What can be read off a graph: values, intercepts, domain and range, symmetry, where the function rises and falls, its extrema, and its average rate of change.
Points, intercepts, domain and range
Find ;.
;.
Is pos/neg? Positive. :negative.
For what is ?;. (That’s finding all the -intercepts.)
For what is ? and , so on .
For what is ? and , so on .
Domain: ,.
Range: ,.
-intercepts: ,,.
-intercepts: .
How many times does cross the graph? Twice: once at and once at .
For what does ? and , so .
For what does ? and , so .
(a) ,.,.
(b) ,.,.
(c) ,.,.
(d) ,.,.
The same questions for a function given by a formula:
Example 8.3.1.
Show solution
;.
: find .
Check: ✓, so the point is .
Domain: ,.
-int: set :.
-int: set :.
Even and odd functions
Definition 8.3.2.
Even: symmetric about the -axis: .
Odd: symmetric about the origin: .
(a) Even: reflect across the -axis even. ,.
(b) Odd: rotate odd. ,.
(c) Neither. ,.
(d) Odd: rotate .,.
Note.Even make opposite inputs have the same outputs (ex: and ).
Odd make opposite inputs have the opposite outputs (ex: and ).
Example 8.3.3.
Show solution
so :even function.
Example 8.3.4.
Show solution
even function.
Example 8.3.5.
Show solution
odd function.
Example 8.3.6.
Show solution
Neither. for an input like , but then is a complex # / DNE, so can’t equal or .
Example 8.3.7.
Show solution
But: , which is not either. Thus neither. With numbers:
Example 8.3.8.
Show solution
so is an odd function.
Increasing and decreasing
Definition 8.3.9.Increasing: for ,. From left to right, the graph climbs.
Decreasing: for ,. From left to right, the graph falls.
Note. These are all given by open intervals on the -axis.
No ; only .
No ; only .
(a) Falling on , rising on , falling on , rising on .
INC: . DEC: .
These are not coordinates: they are intervals.
(b) INC: . DEC: .
(c) INC: . DEC: .
(d) INC: . DEC: N/A (no fall).
Extrema
Definition 8.3.10 (Local/relative max). On an open interval around , if , then is a local max.
It is where the function changes from increasing to decreasing.
Definition 8.3.11 (Local/relative min). On an open interval around , if , then is a local min.
It is where the function changes from decreasing to increasing.
Note. Local min and max never occur at endpoints.
Definition 8.3.12.Absolute maximum: the highest output on the graph.
Absolute minimum: the lowest output on the graph.
Note. Every continuous function on a closed interval will have an abs. max and an abs. min.
Abs. max and min can occur at endpoints.
Graph
LMIN
LMAX
AMIN
AMAX
(a)
at
at
at
at
(b) (not a cont. function)
none
none
at
none
For (b): remember LMIN and LMAX cannot be at endpoints.
Graph
LMIN
LMAX
AMIN
AMAX
(c)
at
at
at
at
(d)
at
at
none
at
(e)
at and at
at
at
none
(f)
none
none
none
none
For (d): AMIN is none because the graph goes to , but is not a value it reaches.