Algebraic Mathematics of Linear Inequality & System of Linear InequalityJacqueline Chau
A brief, yet thorough look into the Linear Inequality & System of Linear Inequality and how these Math Concepts would be useful in solving our everyday life problems.
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Algebraic Mathematics of Linear Inequality & System of Linear InequalityJacqueline Chau
A brief, yet thorough look into the Linear Inequality & System of Linear Inequality and how these Math Concepts would be useful in solving our everyday life problems.
Enroll for FREE MCA TEST SERIES and FREE MCA MOCK TEST
For more details on MCA entrance and sure shot success,
Paste this link: http://www.tcyonline.com/india/mca_preparation.php
TCYonline
No. 1 Testing Platform
Try to download if possible. If you do so, you will be able to click the arrows and see the some of the objects in the slides move. The presentation is more interractive if you download the Powerpoint and open it on your device.
Try to download if possible. If you do so, you will be able to click the arrows and see the some of the objects in the slides move. The presentation is more interractive if you download the Powerpoint and open it on your device.
1. Graph the exponential function by hand. Identify any asymptotes.docxjackiewalcutt
1. Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. (Enter NONE in any unused answer blanks.)
Equation of horizontal asymptote:
Equation of vertical asymptote:
Value of y-intercept
Value of x-intercept
The function is
2. Use the graph of y = 2x to match the function with its graph.
A
B
C
D
y = 2x – 4
y = 2x – 5
y = 2x + 4
y = 2–x
3. Use the graph of f to describe the transformation that yields the graph of g. Then sketch the graphs of f and g by hand.
f(x) = −2x, g(x) = 5 − 2x
The graph of g(x) = 5 − 2x is a vertical shift five units downward of f(x) = −2x.
The graph of g(x) = 5 − 2x is a horizontal shift five units to the left of f(x) = −2x.
The graph of g(x) = 5 − 2x is a vertical shift five units upward of f(x) = −2x.
The graph of g(x) = 5 − 2x is a horizontal shift five units to the right of f(x) = −2x.
Sketch the graphs of f and g.
4. Use a graphing utility to construct a table of values for the function. (Round your answers to three decimal places.)
x
f(x) = 5x − 3
-1
0
1
2
3
Sketch the graph of the function.
Identify any asymptotes of the graph. (Enter NONE in any unused answer blanks.)
vertical asymptote
x
=
horizontal asymptote
y
=
5. Use a graphing utility to construct a table of values for the function. (Round your answers to three decimal places.)
x
g(x) = 4 − e−3x
-4
-3
-2
-1
0
Sketch the graph of the function.
Identify any asymptotes of the graph. (Enter NONE in any unused answer blanks.)
vertical asymptote
x
=
horizontal asymptote
y
=
6. Fill in the blank.
If
x = ey,
then y = .
7. For what value of x is
ln x = ln 9?
x =
8. Write the logarithmic equation in exponential form. For example, the exponential form of
log5 25 = 2 is 52 = 25.
log2 512 = 9
=
9. Write the logarithmic equation in exponential form. For example, the exponential form of log5 25 = 2 is 52 = 25.
log
1
100,000,000
= -8
=
10. Write the exponential equation in logarithmic form. For example, the logarithmic form of
23 = 8 is log2(8) = 3.
43/2 = 8
log
=
11. Use a calculator to evaluate the function at the indicated value of x. Round your result to three decimal places.
f(x) = log10(x) x = 4/5
12. Solve the equation for x.
log10(102) = x
13. Write the logarithmic equation in exponential form. For example, the exponential form of ln(5) = 1.6094... is e1.6094... = 5. (Do not use ... in your answer.)
=
14. Write the exponential equation in logarithmic form. For example, the logarithmic form of
e2 = 7.3890 is ln 7.3890 = 2.
(Do not use ... in your answer.)
e2.2 = 9.0250
ln
=
15. Use the properties of natural logarithms to rewrite the expression.
5 ln(e5)
16. Use the properties of logarithms to rewrite and simplify the logarithmic expression.
ln
9
e9
17. Use the properties of logarithms to expand the expression as a ...
ENGR 102B Microsoft Excel Proficiency LevelsPlease have your in.docxYASHU40
ENGR 102B: Microsoft Excel Proficiency Levels
Please have your instructor or TA initial each level as you complete it. If you need additional help, ask the TAs or use the help guide within Excel.
Once you master Excel Levels I through IV, you can note Excel as a skill on your resume!
Please see D2L Content for this week for your Excel Homework assignment (individual), which is due via D2L Dropbox by the due date specified in the D2L News for your section.
