and inequalities
#1 โ 10 : Find the
a) interval(s) where the graph is increasing.
b) interval(s) where the graph is decreasing.
c) the coordinates of local maximum point, if any
d) the local maximum value
e) the coordinates of the local minimum point if any
f) the local minimum value
#1 โ 10 : Find the
a) interval(s) where the graph is increasing.
b) interval(s) where the graph is decreasing.
c) the coordinates of local maximum point, if any
d) the local maximum value
e) the coordinates of the local minimum point if any
f) the local minimum value
13 - 16 : Determine if the function defined in the table is increasing or decreasing
x f(x)
x g(x)
x h(x)
x k(x)
a) Coordinates of the absolute maximum point.
b) Value of the absolute maximum
c) Coordinates of the absolute minimum point
d) Value of the absolute minimum
