Pre Calc Session 4 Mid Term Practice

1
Pre Calc Session 4Mid Term Practice

• Review Activity

2
Functional Concepts
Polynomial Jargon
Rational Functions
Motion Sickness?
Family Composition
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3

10 points


Find the inverse to f(x) ?x 6.

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4
A N S W E R
10 points
y ? x 6 y 6 ?x 8(y 6) 3x 8y/3 16
x 8x/3 16 y
f-1(x) 8x/3 16
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5

20 points
Consider the rational function R(x)
5x
- 6
x-4
Describe in detail what makes this a function and
determine if it is one to one.
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6
A N S W E R
20 points
Regardless of any element of the domain you
substitute for x, there will be at most one range
element to which it maps. This function is 1-1
because any range element that is derived will be
unique to the given domain element from which it
came so that no other element of the domain will
give the same element from the range. ie. this
functions inverse will still satisfy the
criteria to be a function.
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7

30 points

Consider the rational function R(x)
2
3
x-1
Algebraically find the domain and the range for
the inverse of this rational function.
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8
A N S W E R
30 points
2
1. (switch x y)
The inverse is f-1(x)
x-3
The domain of this inverse clearly does not
include x3, so it is Dxxx3. The range of
this inverse is the same as the domain of the
original function which clearly does not include
1, so the Ryyy1.
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9

40 points
Which graph is the inverse of the following
function? (After restricting the domain.)
a
b
c
d
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10
A N S W E R
40 points
The original function is yx22 and is not 1-1,
so restricting the domain to xgt0, the inverse
will be y vx-2.
a
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11

50 points
Given the functions f and g described below, find
f o g(x).
x2
f(x) 5x-2
g(x)
5
Give an explanation of why this result happened
in this way.
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12
A N S W E R
50 points
x2
)
(
- 2
x2-2 x
5
f o g(x)
5
I started with x as the input and then proceeded
to arrive at x after doing the composition, f o
g. This indicates that f and g may be inverses.
In fact, they are inverses in this situation
because g o f will also give me a result of x,
too.
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13

10 points
Describe the end behavior of the polynomial
P(x) -2x3x2-3x1
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14
A N S W E R
10 points

As x -8, y 8 As x 8, y -8




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15

20 points
What is the greatest number of negative roots
possible for P(x) 4x3 - 2x2 5x - 1
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16
A N S W E R
20 points
P(-x) -4x3 - 2x2 - 5x 1 has no sign changes
so there will not be any negative roots at all!
(DesCartes Rule of Signs)

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17

30 points
How many absolute max mins are there? How many
local max mins are there?
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18
A N S W E R
30 points
There is an absolute max in the vicinity of x-5
and there is a local max near x3. There is no
absolute minimum because of the end behavior at
both ends while there does seem to be a local
minimum close to x-1. Therefore, the answer to
the question is 1,0,1, and 1, respectively.
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19

40 points
What is the 4th degree polynomial whose roots are
2i, 1 and ¼ ?
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20
A N S W E R
40 points
(x2i)(x-2i)(x-1)(4x-1) (x24)(4x2-5x1)
4x4 5x3 x2 16x2 20x 4 4x4 5x3 17x2
20x 4

You could also use the SPC, DPC, TPC, and QPCs
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21

50 points
Which of the following could be a factor of a
polynomial given the following synthetic division.
1 3x-1 2 2x-1 3 2x1 4 2x3
0
-1
-2
-3
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22
A N S W E R
50 points
location principle
0

-1
-2
-3
There is a Real root between 0 -1 so the only
choice that could yield this zero would be 2x1.
(root -½)
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23

10 points
The graph of the function is given by the
equation below. Find k.
2x2-x-3
f(x)
(x1)(x-k)
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24
A N S W E R
10 points
k4 because there is a vertical asymptote at x4.

