Defining the Absolute Value Function
•
To investigate the concept of absolute value
•
To construct and interpret graphs of
absolute-value functions.
•
To learn the piecewise definition of the
absolute value function
•
To evaluate expressions containing absolute
value.
LESSON 7.5
Cal and Al both live 3.2 miles from school, but in
opposite directions.
If you assign the number 0 to the school, you can
show that Cal and Al live in opposite directions
from it by assigning +3.2 to AI's house and -3.2 to
Cal's house. For both Cal and Al, the distance
from school is 3.2 miles.
Distance is never negative, but distance is often found by
subtracting, which sometimes gives a negative result.
For this reason, it is useful to have a function that turns opposite
numbers into the same positive number.
The absolute value of a number is its size, or magnitude,
regardless of whether the number is positive or negative.
A way to visualize the absolute value of a number is to picture
its distance zero on a number line.
A number and its opposite have the same magnitude, or
absolute value, because they're the same distance from zero.
For example, 3.2 and -3.2 are both 3.2 units from zero, so they
both have an absolute value |3.2|=3.2 and |-3.2|=3.2 .
Evaluate each expression.
5  5
2 17  3
6  6
8
4
4
0
Absolute value is useful for answering questions
about distance, pulse rates, scores, and other
data values that lie on opposite sides of a central
point 5 mean.
The difference of a data point from the mean of its
data set is calla: deviation from the mean.
In this investigation you will learn how the
absolute-value function tells how much an item
of data or a whole set of data deviates from the
mean.
Collect at least 10 pulse rates from your class.
Record the data in a table and enter the
numbers into list L1 on your calculator.
Calculate the mean of the pulse rate using the
graphing calculator.
Find the difference between each data
point and the mean of the data in list LI.
Use L2 to find the difference between the
values in L1 and the mean of the data.
Record these numbers in a second row of
your table. What do these numbers
represent?
Make a dot plot of the list L1 data and
note the distance from each data point
to the mean.
Record your results in a third row of
your table and enter them into list L3.
How are these entries different from
those in list L2? How are they alike?
Next, plot points in the form (L2, L3).
What numbers are in the domain and
range of the graph?
Use the trace function on your calculator
and use the arrow keys to step through
the data points. Which input numbers
are unchanged as output numbers?
Which input numbers are changed, and
how?
Does it make sense to connect these
points with a continuous graph? Why or
why not?
How does this graph compare to the graph of
Y1 = abs(x) on your calculator?
Find the mean of the deviations stored in list
L2. Compare it to the mean of the distances
stored in list L3. Which do you think is a
better measure of the spread of the data?
In your own words, write the rule for the
function you graphed in Step 8.
What number is output as y when the input,
x, is positive or equal to zero?
What number is output when x is negative?
How can you use operations to change these
numbers?
The absolute-value function is defined
by two rules.
if x  0
x
x 
 x if x  0

Solve each equation or inequality
symbolically.
x  7  12
x  2  7  12
x  2  7  12