Download math 120 first day`s review

Math 120
First Day Review, And Homework
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Name:_______________________________________________________________
You are to work together to correctly complete as many problems as you can.
Before the end of this lecture, the instructor will clearly identify which problems need to
be done as homework. This list will then be posted to the class website (under the
"Lectures and Homeworks" page). Each student must submit their own, individual
homework. This means that each student must submit their own, individual packet (with
the specified problems completed) for this first homework assignment.
Where possible, you should check your work: when you arrive at an answer
(e.g., "X = 4"), make sure that your answer is correct, on your own (e.g., go back to the
original equation, and plug in "4" for X, and make sure that the math works out
correctly).
YOU MUST HAND IN THE CORRECTLY COMPLETED PROBLEMS ON THURSDAY,
AT THE START OF CLASS.
Failure to do so will result in no credit being given for this homework assignment
Distance Formula
From Section 1.1:
1) Find the distance between P1 = (-2, 3.5) and P2 = (4, 7.25)
Midpoint Formula
2) Find the midpoint between 1,2 and 3,4
From Section 1.1:
3 7
 13 7 
3) Find the midpoint between  ,  and  , 
4 8
 4 8
From Section 1.1:
4) For each of the following, figure out if the given point satisfies the give equation. SHOW YOUR WORK!
A.
1
x4
2
(2,5)
y
B.
1
x4
2
(2,6)
y
C.
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1 2
x 4
2
(-13, -88.5)
y
D.
1 2
x 4
2
(4,12)
y
Math 120
First Day Review, And Homework
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Explain (in English) what is meant when someone says "This point satisfies that equation"?
Graphing a Linear Function
From Section 1.1/1.2:
5) First do some algebra so you can put these
into your graphing calculator, then graph them
(using your calculator).
From Section 1.1/1.2
Sketch the function here:
X
2x  3y  3
10
9
8
7
6
5
4
3
2
1
0
-1 0 1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
2 3 4 5 6 7 8 9 10
Y
Solving Quadratic Equations Algebraically : Review
Goal: Review factoring, completing the square, and the quadratic formula
From Section 1.2 (4e)
6) Write out the quadratic equation, the quadratic formula, the discriminant, and a quick description of what the
discriminant tells us, here:
7) Solve for x, using the quadratic formula
x 2  5x  4  0
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Math 120
First Day Review, And Homework
8) Solve for x, using the quadratic formula
x 2  25  0
9) Solve for x, using the quadratic formula
v 2  7v  6  0
10) Solve for x, by factoring:
x 2  25  0
11) Solve for x by factoring:
x 2  4x
12) Solve for x by factoring
v 2  7v  6  0
13) Solve for x by factoring (try factoring by grouping)
x3  x 2  4x  4  0
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Math 120
First Day Review, And Homework
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14) Solve for x by completing the square
x 2  5x  4  0
Solving Equations With Radicals: Algebra & Graphically
Goal: Remember how to solve equations with radicals in them, and double-check your work with your
calculator.
From Section 1.5 (4e & 3e)
15) Solve the following problem algebraically:
z 5
16) Solve the following problem algebraically:
y 3 5
17) Solve the following problem algebraically:
2y  2
3
2
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Math 120
First Day Review, And Homework
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18) Solve the following problem algebraically:
2x  3  x  1  1
Functions Review
Goal: Recall how functions and relations differ ; apply your intuitive understanding to functions written using
function notation.
From Section 2.2 (4e)
19)
20)
Is the relation pictured below also a function?
Is the relation pictured below also a function?
<Formal Names>
Robert
Joseph
<Nicknames>
Robert
Bob
Rob
Joe
Joseph
Ralph
Ralph
21)
Which of the following relations are functions?
A) (1,4), (2, 5), (3, 6), (4,7), (5, 8)
B) (1, 4), (2, 4), (3, 4), (4,5), (5, 4)
C) (1, 4), (2,17), (1, 16), (-1, 13), (11, 12)
<Nicknames>
<Formal Names>
Robert
Robert
Bob
Rob
Joseph
Joe
Joseph
Ralph
Ralph
22) Does the following graph show a function?
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Math 120
First Day Review, And Homework
23) Does the following graph show a function?
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24) Does the following graph show a function?
Function Notation & Domain
From Section 2.2 (4e)
Goal: Get comfortable with function notation , start looking at how to figure out what the domain is.
24
24
x  2
x 2 , find the value of
26) If f(x) =
2
( x  1)
25) If f(x) = x
, find the value of
f(6)
f(2) + 2
f(-4)
What is the domain of X?
What is the domain of X?
(Hint: You can answer this question by asking "What
values of X WON'T work? Your answer is then "All
numbers, EXCEPT a, b, or c")
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Math 120
27) Given f(t) =
First Day Review, And Homework
4  3t
28) Given f(t) =
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2  5t
(Notice that we can define a function on t (rather than x) as f(t)
– just make sure to use t, not x, on the right side.)