If you use a Mac, please be sure to submit your homework in a format that the grader and instructor can open on a PC.
Level I: Basic Functions Initials _______
1. Calculating an Average: Calculate the arithmetic average of the 5 values listed below. Enter the values in cells A2 through A6. Place a descriptive label in cell A1.
3.6, 3.8, 3.5, 3.7, 3.6
First, calculate the average the long way, by summing the values and dividing by 5:
You will enter the following formula into a blank cell to accomplish this:
=(A2+A3+A4+A5+A6)/5
Second, calculate the average using Excel’s AVERAGE( ) function by entering the following formula in a cell:
=AVERAGE(cellrange)
Replace the “cellrange” with the actual addresses in your spreadsheet of the range of cells holding the five values (i.e., for this problem, the cell range is A2:A6).
2. Determining Velocities (in kph): Some friends at the University of Calgary are coming south for spring break. Help them avoid a speeding ticket by completing a velocity conversion worksheet that calculates the conversion from mph to kph in increments of 10 from 10 to 100. A conversion factor you will need is 0.62 miles/km; you will need this factor to convert from miles/hour to km/hour. Place the conversion factor in its own cell and then reference it in your conversion calculations using absolute cell referencing (e.g., $C$2). Refer to the CBT video on Absolute and Relative Cell Referencing from the “Preparation for the Excel Workshop” assignment if you don’t remember how to do this.
Level II: Advanced Functions Initials _______
1. Projectile Motion I: (See following page for Fig. 1 Excel chart) A projectile is launched at the angle 35o from the horizontal with a velocity equal to 30 m/s. Neglecting air resistance and assuming a horizontal surface, determine how far away from the launch site the projectile will land.
To answer this problem, you will need:
1. Excel’s trigonometry functions to handle the 35o angle, and
2. Equations relating distance to velocity and acceleration
When velocity is constant, as in the horizontal motion of our particle (since we’re neglecting air resistance), the distance traveled is simply the initial horizontal velocity times the time of flight:
(Equation 1)
What keeps the projectile from flying forever is gravity. Since the gravitational acceleration is constant, the vertical distance traveled becomes
(Equation 2)
Because the projectile ends up back on the ground, the final value of y is zero (a hor ...
Page 1 of 7 Pre‐calculus 12 Final Assignment (22 mark.docxkarlhennesey
Page 1 of 7
Pre‐calculus 12
Final Assignment (22 marks)
Each question is worth 1 mark. You must show all your work to obtain full marks.
Marks will be deducted for no work shown.
1. What happens to the graph of 1 if the equation is changed to 1?
2. The graph of y = √ undergoes the transformation (x, y) ( 3,2 5)x y . What is the resulting
equation?
3. Determine the equation of the polynomial in factored form of the least degree that is symmetric
to the y‐axis, touches but does not go through the x‐axis at (3, 0), and has P(0) = 27
4. Determine the measure of all angles that satisfy the following conditions. Give exact answers.
csc =2 in the domain 2 2
Page 2 of 7
5. Solve: 3cos² 8cos 4 0, over all real numbers
6. Use factoring to help to prove each identity for all permissible values of x. Must state
restrictions over all real numbers.
2sin sin
tan
cos sin cos
x x
x
x x x
Page 3 of 7
7. In a population of moths, 78 moths increase to 1000 moths in 40 weeks. What is the
doubling time for this population of moths?
8. Solve the following equation: log 3 log 5 2
9. Solve for x algebraically: 5 2 3 . State your answer to the nearest hundredth.
10. A radioactive substance has a half‐life of 92 hours. If 48g were present initially, how long will it
take for the substance to decay to 3g? Show algebraically.
Page 4 of 7
11. Given the following two functions √ 1 and 1, evaluate
3 .
12. A sample of 5 people is selected from 3 smokers and 12non‐smokers. In how many ways can
the 5 people be selected?
13. Given the functions 7 and
√
, determine an explicit equation for
, then state its domain.
14. Determine the 4th term of 3 2 .
15. Solve by algebra √13 1 0
Page 5 of 7
16. Determine the domain, range, and intercepts of 2√4 2 3. Graph the function.
17. For the graph of , determine an non‐permissible values of , write the coordinates of
any hole and write the equation of any vertical asymptote.
Page 6 of 7
18. Sketch the graph of 3 4 5. State the domain, range, and equation of the
horizontal asymptote.
19. Suppose you play a game of cards in which only 5 cards are dealt from a standard 52 deck. How
many ways are there to obtain at least 3 cards of the same suit? An example of a hand that
contains at least 3 cards of the same suit is 4 hearts and 1 club.
