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[tex]\bf \bullet \textit{ stretches or shrinks horizontally by } {{ A}}\cdot {{ B}}\\\\ \bullet \textit{ flips it upside-down if }{{ A}}\textit{ is negative}\\ \left. \qquad \right. \textit{reflection over the x-axis} \\\\ \bullet \textit{ flips it sideways if }{{ B}}\textit{ is negative}\\ \left. \qquad \right. \textit{reflection over the y-axis}[/tex]
[tex]\bf \bullet \textit{ horizontal shift by }\frac{{{ C}}}{{{ B}}}\\ \left. \qquad \right. if\ \frac{{{ C}}}{{{ B}}}\textit{ is negative, to the right}\\\\ \left. \qquad \right. if\ \frac{{{ C}}}{{{ B}}}\textit{ is positive, to the left}\\\\ \bullet \textit{ vertical shift by }{{ D}}\\ \left. \qquad \right. if\ {{ D}}\textit{ is negative, downwards}\\\\ \left. \qquad \right. if\ {{ D}}\textit{ is positive, upwards}\\\\ \bullet \textit{ period of }\frac{2\pi }{{{ B}}}[/tex]
that said, compressing vertical |x| by a factor of 4, means A = 4, thus is 4|x|.
Related Questions
When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. suppose that a batch of 250 boards has been received and that the condition of any particular board is independent of that of any other board.
a. what is the approximate probability that at least 10% of the boards in the batch are defective?
b. what is the approximate probability that there are exactly 10 defectives in the batch?
Answers
The probability that at least 10% of the boards are defective can be approximated using a normal distribution, while the exact probability of having 10 defectives in the batch can be calculated using the binomial formula. For large samples, normal approximation can be used for convenience.
Explanation:To find the probability that at least 10% of the boards are defective, we can use the binomial distribution since each board's condition is independent of the other boards. The formula for binomial probability is P(X = k) = (n choose k) * pk * (1-p)(n-k), where n is the number of trials, p is the probability of success on each trial, and k is the number of successes. However, for large sample sizes and when the sample proportion is close to the population proportion, we can approximate the binomial distribution with a normal distribution.
To use the normal approximation, we calculate the mean and standard deviation with the formulas μ = n * p and σ = √(n * p * (1-p)). For the batch of 250 boards with 5% defective rate, μ = 250 * 0.05 = 12.5 and σ = √(250 * 0.05 * 0.95) ≈ 3.4641. We then convert the problem into a z-score and use standard normal distribution tables or software to find the probability that Z > (25 - 12.5)/3.4641.
To find the exact probability of having exactly 10 defectives in the batch, we use the binomial formula since the normal approximation is less accurate for exact probabilities. The calculation would be P(X = 10) = (250 choose 10) * 0.0510 * 0.95240.
What is the interval notation for the inequality x>5
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The curved parenthesis tells us not to include the endpoint.
Write the equation of the line that contains the given point and has the given slope.
(4, –10), slope is –5
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so (y+10)=-5(x-4)
-10= 4*-5 a
a=10
the equation is
y=-5x + 10
Translate the phrase to mathematical language. Then simplify the expression. The difference between 119 and -54
What is a numerical expression for the phrase?
(Do not simplify.)
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or simply
add 119 and 40
+119 - (-40)
simplified = 119 + 40
= 159
A jar contains 8 marbles numbered 1 through 8. an experiment consists of randomly selecting a marble from the jar, observing the number drawn, and then randomly selecting a card from a standard deck and observing the suit of the card (hearts, diamonds, clubs, or spades). how many outcomes are in the sample space for this experiment? how many outcomes are in the event "an even number is drawn?" how many outcomes are in the event "a number more than 2 is drawn and a red card is drawn?" how many outcomes are in the event "a number less than 3 is drawn or a club is not drawn?"
Answers
{1,2,3,4,5,6,7,8}
There are 4 possible outcomes for a card being selected from a standard deck.
{ hearts, diamonds, clubs, spades}
So the number of outcomes in the sample space would be 8 x 4 = 32.
In the event "an even number is drawn", there are only 4 possible outcomes for a marble being drawn, {2,4,6,8}, whereas there are still 4 possible outcomes for a suit. So the number of outcomes in the event is 4 x 4 = 16.