(Notice that we can define a function on t (rather than x) as f(t) –
just make sure to use t, not x, on the right side.)
find the value of f(2)
find the value of f(2)
find the value of f(-2)
find the value of f(-2)
find the value of f(-10)
Suppose that we want our function to only produce real
numbers (i.e., no complex numbers). In other words, if
we want our function to have a range of only real
number, what must the domain of be?
Suppose that we want our function to only produce
real numbers (i.e., no complex numbers). In other
words, if we want our function to have a range of
only real number, what must the domain of be?
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Math 120
First Day Review, And Homework
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Properties of quadratic functions
Goal: Given a quadratic function, identify the axis of symmetry, vertex, and other properties of it’s
graph.
From Section 3.1 (4e), 2.3 (3e)
29) For the following parabola:
f x   x 2  4 x  3
X
Determine if the parabola opens up or
down:
Find the y-intercept:
10
9
8
7
6
5
4
3
2
1
0
-1 0 1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
2 3 4 5 6 7 8 9 10
Y
Find the x-intercepts (if any):
Find the vertex:
Then, draw the graph (above)
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Math 120
First Day Review, And Homework
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30) Revenue (we’ll use R as the variable for revenue) can be expressed as the price per unit (let’s
use P for price), multiplied by the number of units sold (let’s use x for the number of units).
If the price we can charge is given by the following equation(in terms of units sold):
1
P   x  20
10
(but only if x between 0 and 200, inclusive)(i.e., the domain of the function is: 0  x  200 )
Express revenue as function of just the number of units sold:
Find the maximum revenue one can make selling this particular product.
How many units does one sell to reach that maximum?
What is the price per unit?
31) Much like in the prior exercise, assume that you’re trying to find the maximum amount of money
that can be made from selling lamps. The Price function is given as:
1
P   x  100 for 0  x  600
6
Express revenue as function of just the number of units sold:
Find the maximum revenue one can make selling this particular product.
How many units does one sell to reach that maximum?
What is the price per unit?
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Math 120
First Day Review, And Homework
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X
Average Rate Of Change
From Section 2.4 (4e), 3.2 (3e)
Goal: Identify the rate of change of a function, given it's picture ; intuitively understand ARoC
32) What is the average rate of change
between x = -8 and x = -4?
10
9
8
7
6
5
4
3
2
1
0
-1 0 1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
Draw the secant line between these
two points on the graph to the left
2 3 4 5 6 7 8 9 10
What is the average rate of change
between x = 2 and x = 3?
Y
X
Draw the secant line between these
two points on the graph to the left
33) What is the average rate of change
between x = -8 and x = -4?
10
9
8
7
6
5
4
3
2
1
0
-1 0 1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
Draw the secant line between these
two points on the graph to the left
2 3 4 5 6 7 8 9 10
What is the average rate of change
between x = 4 and x = 6?
Y
Draw the secant line between these
two points on the graph to the left
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Math 120
First Day Review, And Homework
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34) Examine the picture to the left. For
these questions, you may assume that
the interval to be considered is
pictured in it's entirely
Where is the function ascending?
Where is the function descending?
Where is the function constant?
How many local maxima are there? What are they?
How many local minima are there?
What are they?
X
Operations on Functions
From Section 2.2 (4e), 3.5 (3e)
Goal: Intuitively understand what adding,subtracting, etc a function represents ; be able to add, subtract, etc functions
35) Given the functions
f x   3( x  3) 2  2 and
10
9
8
7
6
5
4
3
2
1
0
-1 0 1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
g x   4( x  2) 2  1
First, sketch out f and g on the grid to
the left.
2 3 4 5 6 7 8 9 10
Next, sketch out  f  g x
Using a formula, what is  f  g x ?
Y
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X
Math 120
First Day Review, And Homework
10
9
8
7
6
5
4
3
2
1
0
-1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
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36) Given the functions:
f x   3( x  3) 2  2
g x  4x  1
First, sketch out f and g on the grid to
the left.
2 3 4 5 6 7 8 9 10
Next, sketch out  f  g x
Using a formula, what is  f  g x  ?
Y
37) Given the functions:
f x   3( x  3) 2  2
g x  4x  1
Using a formula, what is  f  g x  ?
38) Given the functions:
f x   12 x 2  24
g x   4 x
f
Using a formula, what is   x  ?
g
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Math 120
First Day Review, And Homework
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39) Given the following pair of functions:
f x   x 2  1
g x   x  5
find
a)  f  g x
b) g  f x
c)  f  g 4
d) g  f 4
e) What is the domain of  f  g x ?
e) What is the domain of g  f x ?
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