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25

20 points
Name the slant asymptote which is indicated on
the graph of

2x2 5x - 2

R(x)

x 2
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26
A N S W E R
20 points
- 4
2x 1
x2
As x 8 , y 2x1
x2
2x2 5x - 2

It will always get close to, but never touch the
line y2x1 because of the remainder, -4.
2x2 4x
x - 2
x 2
-4
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27
30 points

• Write an equation of a rational function with the
  following characteristics.
• A hole at x2
• A vertical asymptote at x-3
• A horizontal asymptote at y1

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28
A N S W E R
30 points
For the end behavior to be getting close to 1 ie.
A horizontal asymptote y 1, the lead term of
the numerator must be x2 because the lead term of
the denominator is already x2. As x gets really
large, the fraction will then get close to 1.
x(x-2)
y

(x-2)(x3)
2 is not in the domain and a common factor in the
numerator indicates a hole
-3 is not in the domain indicating a vertical
asymptote.
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29

40 points

As x -8, y ?



-x2 2x 1

R(x)

x2
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30
A N S W E R
40 points
y will get really close to -1. This is because
the numerator is growing in the negative
direction as fast as the denominator is growing
in the positive direction both have a degree of
2 with opposite coefficients.

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31

50 points
Where is the hole?



x2 4x 3
R(x)
3x 9
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32
A N S W E R
50 points
Since the factor (x3) divides out, the x value
of the hole is -3. When substituting x -3 into
the reduced fraction, you get R(x) - ? . This
means there is a hole at (-3, - ? ).

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33

10 points
V1 is 20 mph at a 40o headingV2 is 20 mph at a
230o directionWhat is their relationship?



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34
A N S W E R
10 points
Opposite vectors because their magnitudes are the
same while their directions are different by 180o.

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35

20 points

What is the horizontal component of a 250 mph
vector at a heading of 0o?

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36
A N S W E R
20 points

It is heading due North, so its horizontal
component would be 0.

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37

30 points
w
a
u
v
b
v w - a ?
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38
A N S W E R
30 points
a

u
v
b
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39

40 points
An airplane is navigating 350 mph due East.
There is a 50 mph wind coming from 45o. Find the
direction of the aircrafts resultant.



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40
A N S W E R
40 points

-6.4o because tan-1 (35.36/314.64)


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41

50 points
A teenager runs horizontally straight off a 48
foot high cliff and allows himself to fall into a
lake. Here are the parametric equations x
10t and y -16t2 48 Why is there
no linear term in the y parameter?
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42
A N S W E R
50 points
The teenager ran off the cliff horizontally so he
had a vertical velocity equal to zero, so the
only things determining his height would be
gravity and the fact that the cliff was 48 ft.
tall to begin with.
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43

10 points
M(a) 220 a M max heart rate a
age E(M) .7(M) E Exercising heart
rate What is the exercising heart rate for a 50
year old person E o M(50)?

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44
A N S W E R
10 points
E o M .7(220-a) which essentially is 3
transformations of y a (1) reflect over the y
axis (2) Translate up 220 (3) Vertically dilate
by scale factor .7 Algebraically, 220-50 170
.7(170) 119 beats per minute

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45

20 points
Consider a vector v which describes the
transformation T15,-8
What is the magnitude of v ?
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46
A N S W E R
20 points
15 units
-8 units

The magnitude is 17
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47

30 points
Describe all transformations indicated in the
order they would be executed on the parent.
f(x) - sin 2(x-p)
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48
A N S W E R
30 points
Right shift p units horizontal dilation sf.5
(compression) reflection over the x axis
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49

40 points
Which graph is y -v-x 4 ?
1
2
What transformations took place to the parent for
this graphical solution to result?
3
4
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50
A N S W E R
40 points
THIRD
T0,4 y -v-x 4
FIRST
ry yv-x

Parent yvx
rx y -v-x
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SECOND
51

50 points
Given the functions f and g below. f(x) 2x22
and g(x) -7 What is h(x) if h(x) f(x)
g(x). Describe the single transformation by which
f(x) becomes h(x).
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52
A N S W E R
50 points
f(x) g(x) 2x2 9 and the single
transformation by which f(x) 2x2 2 becomes
h(x) is essentially T0,7.

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