Answers
Hello there. To solve this question, we have to remember some properties about quadratic equations.
Given the following "quadratic" equation, we want to determine its solutions (if they exist) and the set of said solutions:
\(x^{-2}+6x^{-1}+5=0\)It is not exactly a quadratic equation, but doing the following substitution:
\(y=\dfrac{1}{x}\)(as long the solutions to this equation are not zero, we're good), we'll get
\(\begin{gathered} \left(\dfrac{1}{y}\right)^{-2}+6\cdot\left(\dfrac{1}{y}\right)^{-1}+5=0 \\ \\ y^2+6y+5=0 \\ \end{gathered}\)Solving it using the sum-to-product formula, we're looking for two roots such that
Their sum is equal to -6 and their product is equal to 5
In this case, both numbers have to be negative and satisfy these properties.
It is easy to see that the numbers will be:
\(y=-5\text{ and }y=-1\)Now, we undo the substitution and solve for x:
\(\begin{gathered} \dfrac{1}{x}=-5\Rightarrow x=-\dfrac{1}{5} \\ \\ \dfrac{1}{x}=-1\Rightarrow\,x=-1 \end{gathered}\)Therefore the set of solutions for this equation is:
\(\left\{-\dfrac{1}{5},\,-1\right\}\)The order does not matter in this situation because the set
\(\left\{-1\,-\dfrac{1}{5}\right\}\)Is the same as the other one, because it contains the same elements.
Related Questions
Let B={b 1
…,b n
} be a basis for a vector space ∨. Explain why the B-coordinate vectors of b 1
,…,b n
are the columns e 1
……e n
of the n×n identity matrix. Let B={b 1
,…,b n
} be a basis for a vector space V. Which of the following statements are true? Select all that apply. By the definition of a basis, b 1
,…,b n
are in V. By the definition of a basis, b 1
,…,b n
are linearly dependent. By the definition of an isomorphism, V is isomorphic to R n+1
. By the Unique Representation Theorem, for each x in V, there exists a unique set of scalars c 1
,…,c n
such that x=c 1
b 1
+⋯+c n
b n
. Since b 1
,…,b n
are in V and since for each x in V 1
there exists a unique set of scalars c 1
,…,c n
such that x=c 1
b 1
+⋯+c n
b n
, what is true of each b k
for k=1,…,n ? A. b k
=c 1
b 1
+⋯+c n
b n
for some unique set of scalars c 1
,…,c n
B. b k
=b 1
+⋯+b k
C. b k
=c 1
b 1
+⋯+c k−1
b k−1
+c k+1
b k+1
+⋯+c n
b n
for some unique set of scalars c 1
,…,c k−1
,c k+1
,…,c n
Answers
The correct option is A. bk = c1b1 + ⋯ + cnbn for some unique set of scalars c1, ..., cn.
The columns e1…en of the n×n identity matrix are the B-coordinate vectors of b1,…,bn.
By definition, a vector v in the vector space is expressed in terms of the basis vectors b1, ..., bn as a linear combination of these vectors.
For each basis vector bk, there is a unique set of scalars c1, ..., cn such that bk = c1b1 + ⋯ + cnbn
.In a basis B = {b1, ..., bn} for a vector space V, the following statements are true:By definition of a basis, b1, ..., bn are in V.
By the Unique Representation Theorem, for each x in V, there exists a unique set of scalars c1, ..., cn such that x = c1b1 + ⋯ + cnbn. Therefore, bk = c1b1 + ⋯ + cnbn is true for each bk.
The correct option is A. bk = c1b1 + ⋯ + cnbn for some unique set of scalars c1, ..., cn.
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Line L is tangent to the graph of y = ex at the point (k, ek y-intercept of L is 1/2 (A) 0.405 (B) 0.768 (C) 1.500 (D) 1.560 (E) There is no such value of k
Answers
Since x > 0, the tangent of f'(x) is always positive, and the denominator is always positive. Therefore, f'(x) is always positive, which means that f(x) is always increasing and can only cross the x-axis once. Hence, f(x) has only one root on its entire domain.
To find the y-intercept of line L, we need to determine the equation of line L.
