Transcript x-2
November 1st
copyright2009merrydavidson
Warm Up
1) Complete the Square to change to
Standard Form.
y
=
2(x
–
f(x) = 2x2 – 8x + 9
2) Find ALL roots using the rational root
theorem, factoring and/or quadratic
formula.
f(x) = x4 + 3x3 – 5x2 - 21x + 22
2)2 + 1
x 1, 2, 3 2i
Polynomial Inequalities
x3 + 3x2 – x – 3 < 0
dashed
Where is the
polynomial less
than 0?
(, 3) (1,1)
Let’s do this without
having to graph it first.
Polynomial Inequalities
x3 + 3x2 – x – 3 < 0
Step 1: Find the CRITICAL numbers.
Solutions are critical numbers so
solve for x =.
x = + 1, -3
Since it is “<“ and not “<‘;
x 1, 3
place on the x-axis as a hole.
x3 + 3x2 – x – 3 < 0
F
T
-3
T
-1
F
1
Step 2: Test a point on each interval
in the original inequality to see if it is
“true” or “false”.
The solution is the intervals that turn
out TRUE.
(, 3) (1,1)
Process for solving polynomial
inequalities……
• Pretend it is equal to zero and find zero’s
• Put those on the number line
<, > closed circles
<, > open circles
• Test sections into the ORIGINAL
inequality for T or F
• Write the answer in interval notation for
the true sections
You do this one……
2x3 – 3x2 – 32x > - 48
3
( 4, ) (4, )
2
Rational inequalities
x4
0
x2
Find the solution to a rational
inequality algebraically.
Solve rational inequalities (x-2) x 4
(x-2)
0
Step 1:
Rewrite fractions
x2
with factored denominator
Already done
Step 2:
Find domain restrictions. Put this
number on a number line as a “hole”. x 2
-4
x+4=0
2
x = -4
Step 3.
pretend it is = 0 and find the zero’s.
Put this value on the number line as open or closed
circles.
<, > closed circles
<, > open circles
x4
0
x2
Solve rational inequalities
(, 4] (2, )
T
-4
F
T
2
Step 4: Test intervals using the original inequality.
Step 5.
Write the solution in interval notation.
Process for solving rational
inequalities……
• Find domain restrictions from the denominator
and put a hole on the number line
• Pretend it is equal to zero and find zero’s
• Put those on the number line as open/closed
dots
• Test sections into the ORIGINAL inequality for T
or F
• Write the answer in interval notation for the true
sections
A common mistake is to forget
to do the first step!
2x 7
3
x 5
You do this one……
(,5) [8, )
5
3
x6 x2
You do this one……
(, 14] (6, )
3x
x
3
x 1 x 4
You do this one……
Quiz next time over….
• Find critical numbers for inequalities
• Solve polynomial and rational inequalities
• Graph a rational function including
intercepts, asymptotes and correct end
behavior
• Function composition
• Evaluating a function for a specific value
HW: WS 4-5