Coll Algebra Word Problems Videos 1-15-2011

1.

2.

Coll Algebra 1-14-2011

Date________________ Period____ .

Solving Multi-Step Equations

Solve each equation.

VI. Word Problems

1. One number is 5 more than twice another number. The sum of the numbers is 35. Find the numbers.

2. Ms. Jones invested $18,000 in two accounts. One account pays 6% simple interest and the other pays

8%. Her total interest for the year was $1,290. How much did she have in each account?

3. How many liters of a 40% solution and an16% solution must be mixed to obtain 20 liters of a 22%

solution?

4. Sheila bought burgers and fries for her children and some friends. The burgers cost $2.05 each and the

fries are $.85 each. She bought a total of 14 items, for a total cost of $19.10. How many of each did

she buy?

Word Problems Answers

1. Let x = “another number” forcing 2x + 5 = “One number.” x + 2x + 5 = 35 and x = 10.

“One number” = 25 and “another number” = 10.

2. Let x = the dollars in the account paying 6% interest

Then, 18,000 – x = the dollars in the account paying 8%.

The interest dollars are calculated by multiplying the total dollars in the account by the interest rate.

Hence: .06 x = the interest earned by the first account

.08 (18,000 – x) = the interest earned by the second account. Adding up all the interest,

.06x + .08(18,000 – x) = 1,290. Solving, x = 7,500. So, Ms. Jones has $7,500 in the account paying

6% interest and $10,500 in the account paying 8% interest.

3. Use the following buckets:

From the diagram, we get the equation: .4x + .16 (20 – x) = 20(.4)

x = 5 and the answer is 5 liters at 40% and 15 liters at 16%.

4. Let x = the number of burgers and 14 – x = the number of fries. To get the total amount of money

spent, multiply the number of items by the cost of the item. 2.05 x = the total dollars spent on burgers

and .85 (14 – x) = the total dollars spent on fries. The equation is: 2.05x + .85 (14 – x) = 19.10.

Solving the equation, x = 6. Hence, she bought 6 burgers and 8 fries.

(x)     +   (20 - x) = (20 liters)

40 %        16 %          40 %

College Al 1/14/2011

1.  (2/y)+(5/2) = (2/3y) 

2.  (X²+1)/(x-1) = (X²+4x-3)/(x-1)

Pre Cal-2

Pre Cal: Exponential functions-2

Exponential Function

	(Day 3 & 4)How do transformations of exponential equations affect the function analytically and graphically? (Day 5)How do exponential functions relate to real world phenomena?
	b. Investigate and explain characteristics of exponential functions, including domain and range, asymptotes, zeros, intercepts, intervals of increase and decrea c. Graph functions as transformations of f(x) = ax. d. Solve simple exponential equations and inequalities analytically, graphically,  and by using
Opening –	(Day 3) Introduce general parent exponential function and discuss how inputting different values for a will change the graph. Discuss graphing exponential functions using Graphic Organizer # 1 (Day 4) Teacher models two problems exemplifying dilations of exponential functions. (Day 5) Bellringer: 1st Part: Video on Donald Trump http://www.youtube.com/watch?v=QtKeSlS1zmk 2nd Part: Teacher will introduce the Part 3: The Beginning of a Business learning task. They will review the compound interest formula and its components.
Work session –	(Day 3) : Bacterial Growth Activity (Day 4) Exponential Function Graphic Organizer # 2 Worksheet Then the teacher will discuss the Graphic Organizer. Part 4: Some Important Questions (Day 5) Part 3- The Beginning of a Business
	(Day 4) Have the students share their answers from the work session. (Day 5)Have the students share their answers and how it relates to the standards.

                                                                                                         

Exponential Function

          y = a^x

         

1.      Graph f(x) = 3^x.

Complete the table of values.
x	f (x)
– 4
– 2
0
1
3
4

2.      Graph f(x) = 5^x.

Complete the table of values.
x	f (x)
– 4
– 2
0
1
3
4

3.      Graph f(x) = 3^{x +}  2.

Complete the table of values.
x	f (x)
– 4
– 2
0
1
3
4

How did adding the 2 change
the original function?

4.      Graph f(x) = 5^x – 3.

Complete the table of values.
x	f (x)
– 4
– 2
0
1
3
4

How did subtracting the 3 change
the original function?

