5. absolute value , sq , sq roots
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Absolute Value, Square Roots, and Quadratic Equations
Absolute value can be defined as the
from zero on a
number line.
| 3 | is the point that is 3 units away from
distance
Zero
Geometrically:
| x | = the distance of x from zero.
Algebraically:
x, for x > 0 | x | =
-x for x < 0
Find | x | when x has the following values: -2, 5, .06, 0 , -⅞
x -2 5 .06 0 -⅞
| x |
Solve: | x - 3| = 5 can be read as "x is 5 units from 3 in
either direction"
Algebraically: | x - 3| = 5 means
x -3 = or x -3 =
Geometrically:
either
-
Graph the points
Use your calculator to complete the graph: The Absolute Value function is located in:
Math> NUM>1: abs(
Is f : x | x | a function?
Create a table of values:
x | x |
0
4
-4
9
-9
16
-16
The square root or sign
stands for the square root of a real number.
4 stands for the positive value or
Square Root
radical
The
positive
2
To find the negative square root you need to write: - 4
which equals -2
2xGraph the points
Use your calculator to complete the graph
Complete the following table:
x
0
4
-4
9
-9
16
-16
8
6
4
2
-2
-4
-6
-10 -5 5 10
f x = x
y x
8
6
4
2
-2
-4
-6
-10 -5 5 10
f x = x
2y x
2x
What conclusion can you draw about the solutions to
and | x | ?
Absolute Value- Square Root Theorem
For all real numbers x, = | x |
2xWhat does this mean?
The square root of a squared expression is the
as the absolute value of the expressionsame
Example 1 : Solve the following -
2x = 72
| x | = 72
therefore: x = 72 or x = - 72
x = or x = -
2 72x
Example 2: Solve the following -
34 2 12y
Example 3: Solve the following -
23 18x
Example 4:Example 4: A square and a triangle have the same area. The triangle has base 7 cm. And altitude 6 cm. What is the length of a side of the square?
ClosureClosure• The |x| is
– Geometrically:– Algebraically:
• What can be said about
• How do you solve |2y + 4| = 16?
• How do you solve 4a2 = 28?
2x and x
17
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