The lumpsum amount invested in 18 years ago for the given value Future value and interest is = Rs 2763,18.3268
What about lumpsum?
In mathematics, a lump sum refers to a single, fixed amount of money or quantity of something that is paid or received all at once, rather than being paid or received in installments or over a period of time.
The term "lump sum" can be used in various mathematical contexts, such as in finance, where it can refer to a single payment or investment, or in statistics, where it can refer to a single value or data point.
Define future value:
In mathematics, future value (FV) refers to the value of an investment or cash flow at a specified date in the future, assuming a certain rate of return.
The future value of an investment is the amount of money that an investor would expect to have at a specified date in the future if they were to invest a certain amount of money today, and if that investment earns a certain rate of return over time.
⇒ The formula for calculating the future value of an investment is:
[tex]FV = PV x (1 + r)^n[/tex]
According to the given information:
Interest Rate = 5%
Time Period = 18 years
Future Value = Rs 6649,93.
Using the concept of TVM calculation we have that,
Present value = [tex]\frac{Future Value}{(1+r)^{t} }[/tex]
So, the Present Value = [tex]\frac{664993}{(1+0.05)^{18} } = 2763,18.3268[/tex]
Hence, the Lumpsum Amount is Rs 2763,18.3268
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All nine transactions for Ralston Sports Co. for September, the first month of operations, are recorded in the following T accounts:
Cash
(1)
(7)
(9)
(4)
(8)
(3)
(2)
25,000 (3)
11,900 (5)
9,700 (6)
(8)
James Ralston, Capital
(1)
Accounts Receivable
9,900 (9)
James Ralston, Drawing
7,000
Supplies
12,500
Fees Earned
(4)
(7)
Equipment
9,500
12,500
7,600
10,500
7,000
25,000
9,700
9,900
11,900
Answer:
Step-by-step explanation:
Cash
(1)
(7)
(9)
(4)
(8)
(3)
(2)
25,000 (3)
11,900 (5)
9,700 (6)
(8)
James Ralston, Capital
(1)
Accounts Receivable
9,900 (9)
James Ralston, Drawing
7,000
Supplies
12,500
Fees Earned
(4)
2. What is the y-intercept of the equation y =
a) (0,0)
(0, 12)
хо
Q
3. What is the domain of the equation y =
b) x # -4,0
د در
Ax#4
to
(x-3)(x-4)? (2A.2A)
c) (0,4)
بیرون
(x-3)(x-4) ? (2A.2A)
y=3
c) x #0
d) No y-intercept
4. What is/are the vertical asymptote(s) of the equation y = (x-3)(x-4) (2A.2A)
What we are t
a) x = 3
b) x = 0
x=4
d) x = 0 and x = 4
d) x # 4,0
+8
5. What is the horizontal asymptote of the equation y = (x-3)(x-4) (2A.2A)
b) y = 4
c) y = 1
d) y = 0
X-4 = 0
X=4
Because the x2 term's coefficient is positive, the function will rise without equation bound as x approaches infinity or negative infinity. As a result, no horizontal asymptote exists.
What is equation?An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" asserts that the phrase "2x + 3" equals the number "9".
The y-intercept of the equation y = (0,0) is the point on the line where it meets the y-axis and corresponds to the value of y when x equals 0. As a result, the equation y = 0's y-intercept is (0,0).
An equation's domain is the set of all possible values of x for which the equation is defined.
a) Because there are no constraints on the potential values of x, the domain of the equation y = x2 - 4x is all real integers.
b) The domain of the equation y = 1/(x-4) includes all real numbers except x = 4, which is undefined since division by zero is undefined.
c) Because the term within the square root must be non-negative, the domain of the equation y = sqrt(x-3)(x-4) is x >= 3.
As x approaches a specific value, the graph of the function approaches but never reaches vertical asymptotes. As the denominator of a rational function hits 0, they arise.
Because y = (x-3)(x-4) is not a rational function, it lacks vertical asymptotes.
A function's horizontal asymptote is a horizontal line that it approaches when x approaches infinity or negative infinity.
The dominant term in the formula as x gets extremely large or very tiny is x2, thus we can rewrite the equation as [tex]y = x2 - 7x + 12[/tex] .
Therefore, Because the x2 term's coefficient is positive, the function will rise without bound as x approaches infinity or negative infinity. As a result, no horizontal asymptote exists.
