Download M65, Mod 3, Section 3

M65, Mod 3, Section 3.1 Functions and the Quadratic Function
I. Review of functions:
Ex. 1: Determine if the following are functions. Name the independent and
dependent variables. State the domain and range. In b) the units are 1.
b)
a)
r
s
9
3
y
x
5
3
6
-2
-8
5
 
c) For the function f x  9x  3.7
Use the table function on your calculator to find: f(0.01), f(-2), f(0)
II. Quadratic Functions:
Definition:
Ex. 2: Which of the following are equations of quadratic functions? If it is a
quadratic function, name the coefficients a, b and c.
 
a) g x  5  3 x  2x
b) y  x  3
 
c) f x  3x  14
2
III. Characteristics of the Graphs of Quadratic equations:
Ex. 3: Graph the quadratic equation: f ( x )  x
x
Mod 3, Sec. 3.1, p2
2
f(x)
-3
-2
-1
0
1
2
3
The name of the graph of a quadratic is a _________________.
It has a _________________ shape.
It will have either a ______________ or a _____________________.
It has points symmetric about a line called the______ of _____________.
The ________________ is on the axis of symmetry.
Ex. 4: Graph the quadratic function:
g ( x ) x 2
When the coefficient a is negative the
graph opens ________________and it will have a ______________.
Mod 3, Sec. 3.1, pg 3
Ex. 5. Use your calculator to help sketch the graphs of the following:
y1  x 2
y 2  2x 2
y3   3x 2
Describe in words what effect the numbers
have on the shape:
Ex. 6: Use your calculator to help sketch the graphs of the following:
y1  x 2
1
y2  x2
2
1
y3   x 2
10
Describe in words what effect the numbers
have on the shape:
Ex. 7: Use your calculator to help sketch the graphs of the following:
y1  x 2  2
y2  x2  3
y3   x2  3
Describe in words what effect the constants
have on the graph.
Mod 3, Sec. 3.1, pg 4
Ex. 8: Use your calculator to help sketch the following. Label the vertex. Draw
in and label the Axis of Symmetry:
y1  2 x 2  4 x  5
Trace or use Calc to find the vertex:
Vertex (point) = _____________
A. O. S (an eqn) = ______________
a = _____
b = _____
c = _____
Describe in words what happens to the graph when the equation has a bx term
(a linear term):
Ex. 9: Use your calculator to help sketch the graph of the following:
y1   x 2  5 x  1
Vertex: ___________
A. O. S.:__________
a = _____
b = _____
c = _____
Notice that the axis of symmetry can be found by using the formula:
X=
Once the A. O. S. is found, we can substitute that x value into the equation to find
the y value of the Vertex. It is important that you memorize this formula.
Math 65. Section 3.2a Rules of Exponents
Exponents is a shorthand way of writing _____________ _____________.
Ex : 5 3
5 is the __________ and 3 is the ______________.
It means:
RULES FOR EXPONENTS:
1. Multiplication of the same base: __________ the base, _________ the
exponents.
Ex. 1: x 4  x 2 
Ex. 2: 4 x 2  3 x3 
Ex. 3:  3 pq 2  5 p 4 q  
2. Raise a power to a power: ___________ the base, ____________the
exponents.
Ex. 4:
x 

Ex. 5:
z 

2 3
3 5
Ex. 6:  23  
4
3. Raise a product to a power. Raise each __________ to that power.
Ex. 7:  2x 4  
3
Ex. 8:  2x3  
5
Ex. 9:  3xy 2  
3
Ex. 10:  9x 2 y 3 z 4  
2
M65, Sec. 3.2a, pg2
4. Division of the same base:__________ the base, _______________
the exponents. (_____ ____________ minus _________ ___________)
11:
x4

x2
12:
x3 y 5

xy3
5. Raise a quotient to a power: Raise each part of the fraction to the
______________.
3
13:
 3x 2 

 
 4y 
14:
 2y 
 3 
x 
2
Look at the following pattern and determine the missing exponents and numbers:
3 4  3 3 3 3
3
 3 3 3
3
 3 3
3
3
3