Since line L is tangent to the graph of y = ex at the point (k, ek), the slope of line L must be equal to the slope of the tangent line to y = ex at (k, ek), which is simply ek.
Therefore, the equation of line L is:
y - ek = ek(x - k)
Simplifying, we get:
y = ekx - ek2
To find the y-intercept, we set x = 0 and solve for y:
y = ek(0) - ek2 = -ek2
Therefore, the y-intercept of line L is -ek2.
We need to find the value of k such that the y-intercept of line L is 1/2.
-ek2 = 1/2
Solving for k, we get:
k = ln(sqrt(2))
So the answer is (A) 0.405.
To prove that the function has only one root on its entire domain, we take the derivative of f(x) and show that it is always positive.
f(x) = ln(x) - 1/x
f'(x) = 1/x + 1/x^2 = (x+1)/x^2
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Can someone please help me with this question?
A semicircle has radius 5.6cm. Work out the perimeter of the semicircle.
Answers
Answer:
28.79 cm.
Step-by-step explanation:
The perimeter of a semicircle is equal to the sum of the diameter (which is twice the radius) and half of the circumference of the circle.
The diameter of the semicircle is 2 × 5.6cm = 11.2cm.
The circumference of a full circle with radius 5.6cm is 2π × 5.6cm = 11.2π cm.
Therefore, the circumference of the semicircle is 1/2 of the circumference of the full circle, which is 1/2 × 11.2π cm = 5.6π cm.
The perimeter of the semicircle is the sum of the diameter and the circumference, which is:
11.2cm + 5.6π cm ≈ 28.79cm (rounded to two decimal places)
Therefore, the perimeter of the semicircle is approximately 28.79 cm.
Answer:
pi ( 5.6 ) + 11.2 or approximately 28.79291 cm
Step-by-step explanation:
The outside of the semicircle or the perimeter is 1/2 of the circumference and the length that closes the circle is the diameter.
Perimeter = 1/2 C + d
= 1/2 *pi*d + d
The diameter is 2 times the radius.
= 1/2 ( pi) * (2r) + 2r
= pi r + 2 r
= pi ( 5.6 ) + 11.2
Approximating pi, we get approximately 28.79291
Find the value of x in the triangle shown below.
7
3
C
Answers
Answer:
\( \sqrt{58} \)
Answer:
B
Step-by-step explanation:
Using Pythagoras' identity in the right triangle.
The square on the hypotenuse is equal to the sum of the squares on the other 2 sides , that is
x² = 3² + 7² = 9 + 49 = 58 ( take square root of both sides )
x = \(\sqrt{58}\) → B
What is the slope and y intercept of the graph?
Captionless Image slope = 1 y-intercept = 2 slope = 1/2 y-intercept = 2 slope = 2 y-intercept = 1 slope = 2 y-intercept = -2
Answers
Answer:
y-intercept = 2
slope = 1
Step-by-step explanation:
each time the graph point goes up 1 over 1 giving you the fraction 1/1 which = 1
if you go straight to the point where the line crosses over the y line you'll see that it passes over 2 that's why the y-intercept is 2
following both of these rules that's how you get the y-intercept of 2 and the slope of 1
Answer:
hey
Step-by-step explanation:
Rose is 60 inches tall. how many feet tall is Rose
Answers
Answer:
5 feet
Step-by-step explanation: 60 divided by 12 inches is 5 feet
Help pls I got an email from school Bc I didn’t finish this and I need to finish it before next class and I have like 6 more checks to do after eeeeeee!!!!!
Answers
Answer:
169.25 \(inches^{2}\)
Jerry, Stein, Johnson, and Mary had a competition to see who could profit the most off of their odd jobs during the summer. They discussed their earnings on the first day of school. Afterwards, each of them decided to make bar graphs to plot the different amounts they earned.
Who made the above graph, and why? a. Johnson, because he wanted to make the amount made by each person appear very different. b. Jerry, because he wanted to accurately show the amount made by each person. c. Stein, because he wanted to make it look like he earned significantly more than the others. d. Mary, because she wanted to make the amount made by each person appear reasonably close.
Answers
Option (C) is correct Stein because he wanted to make it look like he earned significantly more than others.