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which of these animals can travel at the greatest distance from sea level?
Answer:
Dragonflies
Step-by-step explanation:
They can fly on top of the water, but also, they have the fastest turns per second for their wings.
Which statement identifies the effect of replacing f (x) with 1/2 f(×) on the graph?
A The curve would remain the same size but would be flipped upside down.
B The curve would remain the same size but would shift to the left.
C The curve would be narrower, but the vertex would be in the same position.
D The curve would be wider, but the vertex would be in the same position.
quiz help me please
Answer:
D
Step-by-step explanation:
The curve will be wider but the vertex would be in the same position.
If a<1, our function will be wider
If a>1, the function will be more narrow
The correct answer is D
Given the following information, fill in the blanks.
x = 14.5, μ> 13, n = 50, o = 5.1 and a = 0.01
Round to TWO decimal places.
Critical Value: type your answer...
z-value: type your answer...
P-value: type your answer...
Reject or fail to reject? choose your answer...
✨
Answer: 46
Step-by-step explanation: i took quiz i swear the answer is
A group of 125 students went on a field trip to a museum. Mr. Shan asked a random sample of 50 students which exhibit was their favorite. Ten students favored the insect exhibit, 15 students favored the space exhibit, and 25 students favored the dinosaur exhibit. Based on the survey results, which inferences about the entire group of 125 students are true? Select TWO correct answers. 1.One-fifth of students favored the insect exhibit. 2.The dinosaur exhibit is most favored. 3.The insect exhibit is favored more than the space exhibit. 4.Only 24% of students favored the space exhibit. 5.Fifty students favored the dinosaur exhibit.
Answer:1 and 2
Step-by-step explanation:
so, they asked 50 students and then that was later broke down to smaller groups.
10,15,25
so, for every 10 students will be one. 1 vote= 10 students.
also, the dinosaur had the greatest number of votes so that explains answer 2.
You mix gas and oil to obtain 2 and 1/2 gallons of mixture for an engine. The mixture is 40 parts gasoline and 1 part oil. How much gasoline must be added bring the mixture to 50 parts gasoline and 1 part oil?
We need to add a volume of 39.69 fluid ounces of gasoline to the mixture to bring it to 50 parts gasoline and 1 part oil.
What is mensuration?Mensuration is the branch of mathematics concerned with the measurement of geometric figures and parameters such as length, volume, shape, surface area, and lateral area.
According to information in the question:
The mixture is 40 parts gasoline and 1 part oil, there are 40 + 1 = 41 parts in total.
Let, the amount of oil in the mixture "x".
We know that there are 2 and 1/2 gallons of the mixture, which is the same as 2.5 * 128 = 320 fluid ounces.
So, the equation which can be formed is
x/41 * 320 = the amount of oil in the mixture (in fluid ounces)
Solving for "x":
x/41 * 320 = x/41 * 50 + 320 - (x/41 * 50)
Multiplying both sides by 41:
x * 320 = 50x + 41(320 - 50x)
Simplifying:
320x = 41(320)
x = 41 * 10 = 410 (fluid ounces of oil in the mixture)
Thus, amount of gasoline in the mixture is
40 parts gasoline / 41 parts total * 320 fluid ounces = 311.22 fluid ounces of gasoline.
Let the gasoline we need be "y", so a equation can be formed as follows
(311.22 + y)/(41 + y) * 320 = 410
Solving for y:
(311.22 + y)/(41 + y) = 410/320
Multiplying both sides by (41 + y) * 320:
311.22 + y = 410/320 * (41 + y) * 320
Simplifying:
311.22 + y = 205.625 * (41 + y)
311.22 + y = 8434.375 + 205.625y
Solving for y:
204.625y = 8123.155
y = 39.69 (fluid ounces of gasoline to add)
Therefore, we need to add 39.69 fluid ounces volume of gasoline to the mixture to bring it to 50 parts gasoline and 1 part oil.
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Find the critical numbers and absolute extrema for….
Final Answer: The critical number is x = DNE. The absolute maximum is x = 0.2, and y = -125.The absolute minimum does not exist on the interval.
To find the critical numbers?we need to find the values of x where the derivative of y equals zero or does not exist. So, we first find the derivative of y: y' = 10x^{(-3)}
The derivative is defined for all x in the interval [0.2, 2], so there are no points where the derivative does not exist.