20. Given , determine , the inverse of .
Page 7 of 7
21. Consider the digits 0, 2, 4, 5, 6, 8. How many 3‐digit even numbers less than 700 can be
formed if repetition of digits is not allowed? Note: the first digit cannot be zero.
22. If and 2 3, determine the value of 1 .
Page 1 of 9
Pre‐calculus 12
Midterm Assignment (24 marks)
Each question is worth 1 mark. You must show al ...
5. Use the “y=“ to match equations with graphs and/or tables. The table below shows various values for x and y. Which equation best describes the relationship between x and y? A. y = −3x + 5 B. y = −5x − 7 C. y = −x + 17 D. y = 3x + 41 Enter the answer choices into the calculator through the “y=“ feature and then look at the tables to find a match. − 28 11 − 16 7 11 − 2 23 − 6 y x
6.
7.
8.
9.
10. Which expression represents the area of a rectangle with sides measuring 2x 2 y 4 z units and 5xy 4 z 3 units? F. 7x 2 y 8 z 3 units 2 G. 7x 3 y 8 z 4 units 2 H. 10x 3 y 8 z 4 units 2 J. 10x 2 y 8 z 3 units 2 Select 3 different numbers to store for x, y, and z. Press the button for the first number, press the “STO” button, press the letter of the alphabet representing the variable, and then enter. Repeat this procedure for each variable.
11. Press the button for the first number, press the “STO” button, and then enter. press the letter of the alphabet representing the variable, Repeat this procedure for each variable.
12. The figure described is a rectangle. Draw and label one. The problem asked which expression represents the AREA. Area of a rectangle is length times width . (The formula is on the math chart.) You are going to enter into the home screen of your calculator the expressions found here for length times width: (2x^2y^4z)(5xy^4z^3) 2x 2 y 4 z 5xy 4 z 3
13. Now, you want to go to TEST (2 nd MATH) Since you want the equal sign and it is already highlighted, press ENTER. Notice, the equal sign appears after your expression for area. And, one by one, type in the answer choices into the calculator just as they appear. 2x 2 y 4 z 5xy 4 z 3
14. If the answer choice has an expression that is EQUAL to the area expression, when you ENTER you will get “1” for an answer. If the answer choice has an expression that in NOT EQUAL to the area expression, when you ENTER you will get “0” for an answer. Since we have a “0”, answer choice F is not correct and we need to repeat this procedure using G. Option F is not the correct answer. Repeating the procedure does NOT mean that you have to retype everything! Press 2 nd ENTER.
15. Notice that the step above just reappears. Now, backspace until you get to the equal sign And type over the old choice with the new choice or delete the old choice and type in the new. ENTER Repeating the procedure does NOT mean that you have to retype everything! Press 2 nd ENTER. Another zero! Repeat with option H. 2 nd ENTER, backspace to the equal, type in option H, and ENTER. Looks like H is the answer. Repeat with J just to be sure!
16. Which expression represents the area of a rectangle with sides measuring 2x 2 y 4 z units and 5xy 4 z 3 units? F. 7x 2 y 8 z 3 units 2 G. 7x 3 y 8 z 4 units 2 H. 10x 3 y 8 z 4 units 2 J. 10x 2 y 8 z 3 units 2 Sure enough, option H is the correct answer. Do you have to do ALL of this work? NO. If you know that 2 times 5 is 10, you could have eliminated entering in choices F and G. If you know the rules for when you add, subtract, and multiply exponents for same bases and trust your own memories and work, then make your selection.
17.