3

3

M65, Sec. 3.2a, pg 3
ZERO AND NEGATIVE EXPONENTS:
6. Any ________________ number raised to the zero power is
________. Zero raised to the zero power is ________________.
Ex. 15:
40 
Ex. 16:
 ab 
Ex. 17:
a 0b 2 
Ex. 18:
6x

x0
0

7. Negative exponents indicates ____________________.
3
Ex. 19:
5
  
2
Ex. 20:
 2m 
Ex. 21:
2m4 
4

Simplify completely. Write with positive exponents only.
Ex. 22:
5m 4

m10
Ex. 23:
 3a 2 
 1  
b 
Ex. 24:
23  24  22 
Ex. 25:
 3    3
4
0 2
3

Some practice:
M65, Sec. 3.2a, pg 4
Simplify each of the following expressions (if possible) using the rules for
exponents. Try writing the expression in expanded form if you get stuck.
1.
x 3  x 3  __________
2.
x 3  x 3  __________
3.
x 2  x 3  __________
4.
x 2  x3  __________
5.
t  t  t  t 2  __________
6.
t  t  t  t 2  __________
7.
n 4  n 2  n  m3  m3  __________
8.
 ab  a b   __________
9.
2a  3a  __________
10.
2a  3a  __________
11.
y 
12.
a b 
13.
w8
 __________
w2
14.
b
 __________
b4
15.
 5 x  2 x   __________
16.
 5x 
2 3
3
 __________
2
2
3 4
2 5 2
3 2
 __________
 __________
Simplify the following by hand. You may check your answers on a calculator.
17.
24  23  __________
18.
2 
19.
42  __________
20.
 4 
21.
32  23  __________
22.
32  23  __________
3 4
2
 __________
 __________
Math 65. Section 3.2b continued. Scientific Notation
Scientific Notation:
I. Review multiplication and division by powers of 10.
Ex.
3.2 100 =
75 100 =
In general:
II. Standard Notation:
Scientific Notation:
Ex. 1: 7,300,000
Ex. 2: 0.000 000 45
Ex. 3:
3.45  106
Ex. 4:
4.56  10 3
Ex. 5:
2.1 104
Ex. 6: -230,000
Ex. 7: - 0.00573
Calculator Notation:
M65, Sec. 3.2b cont, pg 2
Ex. 8: 4.5E  7
means:
Ex. 9: 3.45E  6
means:
Ex. 10: -2,354E4
means:
Ex. 11: -1.57E-2
means:
III. Calculations with calculators:
4.3 EE 6
means
4.3  10 6
It will show up on your
calculator as 4.3E 6
Hints:
1. Use (
) around numerators and denominators.
2. Use EE 6 instead of *10^6
3. Use * between numbers instead of (
).
4. Write your answer in scientific notation using 7.2  10 4 ,
NOT calculator notation which is 7.2E 4 !
5. Be cautious with your negatives. You need to recognize when the
negative applies to the number, a, or to the exponent ,b.
Ex. 12:
1.49 1013
2.75 10  9
Leave your answer in scientific
Notation, rounded to two decimal
places.
M65, Sec. 3.2b, cont. pg 3
Key Strokes:
1.49 EE 13  2.75 EE  9 ENTER
On your calculator it will look like: (1.49E13)/(2.75E-9) = 5.418181818E21
You need to interpret that calculator notation into scientific notation and round
your answer to two decimal places:
 5.42 1021
 4.3 10 5.4 10 
6
Ex. 13
8.2 10  2
5
Round your answer in
scientific notation to two
decimal places.
Key Strokes:
 4.3 EE 6*5.4 EE 5  8.2 EE  2 ENTER
Your calculator will show: (4.3 E 6 * 5.4 E 5)/(8.2 E – 2) = 2.831707E13
 2.83  1013
Use your rules of fractions, multiplication and exponents to check your answer:
M65, Sec. 3.2b cont. pg 4
Use your calculator to find the following: Write out the keystrokes. Put your final
answer in scientific notation, rounded to 2 decimal places:
8.3 10  4.210 
3
7
Ex. 14:
5.43 104
Key Strokes:
Ex. 15:
3.4 10  3
(4.12)(3.75 108 )
Key Strokes:
 3.2 10 5.46 10 
1.12 10 3.5 10 
8
4
Ex. 16:
Key Strokes:
6
2
Math 65. Section 3.3 Square Roots
I. Definition and properties of Square Roots:
A. Taking the Square Root is the Inverse Operation of Squaring.
ex: 52  25 so the square root of 25 is ______.
but  5   25 so the square root of 25 might also be _____.
2
B. Notation:
ex:
a
reads "the principal (or positive) square root
of a"
25
is read "the positive square root of 25" = ____
 25
is read "the negative square root of 25" =_____
C. a must be greater than or equal to zero ( a  0 ) to get a real number
for the square root.
D. Evaluate:
1.
144 
2.
49 
3.
 100 
4.
200 
5.
36 
6.
9 
II. Sets of Numbers:
M65 Sec. 3.3 pg 2
Rational Numbers
Irrational Numbers
* Non-Real Numbers:
III. Multiplying and Approximating Radicals:
M65, Sec. 3.3 pg 3
A. Rule:
Ex. 7:
3 3 
Ex. 8:
 2 8 
Ex. 9:
 2 6
Ex. 10:
12  4 
B. Simplifying under the radical. The radical sign is a __________ symbol. Use
(
) in calculator.
Ex. 11:
Key Strokes:
52  4  3  2  
5
2