To find where y' = 0, we set the derivative equal to zero and solve for x: [tex]10x^{(-3)} = 0[/tex] x = DNE There are no critical numbers since the derivative does not equal zero at any point in the interval.
Next, we check the endpoints of the interval: [tex]y(0.2) = -5/ (0.2^2) =[/tex] [tex]-125y(2) = -5/(2^2) = -1.25[/tex] Since y is a decreasing function on the interval,
the absolute maximum occurs at the left endpoint x=0.2: Absolute maximum: [tex]y(-0.2) = -125[/tex]
Since y is an unbounded function as x approaches zero, there is no absolute minimum on the interval [0.2, 2].
Final Answer: The critical number is x = DNE.The absolute maximum is x = 0.2, and y = -125.The absolute minimum does not exist on the interval.
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5 Question At what point does the MAXIMUM occur on the graph above? * pls help D: ( quiz)
Answer:
The answer would be C (1,4)
Step-by-step explanation:
The maximum refers to the highest point on the graph.
In this graph, the highest point lands 1 point to the right of zero and then four points up from there.
Making the maximum (1,4)
Answer:
Step-by-step explanation:
The maximum occurs at (1,4). The maximum is the highest point a function will reach. You can see on the graph that the highest point occurs when x = 1 and y = 4.
So the maximum is at (1,4)
A trick you can use to easily find it next time, imagine a horizontal line is coming down the graph from the top. When the line touches the function, that will be the maximum
You want to create a simulation of the following scenario:
In country x 50% of people have blood type O, 25% have blood type A, 12.5% have blood type B, and 12.5% have blood type AB.
In country y, 60% have blood type O, 20% have type A, 10% have type B and 10% have type AB.
What is the best way to assign values for a simulation using random digits table?
Choose answer from photo below! A, B, C, or D : this is the answer I want, not just an explanation please, thank you so much! 100 points!
Thank you :)
The best way to assign values for a simulation using random digits table will be option A.
What will be the simulation?The best way to assign values for a simulation using a random digits table would be to use a table with at least 10 digits (0-9) in each row.
For Country X, we would assign Blood type O with the digits 0-5, Blood type A with digits 6-7, Blood type B with digit 8, and Blood type AB with digit 9. If a digit outside of these ranges is generated, it would be ignored.
For Country Y, we would assign Blood type O with the digits 0-5, Blood type A with digits 6-7, Blood type B with digit 8, and Blood type AB with digit 9. Again, if a digit outside of these ranges is generated, it would be ignored.
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A volleyball team wants to order a team T-shirt. The Red Nose company charges $3.50 per shirt plus a $79 flat fee. The green Machine company charges $5 per shirt plus a $40 flat fee. Let T be the total cost and R be the number of shirts
show your full work
Red Nose: t = 4.50r + 55
Green Machine: t = 5r + 40
Find the total number of shirts that would make the total cost for each company the same
Number of shirts:
Cost:
Answer:
To find the total number of shirts that would make the total cost for each company the same, we can set the two equations equal to each other and solve for the number of shirts:
Red Nose: t = 3.50r + 79
Green Machine: t = 5r + 40
3.50r + 79 = 5r + 40
Subtracting 3.50r from both sides, we get:
79 = 1.50r + 40
Subtracting 40 from both sides, we get:
39 = 1.50r
Dividing both sides by 1.50, we get:
r = 26
So the total cost for each company would be the same if 26 shirts are ordered.
To find the total cost for each company when 26 shirts are ordered, we can substitute 26 for r in each equation:
Red Nose: t = 3.50(26) + 79 = $192
Green Machine: t = 5(26) + 40 = $170
Therefore, if 26 shirts are ordered, the total cost would be $192 for Red Nose and $170 for Green Machine.
The shape of the distribution of the time required to get an oil change at a 15-minute oil-change facility is skewed right. However, records indicate that the mean time is 16.4 minutes, and the standard deviation is 4.2 minutes. Complete parts (a) through (c).
Question content area bottom
Part 1
(a) To compute probabilities regarding the sample mean using the normal model, what size sample would be required?
A.
The sample size needs to be greater than or equal to 30.
B.
The sample size needs to be less than or equal to 30.
C.
Any sample size could be used.
D.
The normal model cannot be used if the shape of the distribution is skewed right.