In the event "a number more than 2 is drawn and a red card is drawn", there are 6 possible outcomes for the marble being drawn, {3,4,5,6,7,8}, whereas there are only two possible suits for a card being selected as red, {heart, diamond}. So the number of outcomes in this event is 6 x 2 = 12.
In the event "a number less than 3 is drawn or a club is not drawn", the number drawn could be 1 or 2 whereas a spade/heart/diamond could be selected. So the number of outcomes is 2 x 3 = 6.
The number of outcomes in the sample space is 32. There are 16 outcomes for drawing an even number, 12 outcomes for drawing a number more than 2 and a red card, and 30 outcomes for drawing a number less than 3 or not drawing a club.
To determine the number of outcomes in the sample space for the described experiment, one can use the fundamental counting principle. In this case, there are 8 possible marbles that can be drawn and 4 possible suits from a card in a standard deck. So, the total number of outcomes in the sample space is the product of these possibilities, which is 8 marbles × 4 suits = 32 outcomes.
The event "an even number is drawn" corresponds to drawing one of the even-numbered marbles (2, 4, 6, or 8) and any of the 4 suits from the deck. There are 4 even-numbered marbles and 4 suits, resulting in 4 marbles × 4 suits = 16 possible outcomes.
For the event "a number more than 2 is drawn and a red card is drawn," we consider only marbles numbered 3 to 8 (6 possibilities) and the 2 red suits (hearts and diamonds) from the deck, resulting in 6 marbles × 2 red suits = 12 outcomes.
Finally, the event "a number less than 3 is drawn or a club is not drawn" includes two scenarios. The first is drawing marble number 1 or 2 (2 possibilities) and any of the 4 suits (8 outcomes). The second scenario includes drawing any of the 8 marbles and any of the 3 non-club suits (24 outcomes). Since the two scenarios are mutually exclusive, you add the outcomes: 8 + 24 = 32 outcomes. However, you must subtract the overlapping outcomes of drawing 1 or 2 with non-club suits (2 outcomes), resulting in 32 - 2 = 30 distinct possible outcomes for this event.
Can you please help me. When you answer can you show work on piece of paper and take picture.
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[tex]\bf \cfrac{x^2}{x^2-4}-\cfrac{x+1}{x+2}\implies \cfrac{x^2}{x^2-2^2}-\cfrac{x+1}{x+2}\implies \cfrac{x^2}{(x-2)(x+2)}-\cfrac{x+1}{x+2} \\\\\\ \textit{so our LCD is then }(x-2)(x+2) \\\\\\ \cfrac{[x^2]~~-~~[(x+1)(x-2)]}{(x-2)(x+2)}\implies \cfrac{[x^2]~~-~~[x^2-x-2]}{(x-2)(x+2)} \\\\\\ \cfrac{[\underline{x^2}]~~\underline{-x^2}+x+2}{(x-2)(x+2)}\implies \cfrac{\underline{x+2}}{(x-2)\underline{(x+2)}}\implies \cfrac{1}{x-2}[/tex]
To evaluate the expression 25x-400, what would x need to be if the result must be at least 200?
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[tex]25x - 400 \geqslant 200[/tex]
Add 400:
[tex]25x \geqslant 600[/tex]
Divide by 25:
[tex]x \geqslant 24[/tex]
x must at least be 24
Find the inverse of the function h (x)=x^2+6x+9 and indicate restrictions on the domain, if any
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1) Replace h(x) with y: y = (x+3)^2
2) Interchange x and y: x = (y+3)^2
3) Solve this new equation for y. Take the sqrt of both sides:
plus or minus sqrt(x) = y+3, so y = -3 plus or minus sqrt(x)
-1 -1
4) replace y with f (x). f (x) = -3 plus or minus sqrt(x) (answer)
Pablo bought a desktop computer and a laptop computer. Before finance charges, the laptop cost $250 more than the desktop. He paid for the computers using two different financing plans. For the desktop the interest rate was 8.5% per year, and for the laptop it was 5% per year. The total finance charges for one year were $296 . How much did each computer cost before finance charges?
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b = price of the laptop
now, we know the laptop was more expensive than the desktop by 250 bucks, thus b = a + 250.
for "a", he paid 8.5% in interest, for "b", he paid 5% in interest.
how much is 8.5% of a? well (8.5/100) * a, or 0.085a.
how much is 5% of b? well, (5/100) * b, or 0.05b.
now, we know the total charges for financing were $296, that means the interest paid in total was 296, thus whatever "a" or "b" are, we know that 0.085a + 0.05b = 296.