Exponential Function GO 2

                                               

Exponential Function

          y = a^x

f (x) =  5x + 1		f (x) =  – 5x
x	f (x) =	x	f (x) =
– 2		– 2
– 1		– 1
0		0
1		1
2		2
3		3

f (x) =(0.125)3x			f (x) = (2)5x
x	f (x) =	x		f (x) =
– 2		– 2
– 1		– 1
0		0
1		1
2		2
3		3

f (x) =  5x – 3		f (x) =  3x + 2
x	f (x) =	X	f (x) =
– 2		– 2
– 1		– 1
0		0
1		1
2		2
3		3

Exponential Function GO 3

Function: f(x) = ( – 2 ) 3^x + 2

Domain______________________

Range________________________

Asymptotes____________________

Zeros________________________

y-intercepts___________________

intervals of decrease______________

intervals of increase_______________

rates of change___________________

Function: f(x) = (0.85) 3^{x – 2} – 1

Domain______________________

Range________________________

Asymptotes____________________

Zeros________________________

y-intercepts___________________

intervals of decrease_______________

intervals of increase_______________

rates of change___________________

Function: f(x) = (2) 3^{x – 1}  + 2

Domain______________________

Range________________________

Asymptotes____________________

Zeros________________________

y-intercepts___________________

intervals of decrease_______________

intervals of increase_______________

rates of change___________________

Unit 5 Exponential Functions

Day 5 Quiz

Simplify using the exponential properties.

1.     (x ³ y²) (x³  y⁷)=

2.     X^3y

X^2y

3.     Solve the exponential equation-

(a)  3^x+1 = 27^x+3                                               (b) 9^x+2 = (1/27)^x+12

       Solve each inequality.

4.     8^x ≥ 2^2x+1

5.     4^x ≤ 4^3x-1

6.     #23 p. 130 word problems (McDougal Littell – possibly)

Exponential Decay Experiments

The following are experiments to explore the phenomena of exponential decay.  Each of these experiments will require you to collect materials, take repeated measurements and graph the resulting data.

Experiment 1:  Cooling Water

You will need:  a container of hot water, a watch, and a candy thermometer

1.  For this experiment, you need to measure the temperature of the hot water.  Record this measurement in the table below.  Recheck and record the temperature of the water every minute until the water reaches room temperature.

x (time in minutes)	y (temperature of water)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

2.  Graph the ordered pairs (time, temperature) on graph paper.

3.  Does this data appear exponential?  Why or why not?

4.  Use the initial value you recorded plus one other point from your graph to write an exponential function to fit your curve.

points you chose:  _________________________

f(x) = _________________________________

5.  Using your equation, f(x), complete the following table.  Plot the new values (x, f(x)) on your graph in another color.

x (time)	f(x)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

6.  Sketch your curve f(x).  Does it appear to be a good fit?

7.  What is the “decay factor”?

Part 3:  The Beginning of a Business

How in the world did Linda ever save enough to buy the franchise to an ice cream store?  Her mom used to say, “That Linda, why she could squeeze a quarter out of a nickel!”  The truth is that Linda learned early that patience with money is a good thing.  When she was just about 9 years old, she asked her dad if she could put her money in the bank.  He took her to the bank and she opened her very first savings account.

Each year until Linda was 16, she deposited her birthday money into her savings account.  Her grandparents (both sets) and her parents each gave her money for her birthday that was equal to her age; so on her ninth birthday, she deposited $27 ($9 from each couple).

Linda’s bank paid her 3% interest, compounded quarterly.  The bank calculated her interest using the following formula.

where A = final amount, P = principal amount, r = interest rate, n = number of times per year the interest was compounded

1.  Using the following chart, calculate how much money Linda had on her 16^th birthday.

Age	Birthday $	Amt from previous year plus Birthday	Total at year end
9	27	0	27.81916
10	30	57.81916	90.48352
11	33	123.4835	161.2311
12	36	197.2311	240.3071
13	39	279.3071	327.9643
14	42	369.9643	424.463
15	45	469.463	530.0714

2.  On her 16^th birthday, the budding entrepreneur asked her parents if she could invest in the stock market.  She studied the newspaper, talked to her economics teacher, researched a few companies and finally settled on the stock she wanted.  She invested all of her money in the stock and promptly forgot about it.  When she graduated from college on her 22^nd birthday, she received a statement from her stocks and realized that her stock had appreciated an average of 10% per year.  How much was her stock worth on her 22^nd birthday?