Part 2
(b) What is the probability that a random sample of n=35 oil changes results in a sample mean time less than 15 minutes?
The probability is approximately enter your response here.
(Round to four decimal places as needed.)
Part 3
(c) Suppose the manager agrees to pay each employee a $50 bonus if they meet a certain goal. On a typical Saturday, the oil-change facility will perform 35 oil changes between 10 A.M. and 12 P.M. Treating this as a random sample, there would be a 10% chance of the mean oil-change time being at or below what value? This will be the goal established by the manager.
There is a 10% chance of being at or below a mean oil-change time of enter your response here minutes.
(Round to one decimal place as needed.)
Part 1: a. A. The sample size needs to be greater than or equal to 30.
Part 2: The probability is approximately 0.0013.
Part 3: There is a 10% chance of being at or below a mean oil-change time of 14.9 minutes.
What is sample size?The number of individuals or items in a population that are selected for study.
A. The sample size needs to be greater than or equal to 30.
This is due to the Central Limit Theorem, which states that the sample size of a normal distribution must be 30 or more to produce reliable results.
If a sample size is less than 30, the results may be skewed and not accurately represent the true population mean and standard deviation.
B. The probability is approximately 0.0013.
This probability can be calculated by using the z-score formula and plugging in the mean, standard deviation, and sample size.
The z-score for a sample mean of 15 minutes =-3.3,
which translates to a probability of 0.0013 using the z-score table.
C. There is a 10% chance of being at or below a mean oil-change time of 14.9 minutes.
This can be calculated by using the z-score formula and plugging in the mean, standard deviation, and sample size.
The z-score for a 10% chance= -1.28,
which translates to a mean oil-change time of 14.9 minutes.
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what's the square root of 144
[tex]\sqrt{144} =12\\12*12=144[/tex]
I need help with my homework
The table that represents the function for a lawn being mowed more quickly than by Killian's rate is given as follows:
Second table.
How to obtain the average rate of change?The average rate of change of a function is given by the change in the output of the function divided by the change in the input of the function. Hence we must identify the change in the output, the change in the input, and then divide then to obtain the average rate of change.
Killian can cut grass at a rate of 1000 square feet each 10 minutes, hence his average rate of change is given as follows:
1000/10 = 100 square feet per minute.
For the second table, in 7 minutes, 1100 square feet of lawn is mowed, hence the rate is given as follows:
1100/7 = 157 square feet per minute.
It is more quickly than Killian's as the rate of change is greater.
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a bag of rice has a mass of 3 kilogram s. Jackie buys 17 bags of rice for the school cafeteria . How many grams of rice did jackie buy?
Answer:51,000
Step-by-step explanation:
ther is 1000 grams every kilogram so 17x3 is 51 and 51x1000 is 51,000
SOLVE THIS BUT I DON'T WANT ANSWER I WANT SOLVING AND ANSWER
50PTS
WILL MARK AS BRAINLIEST
50/1.428
Answer:
Alright
Step-by-step explanation:
50/1.428
Multiply the numerator by the denominator to get 50000/1428
Set equation up in long division format
Divide 50000 by 1428 to get 3
Divide 7160 by 1428 to get 5
Divide 200 by 1428 to get 0
Divide 2000 by 1428 to get 1
Divide 5720 by 1428 to get 4
Divide 80 by 1428 to get 0
Divide 800 by 1428 to get 0
35.014
7.2 Puzzle Time
Where Did Columbus Land when He Found America
The answer to the puzzle is (3)O (7)N which makes word ON, (4)T (10)H (1)E which makes word THE, (6)B (8)E (2)A (9)C (5)H which makes the word BEACH
What is a quadrilateral?
A quadrilateral is a 4 sided polygon. On the basis of its properties quadrilaterals are classified as Trapezium,Kite, Parallelogram, Rectangle, Rhombus and square.
Complete the sentence:
1.Parallelogram-(E)
2.Quadrilateral-(A)
3.Supplementary-(O)
4.Bisect-(T)
5.Congruent-(H)
Using the diagram:
6.Given that KG=17 units, KJ=14 units.
As we know from the figure, GH is congruent to KJ and HJ is congruent to side KG.