What is Profit?Profit is the term used to describe the monetary gain experienced when the revenue from a commercial activity outpaces the costs, costs, and taxes incurred to support the activity in question.
Any gains made go back to the company's owners, who can decide whether to keep the money for themselves, give it to shareholders as dividends, or put it to use again in the company.
Jerry, Stein, Johnson, and Mary discussed their earnings on first day of school. Afterwards each of them decided to make bar graph to plot different amounts they earned.
By the graph we can see that Stein made the graph.
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Estimate to the nearest ten, and then subtract. 139 - 29
Answers
The result of the subtraction is 110.
When we round a number to the nearest ten, we are looking for the multiple of 10 that is closest to that number. In this case, 139 is closer to 140 than it is to 130, so we round it up to 140. Similarly, 29 is closer to 30 than it is to 20, so we round it up to 30.
Once we have rounded the numbers to the nearest ten, we can perform the subtraction operation. Subtracting 30 from 140 gives us:
140 - 30 = 110
So, the result of the subtraction after rounding the numbers to the nearest ten is 110.
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A line has a slope of - Which ordered pairs could be points on a parallel line? Select two options.
(-8, 8) and (2, 2)
(-5, -1) and (0, 2)
(-3, 6) and (6,-9)
(-2, 1) and (3,-2)
(0, 2) and (5, 5)
Answers
The ordered pairs that could be points on a parallel line are:
(-8, 8) and (2, 2)
(-2, 1) and (3, -2)
Which ordered pairs could be points on a parallel line?Parallel lines have the same slope. Thus, we have to find ordered pairs with a slope of -3/5.
We have:
slope of the line is -3/5.
Thus, m = -3/5
Formula for slope between two coordinates is;
m = (y₂ - y₁)/(x₂ - x₁)
A) At (–8, 8) and (2, 2);
m = (2 - 8)/(2 - (-8))
m = -6/10
m = -3/5
B) At (–5, –1) and (0, 2);
m = (2 - (-1))/(0 - (-5))
m = 3/5
C) At (–3, 6) and (6, –9);
m = (-9 - 6)/(6 - (-3))
m = -15/9
m = -5/3
D) At (–2, 1) and (3, –2);
m = (-2 - 1)/(3 - (-2))
m = -3/5
E) At (0, 2) and (5, 5);
m = (5 - 2)/(5 - 0)
m = 3/5
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els 9, Gaps, and Peaks on a Dot Plot Runs Scored Per Game The dot plot shows the number of runs scored by the boy's baseball team for each game last season. Which describes the data in the dot plot? Select all that apply. There are two gaps in the data. O There is a cluster between 3 and 6. • • O There is a cluster at 7. . + O There is a peak at 5. 3 4 5 6. 7 There is a gap between 6 and 8. Intro Done C
Answers
Answer:
ciu4rhiydve
Step-by-step explanation:
mrclutch master on fortnite add me yoclapz_yt
Answer:
2. There is a cluster between 3 and 6.
4. There is a peak at 5.
5. There is a gap between 6 and 8.
(d) does the technical condition which states each group should have approximately equal population standard deviations appear to be satisfied on the transformed data? explain.
Answers
Based on the given information, it is unclear what type of data transformation has been performed. It is difficult to determine whether the technical condition of having approximately equal population standard deviations across each group has been satisfied.
However, if the data transformation has not significantly altered the standard deviations of the original data, then it is possible that this technical condition has been met. A statistical test, such as Levene's test, can be used to assess whether the assumption of equal variances has been violated. If the p-value from this test is greater than the significance level (e.g., 0.05), then we can conclude that there is no evidence to suggest that the population standard deviations are significantly different between groups.
Overall, without more information about the type of data transformation and the results of statistical tests, it is difficult to definitively state whether the technical condition has been satisfied on the transformed data.
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math help please! - ( 2/5 )
Answers
Answer:
D
Step-by-step explanation:
it is D because you always want x to be a lone fraction and that would be the first step im 99% sure unless someone can correct me :)
Answer:
d) subtract 15 from both sides
Step-by-step explanation:
Process of elimination:
a) 2.5 and x are being multiplied, not subtracted, so (a) is wrong right off the bat.
b) you may divide 2.5 by both sides, but you want to get "x" on one side first. So you would do (b) first either.
c) to get rid of 15, which we need to somehow, we need to do the opposite operation than the one in the equation. 15 is +15, not negative, and so you would need to subtract to get rid of 15. This is why (c) is also wrong.
