We can use our understanding of functions and our knowledge of the
shapes of curves to evaluate some of the key coordinates on graphs.
We usually want to label the intercepts of our graphs when we make a
sketch. The yintercept will have a zero as its first
(x) coordinate: (0, ), and any xintercept will
have a zero as its second (y) coordinate: ( ,0). To
decide on the missing coordinates we can think in the following way:
Intercepts
The coordinates of the yintercept are 


The coordinates of any xintercept(s) are 


Example
Fill in the missing coordinates:
To solve this problem we first fill in what we know without any
calculations:
The
yintercept is
and the
xintercept will be
The value
x=1 will make
, so the
xintercept is (1,0):
When two graphs intersect, the point of intersection lies on both
graphs, so the coordinates of the point of intersection will satisfy
both of the equations which led to thses graphs. We need to solve the
equations simultaneously, to find the coordinates of the
intersection.
Examples
 Fill in the missing coordinates:
We solve the equations simultaneously by setting them equal to
one another:
Thus the xcoordinates of the two points of intersection
are x=2 and x=1. To get the corresponding
ycoordinates we use either equation to find
y(1)=0 and y(2)=3. The graphs intersect at
(1,0) and (2,3):
 Fill in the missing coordinates:
The sine function completes one full cycle between x=0
and x=2,
so the righthand point has coordinates
(2,0).
The graph reaches its highest value one quarter of the way
through the cycle, at x=. Since
, the coordinates of the lefthand point are
(,1):
(See also
section 14,
especially topic 1,
common
graphs to memorize and topic 2,
key
function values, for more information on the sine function.)
Practice
Be sure you've gone through each step in the examples above before
doing these. Once you've worked them until you're sure that you
understand them, go on to the next section. There are more problems
of this type in the section test at the end of the section.
Note that you can get new practice problems by clicking the "Refresh"
button at the bottom of the practice set.
precal: 11.1  evaluation,graphs & solving
page created: Wed Jan 28 15:09:00 2015
Comments to
mathitc(at)umich(dot)edu
©20032007 Gavin LaRose, Pat Shure /
University of Michigan Math Dept. /
Regents of the University of Michigan