Therefore, GH=KJ=14units- (B)
7.∠GKJ=86° and ∠GHJ=(x+6)°
We know that ∠GKJ is congruent to ∠GHJ
Therefore, x+6 = 86
x=86-6
x=80-(N)
8.Given KH=20 & KF= KH÷2 {diagonal pf parallelogram bisect}
=20 ÷ 2
=10 units-(E)
9.Given ∠HJK=82° & We know ∠HJK+∠GKJ=180° {consecutive angles}
∠GKJ=180-∠HJK
=180-82
=98°-(C)
10.Give that, ∠HJK=82° & we know ∠HGK=∠HJK {opposite angles of parallelogram}
∠HGK=82°-(H)
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Find the Volume of the triangular prism
For the given triangular prism, the volume value is obtained as Option B: 31.5 in³.
What is a triangular prism?
The pentahedron Triangular Prism contains nine different nets. The bases' vertices and edges are connected by means of their three rectangular sides. The rectangular sides of the triangular prism are joined to one another side by side. Every cross-section that is perpendicular to the base faces is a triangle.
The height of the triangular prism is given as 2 inches.
The base of the triangular prism is given as 7 inches.
The length of the triangular prism is given as 4.5 inches.
The volume of a triangular prism is given by the formula -
V = (1/2) × b × h × l
where b is the base of the triangle, h is the height of the triangle, and l is the length of the prism.
Substituting these values into the equation, we get -
V = (1/2) × 7 inches × 2 inches × 4.5 inches
V = 7 inches × 1 inch × 4.5 inches
V = 31.5 cubic inches
Therefore, the volume of the triangular prism is 31.5 cubic inches.
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If $14,000 is invested at 7% annual interest, which is compounded continuously, what is the account balance after 12 years, assuming no additional deposits or withdrawals are made?
the account balance after 12 years of continuous compounding at 7% annual interest, assuming no additional deposits or withdrawals are made, would be $32,447.44.
The formula for continuous compounding is:
A = P[tex]e^(rt)[/tex]
where:
A = the final amount
P = the principal amount (initial investment)
e = the mathematical constant approximately equal to 2.71828
r = the annual interest rate
t = the time in years
Plugging in the given values, we get:
A = 14000[tex]e^(0.0712)[/tex]
A = 14000*[tex]e^0.84[/tex]
A = 14000*2.31796
A = $32,447.44
Compound interest is a type of interest that is calculated not only on the initial principal amount, but also on the accumulated interest from previous periods. In other words, interest is added to the principal amount, and then interest is calculated on the new total amount.
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In ΔVWX, v = 33 inches, w = 57 inches and ∠X=147°. Find ∠W, to the nearest degree.
Smith, from Los Angeles, and Jones, from New York City, are both traveling to Kansas City, beginning their trips at the same time. The graph shows the distance in miles from Kansas City for each of them after they have both traveled for the same number of hours. After how many hours from the start of their journeys were they the same distance from Kansas City?
Answer: The answer should be 20 hours
Step-by-step explanation:
That is when the 2 lines meet up so that is when they are the same difference from Kansas City.
danielle wants to build a sandbox that has a perimeter of 46 feet. The length is 15ft. What is the width of their sand box?
Answer:
8 feet.
Step-by-step explanation:
Let's denote the width of the sandbox as "w".
We know that the perimeter of a rectangle is given by the formula:
Perimeter = 2(length + width)
So, we can plug in the given values:
46 = 2(15 + w)
Now, we can solve for "w":
46 = 30 + 2w
Subtracting 30 from both sides:
16 = 2w
Dividing by 2:
w = 8
Therefore, the width of the sandbox is 8 feet.
What values of y and z make
The value of y and z are 11 and 11.
How to determine the valuesWe can see from the diagram that the two triangles are of equal lengths.
To determine the value of the variables, we have to note that;
In triangle QRS, the adjacent sides is y + 20
In triangle VX, the adjacent side is y = 31
Now, equate the values
y + 20 = 31
collect like terms
y = 31 -20
y = 11
Also, the hypotenuse sides are equivalent, then
y + 3z + 28 = 6y + z - 5
collect the like terms
y - 6y + 3z - z = -5 - 28
add or subtract the values
-5y + 2z = -33
Substitute the values
2z = -33 + 55
2z = 22
z = 11
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Help with this trig identities problems.
1) Given csc Φ = 7/3 and cot Φ = - (2√10)/(3), find sec Φ.
2) Given that sec β = 6/5 and sin β > 0, find tan β and sin β.