That leaves us, with (d), because you need to subtract 15 to isolate "x".
Is `(2,5)` a solution to the system:
`x+y=7`
`2x-3y=-11`
Answers
Answer:
the answer is in the picture.. if you need to actually solve the system please tell me and i will send that too
Help me pls, giving brainliest.multiple choice.
Answers
Answer:
E
Step-by-step explanation:
It has the degree of 9
The leading coefficient is 6
It has 5 terms
Subject: Trigonometry, Evaluation of Functions
Hi, I only need help matching the graphs to the questions :) Brainly only allows me to attach 5/10 of the graphs, if the ones i've attached aren't the answer i posted a similar question before this one with the rest of the graph choices. Sorry!!
1. Find <0 if cos0 = 4/7 and sin0 is negative.
2. Find <0 if sec0 = -13/5 and tan0 is negative.
3. Find <0 is sin0 = 7/10 and cot0 is positive.
4. Find <0 if tan0 = 3/4 and sin0 is negative.
Answers
(i) cosФ = 4/7 and sinФ is negative → It lies in the 4th quadrant 304.8° and the suitable graph is not provided here.
(ii) secФ = -13/5 and tanФ is negative → It lies in the IInd quadrant 112.6° and the suitable graph is J.
(iii) sinФ = 7/10 and cotФ is positive → It lies in Ist quadrant =44.4° and the suitable graph is I.
(iv) tan0 = 3/4 and sin0 is negative → It lies in the IIIrd quadrant 216.9° and the suitable graph is not provided here.
What is quadrant?
The point is in the first quadrant if both x and y were positive. If both x and y are negative, the point will reside in the second quadrant. In the third quadrant, if either x and y are negatives, the point is located. The point is situated in the fourth quadrant if both x and y are positive.(i) cosФ = 4/7 and sinФ is negative
Ф=cos^(-1)(4/7)= 55.2°
It lies in 4th quadrant (360-55.2)°=304.8°
A suitable graph is not provided here.
(ii) secФ = -13/5 and tanФ is negative.
Ф=sec^(-1)(-13/5)=67.4°
It lies in IInd quadrant (180-67.4°)=112.6°
A suitable graph is J.
(iii) sinФ = 7/10 and cotФ is positive.
Ф=sin^(-1)(7/10)=44.4°
It lies in Ist quadrant =44.4°
A suitable graph is I.
(iv) tan0 = 3/4 and sin0 is negative.
Ф=tan^(-1)(3/4)=36.9°
It lies in IIIrd quadrant (180+36.9)=216.9°
A suitable graph is not provided here.
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Use the Recidivism.xls data, construct a 95% confidence interval around the sample mean of AGE in Excel. Copy and paste the Excel output here. Interpret your results.
age
56
24
26
45
64
79
38
40
69
74
34
54
21
74
62
76
59
59
47
66
66
52
53
59
38
30
48
44
23
46
Answers
Using the mean and data given, the 95% confidence interval for the data is approximately (44.25, 59.35).
What is 95% confidence interval?To calculate the 95% confidence interval for the given data, we can use the following formula:
Confidence interval = mean ± (critical value * standard deviation / sqrt(n))
where:
mean is the average of the datacritical value is the value corresponding to the desired confidence level (in this case, 95%)standard deviation is a measure of the variability of the datan is the number of data pointsFirst, let's calculate the mean, standard deviation, and the critical value.
Mean (μ) = (56 + 24 + 26 + 45 + 64 + 79 + 38 + 40 + 69 + 74 + 34 + 54 + 21 + 74 + 62 + 76 + 59 + 59 + 47 + 66 + 66 + 52 + 53 + 59 + 38 + 30 + 48 + 44 + 23 + 46) / 30 = 51.8
Standard Deviation (σ) = √[(∑(x - μ)²) / (n - 1)]
= √[(∑(56 - 51.8)² + (24 - 51.8)² + ... + (23 - 51.8)² + (46 - 51.8)²) / (30 - 1)]
≈ 17.04
To determine the critical value, we need to consider the sample size and confidence level. Since the sample size is 30 and we want a 95% confidence level, we can use a t-distribution with 29 degrees of freedom. Looking up the critical value in a t-table or using a calculator, we find the value to be approximately 2.045.