18. Tammy drew a floor plan for her kitchen as shown below. Which expression represents the area of Tammy’s kitchen floor in square units? F. 6x 2 + 30x + 5 G. 6x 2 + 13x + 5 H. 10x + 12 J. 5x + 6 Once again, this problem refers to AREA of the rectangular kitchen. We already know that means finding length times width. The expression for Area, then, would be (3x + 5)(2x + 1) (3x + 5) units (2x + 1) units
19. Tammy drew a floor plan for her kitchen as shown below. Which expression represents the area of Tammy’s kitchen floor in square units? F. 6x 2 + 30x + 5 G. 6x 2 + 13x + 5 H. 10x + 12 J. 5x + 6 But this time, we are going to use “y=“ instead of STO, since there is only the variable x used in the expressions. Enter the expression (3x + 5)(2x + 1) in y 1 (3x + 5) units (2x + 1) units
20. Tammy drew a floor plan for her kitchen as shown below. Which expression represents the area of Tammy’s kitchen floor in square units? F. 6x 2 + 30x + 5 G. 6x 2 + 13x + 5 H. 10x + 12 J. 5x + 6 One by one, enter the answer choices into y 2 . Then, graph. If both equations have exactly the same graph, the two expressions are equivalent and you found your correct answer. Changing to this option allows you to follow along as the 2 nd function is graphed. F is not the correct answer since the two graphs are different. (3x + 5) units (2x + 1) units
21. Tammy drew a floor plan for her kitchen as shown below. Which expression represents the area of Tammy’s kitchen floor in square units? F. 6x 2 + 30x + 5 G. 6x 2 + 13x + 5 H. 10x + 12 J. 5x + 6 Replace choice F in y 2 with choice G and graph. Did you watch as the little circle made its way around the same parabola? Option G is the correct choice. To be safe, you can check options H & J. If you realize that those two options are linear (no seen exponents for x), their graphs could never be a parabola and thus are not correct answer choices. (3x + 5) units (2x + 1) units
22. Use STATPLOT to compare points or scatter plots. Which point on the grid below best represents the coordinates ? , A. Point K B. Point M C. Point R D. Point U Press the STAT button. Select EDIT Enter 8/3 in L 1
23. Which point on the grid below best represents the coordinates ? , A. Point K B. Point M C. Point R D. Point U When you enter, the calculator will change the fraction into a decimal. Enter 7/3 into L 2 Set the window to the scale in the problem so you can make a good comparison. Go to STAT PLOT (2 nd y=) Enter and turn on the plot by entering again. You should see xlist: L 1 for the x-coordinate & ylist: L 2 for the y-coordinate. And graph. This point has an x-coordinate between 2 and 3. And a y-coordinate between 2 & 3.
24.
25. Use the calculator to solve systems. If the system of linear equations 2x + y = 1 and y = − x + 1 are graphed on the same coordinate grid, which of the following is the solution to this system of linear equations? A. (2, 0) B. (0, 2) C . (0.5, 0) D . Not here 2x + y = 1 is not yet calculator friendly! Get the y by itself by subtracting 2x from each side. y = 1 – 2x Enter both equations using the y= feature. Graph. Adjust the window, if necessary to see the point of intersection.
26. You want to go to the CALC feature (2 nd TRACE) Select intersect since that is what you are looking for. Since there is only 1 point of intersection, Enter when the calculator says “First curve?”, “Second curve?”, and “Guess” The coordinates of the point of intersection, which is the solution, are shown at the bottom of the window. (0, 1) is the point where these two lines intersect. A. (2, 0) B. (0, 2) C . (0.5, 0) D . Not here Looking at the answer choices, the correct solution is not there.
27.
28.
29.
30.
31. Know how to change your table settings. You set the number where you want the table to start You set the scale that you want the values in the table to go by You determine whether you want the table to be filled in automatically as you set it up or to have it wait for you to give it x-values to find.
32. Starting with -3 and going by 1. Starting with -5 and going by 10. Starting with 2 and going by 0.1. Let’s use this equation. You will see different tables for this same function based upon how you set the table to appear.
33. There are different settings you can use on the graphs Makes a regular line Makes a thicker line Makes a regular line and shades above the line Makes a regular line and shades below the line Shows where the graph goes and makes a regular line Shows where the graph goes but makes NO line Makes a dotted line These options come from backspacing and pressing ENTER
35. There are several things you can do with these answer choices to eliminate a few so you won’t have to graph so many. The first thing you should notice is that all of the inequalities have 5 as the y-intercept and a negative slope and no equal sign.
36. Then, whether you plan to use the graphing calculator or not, you need to know that when the inequality sign points to y, as in B and C, the shading is below the line. Since our shading is above the line, we can eliminate these two graphs.
37. For the remaining two choices, you either need to count the slope, starting at the y-intercept or you test the x-intercept of 4 by substituting 4 in for x to see if you get y = 0 or use the calculator to graph and see if the x-intercept is 4. Since today’s tutorial is on calculator usage, that is the method we are going to use.
38. Shaded above To be safe, fraction in ( ). The x-intercept is NOT 4. Wrong choice
39. Shaded above To be safe, fraction in ( ). The x-intercept IS 4. Choice D is verified.
40. This problem can be done a few ways, also. Remember, x-intercepts have y = 0, so you can substitute 0 for y and solve for x. Y-intercepts have x = 0, so you can substitute 0 for x and solve for y. Or you can graph. If you want to do the graphing by hand, remember that there is a blank sheet of graph paper at the end of the math section for you to use as you choose.