Using trigonometric identities, we found that sec Φ = -7/(2√10), sin Φ = 3/7, tan β = √11/5, and sin β = √11/6 for the given values of csc Φ, cot Φ, and sec β.
1. We can start by using the Pythagorean identity to find the values of sin Φ:
[tex]sin^2[/tex] Φ + [tex]cos^2[/tex] Φ = 1
Since csc Φ = 1/sin Φ, we can substitute and solve for sin Φ:
1/(7/3) = sin Φ
sin Φ = 3/7
Next, we can use the fact that cot Φ = cos Φ/sin Φ:
cot Φ = cos Φ/(3/7) = - (2√10)/(3)
Simplifying this expression, we get:
cos Φ = - (2√10)/(3) * (3/7) = - 2√10/7
Finally, we can use the fact that sec Φ = 1/cos Φ:
sec Φ = 1/(- 2√10/7) = -7/(2√10)
2. We can use the fact that sec β = 1/cos β to find the value of cos β:
sec β = 6/5
cos β = 5/6
Next, we can use the Pythagorean identity to find the value of sin β:
[tex]sin^2[/tex] β + [tex]cos^2[/tex] β = 1
sin β = √(1 - [tex]cos^2[/tex] β) = √(1 - 25/36) = √(11/36) = √11/6
Finally, we can use the fact that tan β = sin β/cos β:
tan β = (√11/6)/(5/6) = √11/5
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3/5 = 12/20 and 1/20 = 1/20
Answer:
Step-by-step explanation:
Yes, that's correct!
3/5 can be simplified to 12/20 by multiplying the numerator and denominator by 4:
3/5 x 4/4 = 12/20
And 1/20 is already in its simplest form, so it is equal to 1/20.
D.
At Cameron Elementary, 2 out of every 3 teachers are women. If there are 45 teachers
teaching at Cameron, how many of those are men? (2pts)
Answer: 15 of those teachers are men
Step-by-step explanation: 2 out of 3, is of course written as 2/3
if 2/3 are women, this means 2/3 of 45 is 30, meaning 30 of the teachers are women, so if 30/45 are women, the remaining 15 are men
What is the scale factor?
The scaling factor for given transformation is 3.
What exactly is scaling factor?
The scaling factor, also known as the scale factor, is a value that indicates how much a geometric figure is enlarged or reduced. It is the ratio of the length of a side or dimension of the original figure to the length of the corresponding side or dimension of the new figure. The scaling factor can be greater than 1, which indicates an enlargement, or less than 1, which indicates a reduction. When the scaling factor is 1, the two figures are congruent, which means they have the same shape and size.
Now,
We can find the scale factor by comparing the corresponding side lengths of the two polygons. Let's start by finding the side lengths of the original polygon:
HI: √[(4-0)² + (-5+3)²] = √20
IJ: √[(6-4)² + (0-0)²] = 2
JK: √[(4-6)² + (5-0)²] = √41
KL: √[(0-4)² + (3-5)²] = √20
LH: √[(0-0)² + (-3+5)²] = 2
Now let's find the corresponding side lengths of the dilated polygon:
H'I': √[(12-0)² + (-15+9)²] = √600
I'J': √[(18-12)² + (0-0)²] = 6
J'K': √[(12-18)² + (15-0)²] = √600
K'L': √[(0-12)² + (9-0)²] = √600
L'H': √[(0-0)² + (-9+3)²] = 6
We can see that the side lengths of the dilated polygon are all 3 times as long as the corresponding side lengths of the original polygon. Therefore, the scale factor is 3.
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Help with math problems
The vertex form of the quadratic equations in standard form are, respectively:
Case 9: y = 2 · (x + 2)² - 12
Case 10: y = - (1 / 2) · (x + 3 / 4)² + 33 / 32
Case 11: y = 3 · (x - 4 / 3)² - 16 / 3
Case 12: y = - 3 · (x - 3)²
Case 13: y = (x - 4)² + 3
Case 14: y = (x - 1)² - 7
Case 15: y = (x + 3 / 2)² - 9 / 4
Case 16: 2 · (x + 1 / 4)² - 1 / 8
Case 17: y = 2 · (x - 3)² - 7
Case 18: y = - 2 · (x + 1)² + 10
How to derive the vertex form of a quadratic equationIn this problem we find ten cases of quadratic equation in standard form, whose vertex form can be found by a combination of algebra properties known as completing the square. Completing the square consists in simplifying a part of the quadratic equation into a power of a binomial.