Now, let's calculate the confidence interval:
Confidence interval = 51.8 ± (2.045 * 17.04 / √30)
≈ 51.8 ± 7.55
Therefore, the 95% confidence interval for the data is approximately (44.25, 59.35).
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HELP!!! GEOMETRY! What is ∠ P T R?
Answers
Answer:
Angle PTR and angle PTS are supplementary angles. Lines PQ and RS intersect at an angle less than a right angle. Angle PTR and angle PTS are complementary angles.
Step-by-step explanation:
The sum of a number and two times a smaller number is 62. Three times the bigger number exceeds the smaller number by 116.
Answers
Answer:
10 and 42
Step-by-step explanation:
The difficulty with word problems is translating them into math.
Let's do that
---------------------
The sum of a number and two times a smaller number is 62.
let's call the bigger number b, and the smaller number s
b + 2s = 62
Three times the bigger number exceeds the smaller number by 116
3b = s + 116
-----------------------
Now manipulate one of the equations to isolate the variable
3b = s + 116
Subtract 116 from both sides
3b - 116 = s
substitute for s = 3b - 116 in
b + 2s = 62
b + 2(3b - 116) = 62
Distribute
b + 6b - 232 = 62
combine like terms
7b = 294
Divide both sides by 7
b = 42
to find s plug in b = 42 into
b + 2s = 62
42 + 2s = 62
subtract 42 from both side
2s = 20
divide both sides by 2
s = 10
A patient asks about the purpose of withholding food and fluid before surgery. Which response by the nurse is appropriate?
a)It decreases urine output so that a catheter would not be needed.
b)It prevents overhydration and hypertension.
c)It decreases the risk of elevated blood sugars and slow wound healing.
d)It prevents aspiration and respiratory complications.
Answers
Withholding food and fluids before surgery is done to ensure that the patient's stomach is empty. This helps to minimize the risk of aspiration, which occurs when stomach contents enter the lungs. Aspiration can lead to respiratory complications such as pneumonia, which can be dangerous for the patient.
The appropriate response by the nurse is d) It prevents aspiration and respiratory complications. Withholding food and fluid before surgery is important to prevent aspiration, which occurs when stomach contents enter the lungs during surgery, and can cause respiratory complications. It also helps ensure a clear surgical field. However, the patient will still receive necessary fluids and medications through an IV during surgery to prevent dehydration and maintain blood pressure. It is important to follow the healthcare provider's instructions on pre-operative fasting to ensure the safest surgical experience.
d) It prevents aspiration and respiratory complications.
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the rate at which a pesticide is sprayed, through a large sprinkler system, on a field of vegetables is 2 gallons per hour. the sprinklers run from 3am until 9am each morning. write an integral to represent this situation and evaluate the integral. remember to include units.
Answers
The total amount of pesticide sprayed on the field of vegetables is 12 gallons, over the 6-hour period from 3am to 9am.
To represent the situation, we need to integrate the rate of the pesticide being sprayed over the time interval from 3am to 9am.
Let's set up the integral using the following variables:
t: time in hours, where t=0 corresponds to 3am and t=6 corresponds to 9am.
r(t): the rate of pesticide being sprayed in gallons per hour at time t.
From the problem statement, we know that r(t) = 2 gallons per hour for 3am <= t <= 9am.
Thus, the integral we need to evaluate is:
Integral from 0 to 6 of r(t) dt
= Integral from 0 to 6 of 2 dt
= 2t evaluated from 0 to 6
= 2(6) - 2(0)
= 12 gallons
Therefore, the total amount of pesticide sprayed on the field of vegetables is 12 gallons, over the 6-hour period from 3am to 9am.
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Does 2/6 have a repeating decimal form?
Answers
Answer:
Step-2/6 equals to 0. (3) It is, a so called, 'repeating' or 'recurring' decimal. The repeating decimal 0.3 (Vinculum notation) is also represented as 0.3333...