3.  When Linda graduated from college and got her first job, she decided that each year on her birthday she would purchase new stock in the amount of half what her last stock was worth.  On her 30^th birthday she looked back and saw that her stock had appreciated each year a percent that was half of her age that year.  So on her 23^rd birthday, her stock had appreciated 11.5%; and so on.  What was her stock worth on her 30^th birthday?

Age	Amt from previous year	Amt Linda added	Amt at year end
22	938.73	469.47
23
24
25
26
27
28
29
30	147,888.83

Part 4:  Some Important Questions

All of these examples from Linda’s journey are examples of exponential growth functions… the rumor, compounding interest in a savings account, appreciation of a stock.  Real-life situations tend to have restricted domains.

1.  How is the domain restricted in each of the scenarios?

2.  How would the graph of the rumor be different if the domain was unrestricted?

3.  Graph the function .

4.  What is the range of the function?

5.  Why doesn’t the graph drop below the x-axis?

6.  Now graph .

7.  What is the range of the function?

8.  An exponential function has a horizontal asymptote.  Where is the asymptote located in the graph for #3?  Where is the asymptote located in the graph for #6?

9.   Use your graphing utility to graph the following equations.

Ø 

Ø 

Ø 

Ø 

Ø 

10.  Make some generalizations.  What impact did each of the changes you made to the equation have on the graph?

                                                                        shifts how?

       shifts how?

                       shifts how?

Day 6
E. Q. –	How do exponential functions relate to real world phenomena?
Standard –	MM2A2. Students will explore exponential functions. e. Understand and use basic exponential functions as models of real phenomena.
Opening –	Real World Phenomena Problem-Teacher will walk through how to solve, graph, and discuss the transformations of the exponential functions using Worksheet – Real World Phenomena (Handout)
Work session –	Students will work two problem McDougal Littell Mathematics II textbook.(Compound  interests problems) 1.      Pg 125 mathII example 4 2.      Pg 127 mathII #18-20 3.      Students will make a problem, solve it, graph it & discuss what occurred. (in-class & homework)
Closing –	Students share out work and answer EQ if they can.

Mathematics II - Unit 5

Real World Phenomena Worksheet

INVESTMENTS

Consider a $1000 investment that is compounded annually at three different interest rates:

5%,  5.5%, and 6%.

a.      Write and graph a function for each interest rate over a time period from 0 to 60 years.

b.      Compare the graphs of the three functions.

c.       Compare the shapes of the graphs for the first 10 years with the shapes of the graphs between 50 and 60 years.

24.  Which of these describes the graph of f(x) = 3^x + 4?

A.      It has a vertical asymptote at x = 0.

B.      It has a vertical asymptote at x = -4.

C.      It has a horizontal asymptote at y = 0.

D.      It has a horizontal asymptote at y = 4.

25.  What is the asymptote of the graph of f(x) = 2^x? 

A.      x-axis

B.       y-axis

C.       y = 1

D.      y = -1

26.  How would you translate the graph of f(x) = 5^x to produce the graph of

   f(x) = 5^x – 3?  

1. translate the graph of f(x) = 5^x left 3 units

2. translate the graph of f(x) = 5^x right 3 units

3. translate the graph of f(x) = 5^x up 3 units

4. translate the graph of f(x) = 5^x down 3 units

29.  Find the y-intercept of the graph of y = -3(7^x) .

2.   D                                    23.  A 

3.   D                                    24.  D  

                        

4.   A                                25.  A  

5.   C                                26.  D  

6.   D                                  27.  C  

7.   D                                   28.  B  

8.   B                                   29.  C  

9.   A                                   30.  B  

10. D                                  31.  D  

11. A  

12. D  

13. D  

14. A  

15. A  

16. B  

17. D  

18. B  

19. C  

20. A  

KNOW	HOW	NOW
Function		One-to-Oneness
Domain		Inverse
Range		Inverse Relation
Intercepts		Inverse Function
Maximum		Composition
Minimum		Composite
Linear Function		Horizontal Line Test
Quadratic Function		Power Function
Cubic Function		Restricted Domains
Vertical Line Test		nth Root