 4  3  2 enter
Ex. 12:
42  4  3  2 
Ex. 13:
102  4  2  5  
Ex. 14:
 4
2
 4  2 3  
M65, Sec. 3.3 pg 4
IV. Simplifying Radicals. You will need to know the perfect square of the
numbers 1 – 12.
A. Rule: the converse of the rule is also true:
ab 
Ex.
10  15  150
Check with calculator
B. Simplify, leave as exact, but check with a calculator.
Ex. 15:
150  25  6  25  6  5 6
Ex. 16:
18  3  54 
Ex. 17:
2 14  5 2  2  5 14  2 
Ex. 18:
2 3 3 6 
Ex. 19:
3 8 2 2 
Ex. 20:
4 5  2 10 
Math 65, Section 3.4 Square Root Functions:
I. Review:
A. Give an example of each
1. Linear function:
2. Quadratic function:
B. Domain and Range:
1. Domain
2. Range
C. Exponent Rules:
1. 0 exponent
2. Negative exponent
D. Square Roots:
1. Ball-park Approximate:
2. Simplify – exact
3. Calculator – approximate
also try 50^(1/2)
II. Graphing a square root function:
A. By hand: f(x)  x
x
f (x)
D:
R:
M65, Sec 3.4 pg 2
B. On the Calculator
y1  x
y2 
1
x
2
y3  5 x
y4   x
C. Horizontal and Vertical Shifts:
Ex. 1: Graph by hand: y  x  2
x
y
D:
R:
Ex. 2: Graph by hand: g ( x )  x  4
x
g(x)
D:
R:
M65, Sec. 3.4, pg 3
D. On the Calculator: describe the translations (shifts, stretches,
reflections, etc.) and give the Domain and Range for each.
y1  x
y2  x  1
y3  3 t
y4  r  3
y5  
1
x 5
2
y 6  4  2x
III. Applications of Square Root Functions:
Ex. 3: The side of a square is related to the area.
L( A)  A
When the area is known, we can estimate
the length of the side.
a) Estimate the length of the side of a square with area of 48ft 2
b) Estimate the length of the side of a square with area of 1.97m 2
M65, Sec. 3.4 pg 4
Ex. 4. Velocity of a Nissan pickup is related to its stopping distance by the
equation: v  4.27 d
where d is the stopping distance in feet and
v is the velocity in miles per hour.
a) How fast was he going if it took him 85 feet to stop?
b) Sketch a graph. Use the window:
x   10,500  100 ' s
y   10,100  10 ' s
c) From the graph, estimate the distance required to stop when traveling at 65
mph.
Ex. 5: The time, t, in seconds that it takes a pendulum to swing back and forth
one, is given by:
t (L)  2
L
32
Where L is the length of the pendulum in feet.
a) Find the time a 128-foot pendulum takes to make one back and forth swing.
b) Sketch a graph. Use the window:
x   10,150   10 ' s
y   1,20   1' s
c) From the graph, estimate the length of a pendulum which takes about 10
seconds to make a back and forth swing.