The two forms are introduced below:
Standard form
y = a · x² + b · x + c
Where a, b, c are real coefficients.
Vertex form
y - k = C · (x - h)²
Where:
C - Vertex constant(h, k) - Vertex coordinates.Now we proceed to determine the vertex form of each quadratic equation:
Case 9
y = 2 · x² + 4 · x - 4
y = 2 · (x² + 2 · x - 2)
y = 2 · (x² + 2 · x + 4) - 12
y = 2 · (x + 2)² - 12
Case 10
y = - (1 / 2) · x² - 3 · x + 3
y = - (1 / 2) · [x² + (3 / 2) · x - 3 / 2]
y = - (1 / 2) · [x² + (3 / 2) · x + 9 / 16] + (1 / 2) · (33 / 16)
y = - (1 / 2) · (x + 3 / 4)² + 33 / 32
Case 11
y = 3 · x² - 8 · x
y = 3 · [x² - (8 / 3) · x]
y = 3 · [x² - (8 / 3) · x + 16 / 9] - 3 · (16 / 9)
y = 3 · (x - 4 / 3)² - 16 / 3
Case 12
y = - 3 · x² + 18 · x - 27
y = - 3 · (x² - 6 · x + 9)
y = - 3 · (x - 3)²
Case 13
y = x² - 8 · x + 19
y = (x² - 8 · x + 16) + 3
y = (x - 4)² + 3
Case 14
y = x² - 2 · x - 6
y = (x² - 2 · x + 1) - 7
y = (x - 1)² - 7
Case 15
y = x² + 3 · x
y = (x² + 3 · x + 9 / 4) - 9 / 4
y = (x + 3 / 2)² - 9 / 4
Case 16
y = 2 · x² + x
y = 2 · [x² + (1 / 2) · x]
y = 2 · [x² + (1 / 2) · x + 1 / 16] - 2 · (1 / 16)
y = 2 · (x + 1 / 4)² - 1 / 8
Case 17
y = 2 · x² - 12 · x + 11
y = 2 · (x² - 6 · x + 9) - 2 · (7 / 2)
y = 2 · (x - 3)² - 7
Case 18
y = - 2 · x² - 4 · x + 8
y = - 2 · (x² + 2 · x - 4)
y = - 2 · (x² + 2 · x + 1) + 2 · 5
y = - 2 · (x + 1)² + 10
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6.
CRITICAL THINKING Consider AJKL.
Rotate AJKL 90° clockwise about the origin.
How are the x- and y-coordinates of AJ'K'L'
related to the x- and y-coordinates of AJKL?
a.
The x- and y-coordinates of AJ'K'L' are related to the x- and y-coordinates of AJKL, as x-coordinate of AJ'K'L' = -y-coordinate of AJKL and y-coordinate of AJ'K'L' = x-coordinate of AJKL.
Explain coordinate plane?We create the x-axis and y-axis in the coordinate plane. the origin, where the x- and y-axes cross, and the coordinates ( 0,0 ). The horizontal axis represents the x-axis. The vertical axis represents the y-axis. The first and second coordinates are for the horizontal and vertical axes, respectively. On the coordinate plane, coordinates are expressed by ( x ,y ). The x-coordinate stands for the first coordinate.
When a point is rotated 90° clockwise about the origin, its x-coordinate becomes the negative of its original y-coordinate, and its y-coordinate becomes the positive of its original x-coordinate.
So, for point A(x,y), its new coordinates after a 90° clockwise rotation about the origin would be A'(-y,x). Similarly, for point J, K, and L, their new coordinates would be J'(-yj,xj), K'(-yk,xk), and L'(-yl,xl) respectively.
Therefore, the x- and y-coordinates of AJ'K'L' are related to the x- and y-coordinates of AJKL as follows:
x-coordinate of AJ'K'L' = -y-coordinate of AJKL
y-coordinate of AJ'K'L' = x-coordinate of AJKL
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Write Slope=3 y-intercept=-1/2 in slope intercept form
Answer:
y = 3x - [tex]\frac{1}{2}[/tex]
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
here m = 3 and c = - [tex]\frac{1}{2}[/tex] , then
y = 3x - [tex]\frac{1}{2}[/tex] ← equation of line