Transcript Bellwork: Simplify each, without a calculator

1.
Solve by factoring: 2x2 – 13x = 15.
2.
Solve by quadratic formula: 8x2 – 3x = 10.
3.
Find the discriminant and fully describe the
roots: 5x2 – 3x.
4.
Solve algebraically or graphically:
– 2x – 15> 0
Algebra II
x2
1
Graphing Polynomial
Functions
Algebra II
f(x) = an
n
x
+ an-1
n-1
x
+ ... + a1
1
x
+ a0
where an ≠ 0
Example:
f(x) = 3x4 – 2x3 + 5x – 4
Algebra II
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
Whole numbers therefore
exponents are all ______________
Positive
all __________________

Real numbers
all coefficients are___________________

Leading coefficient
an is called the _____________________

Constant term
a0 is called the _____________________

degree
n is equal to the ____________________
highest
(always the _______________
exponent)
Algebra II
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Standard Form means that the polynomial
is written in _____________Descending
order of
Exponents
_____________
Algebra II
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Standard Form
Example
Degree Name
f(x) = a0
f(x) = a1x1 + a0
f(x) = a2x2 + a1x1 + a0
f(x) = a3x3 + a2x2 + a1x1 + a0
f(x) = a4x4 + a3x3 + a2x2 + a1x1 + a0
Algebra II
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1. f(x) =
2. f(x) =
Yes
f(x) = –3x4 + ½x2 – 7
2
4
½ x – 3x – 7
D: 4
LC: -3
C: -7
N: Quartic
No
x3 + 3x exponents are not whole
numbers
3. f(x) = 6x2 + 2x-1 + x
4. f(x) = -0.5x + πx2 – √2
Algebra II
No
exponents are not whole
numbers
Yes
f(x) = πx2 - 0.5x – √2
D: 2
LC: π
C: –√2
N: Quadratic
7
Direct Substitution means to:
Plug the value into the equation and solve
_____________________________________
____
Algebra II
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f(x) =
–
+ 7x – 11
g(x) = – x4 + 3x2 + 2x + 7
p(x) = – x(2x – 3)(x + 7)
3
3x
1. p(2)
–18
Algebra II
2. g(3)
– 41
2
2x
3. f(-2)
–57
4. g(-3)
–53
9
 Lets
type each in the calculator
and look for:
y=x
y = x2
3
y=x
4
y=x
y = x5
Algebra II
10
End behavior is what the y values are doing as
the x values approach positive and negative
infinity.
It is written: f(x)
_____ as x
-∞, and
f(x)
_____ as x
∞
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
even
If the degree is __________
the ends of the
same
graph go in the _________
direction.

odd
If the degree is __________
the ends of the
opposite directions.
graph go in the _________

Leading coefficient to see what
Look at the ________________
direction the graph is going in.
Algebra II
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1. f(x) = 3x4 – 2x2 + 5x – 8
D:
4, even
2. f(x) = -x2 + 1
D: 2, even
LC: 3, positive
LC: -1, negative
End Behavior:
End Behavior:
∞
-∞
f(x) --->____
∞ as x ---> ∞
-∞
f(x) --->____as
x ---> -∞
-∞ as x ----> ∞
f(x) --->____
f(x) --->____ as x ---->
Algebra II
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3. f(x) = x7 – 3x3 + 2x
D:
4. f(x) = -2x6 + 3x – 7
7, odd
D: 6, even
LC: 1, positive
LC: -2, negative
End Behavior:
End Behavior:
-∞
-∞
f(x) --->____
∞ as x ---> ∞
-∞
f(x) --->____as
x ----> -∞
-∞ as x ----> ∞
f(x) --->____
f(x) --->____ as x ---->
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6. f(x) = 4x3 + 5x7 – 2
5. f(x) = -4x3 + 3x8
D:
8, even
D: 7, odd
LC: 3, positive
LC: 5, positive
End Behavior:
End Behavior:
∞
-∞
f(x) --->____
∞ as x ---> ∞
-∞
f(x) --->____as
x ----> -∞
∞ as x ----> ∞
f(x) --->____
f(x) --->____ as x ---->
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1.
Make a table of values from -3 to 3
2.
Plot the points
3.
Connect with a smooth curve
**(use arrows to demonstrate end behavior)**
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1. f(x) = – x + 1
3
x
-3
-2
-1
0
1
2
3
Algebra II
y
28
9
2
1
0
-7
-26
17
Algebra II
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2. f(x) = x3 + x2 – 4x – 1
x
-3
-2
-1
0
1
2
3
Algebra II
y
-7
3
3
-1
-3
3
23
19
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3. f(x) = –x4 – 2x3 + 2x2 + 4x
x
-3
-2
-1
0
1
2
3
Algebra II
y
-21
0
-1
0
3
-16
-105
21
Algebra II
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4. f(x) = x5 – 2
x
-3
-2
-1
0
1
2
3
Algebra II
y
-245
-34
-3
-2
-1
30
241
23
Algebra II
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Answer each:
f(x) > 0
f(x) < 0
f(x) is increasing
f(x) is decreasing
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Answer each:
f(x) > 0
f(x) < 0
f(x) is increasing
f(x) is decreasing
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



f is increasing when
x < 0 and x > 4
f is decreasing when
0<x<4
f(x) >0 when -2 < x < 3
and x >5
f(x) < 0 when x < -2 and
3<x<5
Use the graph to describe the degree and the leading
coefficient of f.
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



f is decreasing when
x < -1.5 and x > 2.5
f is increasing when
-1.5 < x < 2.5
f(x) >0 when x < -3 and 1
<x<4
f(x) < 0 when -3 < x < 1
and x > 4
Use the graph to describe the degree and the leading
coefficient of f.
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


f is increasing when
x < -1 and 0 < x < 1
f is decreasing when
-1 < x < 0 and x > 1
f(x) < 0 for all real
numbers
Use the graph to describe the degree and the leading
coefficient of f.
Algebra II
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The estimated number V (in thousands) of electric
vehicles in use in the United States can be modeled by
the polynomial function
v(t) = .151280t3 - 3.28234t2 + 23.7565t – 2.041
Where t represents the year, with t = 1 corresponding to
2001.
a.Use a graphing calculator to graph the function for the
interval 1 < t < 10. Describe the graph.
b.What was the average rate of change in the number of
electric vehicles in use from 2001 to 2010?
Algebra II
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The number of students S (in thousands) who graduate in
four years from a university can be modeled by the
function S(t) = -1/4t3 + t2 + 23, where t is the number of
years since 2010.
a. Use a graphing calculator to graph the function for the
interval 0 < t < 5. Describe the behavior of the graph
on this interval.
b. What is the average rate of change in the number of
four-year graduates from 2010 to 2015?
Algebra II
31
1. Decide whether the function
is a polynomial function. If it is,
write the function in standard
from and state the degree and
leading coefficient:
3. Give the end behavior
for the function:
f (x)= 9x - 3x 2 +5
4. Graph: y = 2x3 – 1
1
2 1 2
f (x)= x + - x
3
3 6
2. Use direct substitution to
find f(-1) for the function:
f (x)= -2x 5 +3x 3 - 2x +5
Algebra II
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