41. Since this tutorial is about using the calculator, that is the way we are going to do this problem. The given equation is not calculator friendly. We need to put the equation in y = form. Remember, there is an understood -1 in front of y, due to the subtraction sign. You do NOT have to put the equation in slope-intercept form, just calculator friendly form. The calculator will do the rest.
42. 2x – y = 8 is 2x – 1y = 8. Subtracting 2x, we get -1y = 8 – 2x Then, dividing by -1, we get the calculator friendly form y = (8 – 2x)/-1 You absolutely MUST have the parentheses around the numerator!
43. The y-intercept is negative. We have no choices with negatives. Let’s eliminate the y-intercept choices. The x-intercept appears to be 4
44. You can either substitute the r-values, by hand or on the calculator home screen, one-by-one to make sure that you get the corresponding n-values. And yes, you must check all of them until you find a value that does not work. Or, you can type the answer choices in y = and match the table of values. Let n = y and r = x, and you will be just fine.
45. Not answer choice A Not answer choice B ALL 4 of the ordered pairs match. This is the one!
46. There is only one variable in these expressions. Put the problem’s expression in y 1 and the answer choices, one-by-one, in y 2 . Remember, you want matching graphs. Allows me to watch as the graph is plotted. They matched! Check the others to be sure, though.
47. NOPE Doesn’t look like it, but let’s adjust the window. NO Definitely not this one, either. Looks like F is the correct answer choice.
48. These expressions have more than 1 variable. We want to store a number into the calculator for each of these variables and use the TEST feature. Remember, you pick whatever numbers you want, except 0, 1, or -1.
49. Type in the expression for the problem. BE CAREFUL! You need extra parentheses around the entire numerator and around the entire denominator.
50. You now want TEST (2 nd MATH) and enter to get the equal sign. Now, type in your answer choices, one at a time. Again, extra parentheses around the numerator and the denominator. Remember, when you get a zero, the two expressions are NOT EQUAL
51. Press 2 nd ENTER to get the last step again. Backspace to the equal sign and either delete or type over the old option. We have a 1. That means the two expression ARE EQUAL
52. Verify that the two remaining answer choices are NOT EQUAL before you select G as the correct answer. Be careful that you correctly type in all expressions! Remember to put those extra parentheses around both the numerator and the denominator.
53. Since these problems are kind of complicated to do on the calculator and since you probably haven’t used this procedure very much, here is another problem. You can either use those same 3 values that you already have stored in for a, b, and c, or you can change them.
54. Remember, correctly type in the expression from the problem, using an extra set of parentheses around both the numerator and the denominator. Then, go to TEST, select the equal sign, and type in the answer choices, one at a time. Extra parentheses are needed around entire numerators and entire denominators. NOT A
55. 2 nd ENTER to get the previous operation. Backspace to the equal sign and enter the next answer choice. NOT B Not C, either
56. If this graph is shifted UP, the y-intercept/vertex should be higher. Logically, you should eliminate J because -8 is lower than -3.
57. Let’s type the original function in y 1 and the answer choices, one at a time, in y 2 and see which parabola shifted UP 5 units.
58. Answer choices will have a thicker line Count the hash marks. The new graph shift up 8 units. Too high!
59. Count the hash marks. The new graph shift up 5 units. This is it!
60.
61. If you count, you can see that between 0 and 1, there are 4 spaces—on each axis. That means that the grid is divided into fourths. T is located on the 3 rd space past 0 on the x-axis so its x-coordinate is ¾ . That means we are looking at options G and H.
62. T is located on the 5 rd space below 0 on the y-axis so its y-coordinate is -5/4 . That means the correct option is G.
63. Now, if you have no clue about these points, you will want to use the STAT button on your calculator. Adjust the window on your calculator to match the scale here. You are going from -2.5 to 2.5 by ¼ or .25 on each axis.
64. Recall, when you enter fractions into the calculator, they are changed into decimal form. Be sure that the STAT PLOT is turned on with the proper lists.
68. We can solve this equation the “traditional” way—using the “undo” process.
69. Alternate method We can solve this equation by using the table feature of the graphing calculator. Enter the equation. Go to the table. Scroll down the table until you find 104 in the y-column
70. Alternate method Or we could use the graph and CALC features of the graphing calculator Enter the equation in y 1 and 104 in y 2 . Adjust the window — You need y max to be higher than 104 Graph You need to be able to see in the window where the two lines intersect. That place looks way off to the right. Adjust the window again. Let’s try the x max at 50.
71. Alternate method Enter again for the second curve? And guess? Press 2 nd TRACE so that you get CALC. Now, select Intersect. Move the cursor to be close to the point of intersection.