0715812 avatar
0.33333333333333 2/6 is equal to 0.33333333333333 in decimal form. Use our fraction to decimal calculator to convert any fraction to a decimal and to know if it is a terminating or a recurring (repeating) decimal.by-step explanation:
The fraction 2/6 have repeating decimal form.
What is Fraction?The fractional bar is a horizontal bar that divides the numerator and denominator of every fraction into these two halves.
The number of parts into which the whole has been divided is shown by the denominator. It is positioned in the fraction's lower portion, below the fractional bar.How many sections of the fraction are displayed or chosen is shown in the numerator. It is positioned above the fractional bar in the upper portion of the fraction.We have the fraction.
If we divide 2/6 we get
2/6
= 1/3
= 0.3333333333333333
Here, after decimal the number 3 is repeating.
Thus, 2/6 gives repeating decimal.
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Add.
(6x³ + 3x² − 2) + (x³ - 5x² − 3)
Express the answer in standard form. (Please and thank you)
Answers
Answer:
\(\\\sf7x^3 - 2x^2 - 5\)
Step-by-step explanation:
\(\\\sf(6x^3 + 3x^2 - 2) + (x^3 - 5x^2 - 3)\)
Remove parenthesis.
6x^3 + 3x^2 - 2 + x^3 - 5x^2 - 3
Rearrange:
6x^3 + x^3 + 3x^2 - 5x^2 - 2 - 3
Combine like terms to get:
7x^3 - 2x^2 - 5----------------------------------------
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Hope this helps! :)
Answer:
7x³ - 2x² - 5
Step-by-step explanation:
(6x³ + 3x² - 2) + (x³ - 5x² - 3)
Remove the round brackets.
= 6x³ + 3x² - 2 + x³ - 5x² - 3
Put like terms together.
= 6x³ + x³ + 3x² - 5x² - 2 - 3
Do the operations.
= 7x³ - 2x² - 5
____________
hope this helps!
3 * 2x * 4
simplify
in my textbook answer is shown as 24x i need to know it is 24x
* represents multiply symbol
Answers
Answer:
yes
Step-by-step explanation:
24 * 2 represents 24 × 2
I am not a fan of the use of the asterix to represent multiplication
The sum of two numbers is 45 and their product is 500. What are the two numbers?
Answers
Answer:be A & B A + B = 45. A = 45 - B (1) A x B = 500. But by (1), (45 - B) x B = 500 -B² + 45B = 500 -B² + 45B - 500 = 0
Step-by-step explanation:
tell whether x and y are in a proportional relationship if also write an equation that represents the relationship. for all for give a different explanation 20 points question
Answers
Answer: Two quanties are said to have a proportional relationship if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio. ... inversely proportional: y is inversely proportional to x of y=k/x.
5^2-log(5)5
pls help
Answers
Answer:
5
Step-by-step explanation:
log(5)5=1
5^2-1 = 5^1 = 5
"So for the following question, I got 6 switches (well, 5.025883189, but rounded up to the next whole number/whole switch), but my professor got 5 switches. I think it's because he changed the numbers for the question he assigned to us, but the answer he is using for his key is from an identical problem in our book with slightly different numbers. Image transcription textAssume that a electronic switch operates properly 60% of the time. What is the smallest
number of switches that can be wired in parallel to ensure the system functions (at least
one switch works) 99% of the time? ... Show moreLet me know if I'm crazy and my prof is right?"
Answers
The smallest number of switches that can be wired in parallel to ensure the system functions (at least one switch works) 99% of the time is 5 switches. your professor is right in stating that 5 switches are needed.
Based on the given question, the goal is to determine the minimum number of switches that should be wired in parallel to ensure the system functions at least 99% of the time.
To calculate this, we can use the concept of probability. Given that each switch operates properly 60% of the time, we can calculate the probability of a switch not functioning (40%).
To ensure the system functions at least 99% of the time, we need to calculate the probability of all the switches failing.
To find the probability of all switches failing, we can multiply the probability of each individual switch failing together.
Let's denote the number of switches as 'n'. So, the probability of all switches failing can be calculated as (0.4)^n.
We need to find the minimum value of 'n' for which this probability is less than or equal to 1% (0.01).
So, we can set up the inequality: (0.4)^n ≤ 0.01.
To solve this inequality, we can take the logarithm of both sides.
log((0.4)^n) ≤ log(0.01).
Using the logarithmic properties, this simplifies to:
n * log(0.4) ≤ log(0.01).
Now, we can solve for 'n':
n ≥ log(0.01) / log(0.4).
Calculating this value, we get n ≥ 4.124.
Since the number of switches must be a whole number, we need to round up to the next whole number. Thus, the minimum number of switches required is 5.
Therefore, your professor is correct in stating that 5 switches are needed to ensure the system functions at least 99% of the time.
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what is the y-intercept of the equation 6x-3y=36
Answers
Answer:
(0, -12)
Step-by-step explanation:
Using the slope-intercept form, the y-intercept is 12.
Explain your reasoning
Answers
The function g(x) can be derived from f(x) by vertical dilation and vertical translation. f(x) = (13 / 4) · x³ - (1 / 2) · x² + (7 / 4) · x + 3 / 2 and g(x) = (125 / 7306) · x³ - (125 / 47789) · x² + (875 / 94978) · x + 375 / 47489 + 2.502.
What kind of rigid translations should we use to modify a function and generate another one?
In this question we must infer what kinds of rigid transformations were used to obtain g(x) in terms of f(x). Points A and B belong to the function f(x) and points A' and B' lie on the curve of function g(x). Rigid transformations are transformations applied on functions such that Euclidean distance is conserved.
We noticed the use of rigid transformations on f(x) such that g(x) is found:
Vertical dilation
f'(x) = k · f(x), where k > 0
Vertical translation
g(x) = k · f(x) + c, where c is the translation factor.
We notice that both f(x) and g(x) are both cubic functions. The former has a vertex at (x, y) = (0, 1.5). First, we determine the coefficients of f(x) by solving the following system of linear equations:
(x₁, y₁) = (1, 6), (x₂, y₂) = (- 2, - 30), (x₃, y₃) = (0, 1.5), (x₄, y₄) = (- 1, - 4)
a + b + c + d = 6
- 8 · a + 4 · b - 2 · c + d = - 30
d = 1.5
- a + b - c + d = - 4
The solution of the system is (a, b, c, d) = (13 / 4, - 1 / 2, 7 / 4, 3 / 2). Then, the expression for the function f(x) is f(x) = (13 / 4) · x³ - (1 / 2) · x² + (7 / 4) · x + 3 / 2.
Now we evaluate the function f(x) at x = 7 and x = - 8:
f(7) = (13 / 4) · 7³ - (1 / 2) · 7² + (7 / 4) · 7 + 3 / 2
f(7) = 1104
f(- 8) = (13 / 4) · (- 8)³ - (1 / 2) · (- 8)² + (7 / 4) · (- 8) + 3 / 2
f(- 8) = - 3417 / 2
Now we determine the coefficient of dilation and the translation factor:
f(7) = 1104, f(- 8) = - 3417 / 2, g(7) = 2.5, g(- 8) = - 6.5
2.5 = 1104 · k + c
- 6.5 = - 3417 · k / 2 + c
The solution of the system of linear equations is (k, c) = (250 / 47489, 2.494). Then, we find that the expression for g(x) is g(x) = (250 / 47489) · [(13 / 4) · x³ - (1 / 2) · x² + (7 / 4) · x + 3 / 2] + 2.494, whose simplest form is:
g(x) = (125 / 7306) · x³ - (125 / 47789) · x² + (875 / 94978) · x + 375 / 47489 + 2.494
g(x) = (125 / 7306) · x³ - (125 / 47789) · x² + (875 / 94978) · x + 375 / 47489 + 2.502
The function g(x) can be derived from f(x) by vertical dilation and vertical translation. f(x) = (13 / 4) · x³ - (1 / 2) · x² + (7 / 4) · x + 3 / 2 and g(x) = (125 / 7306) · x³ - (125 / 47789) · x² + (875 / 94978) · x + 375 / 47489 + 2.502.
To learn more on rigid transformations: https://brainly.com/question/1761538
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