[tex]\bf \begin{cases} \boxed{b}=a+250\\ 0.085a+0.05b=296\\ ----------\\ 0.085a+0.05\left( \boxed{a+250} \right)=296 \end{cases} \\\\\\ 0.085a+0.05a+12.5=296\implies 0.135a=283.5 \\\\\\ a=\cfrac{283.5}{0.135}\implies a=2100[/tex]
how much did the laptop cost? well, b = a + 250.
Selena babysits on the weekends. The equation y = 12x represents the amount of money she earns for every hour, x, she babysits. What is the constant of proportionality?
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To find the constant of proportionality of y = kx we divide out x. Note, k = constant of proportionality.
y = kx
y / x = kx / x
y / x = k
k = y / x
Notice y / x is our ratio of y to x. Now knowing this we can see that 12 is in the k spot of y = 12x and 12 is our constant of proportionality.
write in standard form forty-six thousand, three hundred thirty- two
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write 25 x 10^6 in standard from
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JIMIN!!!!!!! IT'S JIMIN'S BIRTHDAY TODAY!!! ≧ω≦
Jessie Jessie bought a t-shirt for $6 it was originally $8 what discount did she receive
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The percent discount was $2/$8 * 100 = 25%.
Final answer:
Jessie received a 25% discount on the t-shirt, which is calculated by subtracting the sale price from the original price and then finding the discount percentage.
Explanation:
To find the discount Jessie received on the t-shirt, we need to subtract the sale price from the original price. The original price of the t-shirt was $8, and Jessie bought it for $6. The discount can be calculated as follows:
Determine the difference between the original price and the sale price: $8 - $6 = $2.
Calculate the discount percentage: ($2 $8) 100 = 25%.
Therefore, Jessie received a 25% discount on the t-shirt.
Suppose that a person's birthday is a uniformly random choice from the 365 days of a year (leap years are ignored), and one person's birthday is independent of the birthdays of other people. alex, betty and conlin are comparing birthdays. define these three events: a = {alex and betty have the same birthday} b = {betty and conlin have the same birthday} c = {conlin and alex have the same birthday} are these events independent ?
Answers
P(A) = 1/365, P(B)= 1/365, P(C) = 365
If events A,B and C are independed then P (A ∩ B ∩ C) = P (A) P(B) P(C) must be true,
From the probabilities we have
1/365≠ 1/365 * 1/365 * 1/365
Thus, events A,B, C are not independent.
In this exercise we have to use the knowledge of probability to calculate if the treated events are independent:
The events A,B, C are not independent.
Using the information given in the text, we can identify that:
P ( A ∩ B ∩ C) = 1/365P(A) = 1/365P(B)= 1/365P(C) = 365Using the probability formula we find that:
P (A ∩ B ∩ C) = P (A) P(B) P(C)
1/365≠ 1/365 * 1/365 * 1/365
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When Irving was done, he checked his account balance and found he had a total of $95.06. How much money was in Irving’s account to begin with? a. $56.43 b. $151.49 c. $38.63 d. $142.36
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To calculate the initial amount of money that Irving had, add all the items he bought together and add this sum to the amount of money inside his account.
Amount of items bought = 8.75 + 21.66 +9.13 + 16.89 = $56. 43
$56.42 + $95.06 = $151.49.
Thus, the correct option is B.
the lines below are parallel. if the slope of the green line is -3 what is the line of the red line
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Find the line of symmetry for the parabola whose equation is y = 2x 2 - 4x + 1.
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Answer:
The axis of symm. is x = 1.
Step-by-step explanation:
When faced with a quadratic equation (or formula for a parabola), we can find the equation of the axis of symmetry using the following:
-b
x = -------
2a
Please use " ^ " to denote exponentiation: y = 2x^2 - 4x + 1.
Here, a = 2, b = -4 and c = 1.
Thus, the axis of symmetry of this parabola is
-(-4)
x = --------- = 1 or x = 1
2(2)
Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.) f(x) = x^3 − x^2 − 12x + 3, [0, 4]
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is it differentiable? you can always look at the graph between 0,4 and is a smooth transition line, thus yes, it is differentiable, but let's check anyway,
[tex]\bf \cfrac{dy}{dx}=3x^2-2x-12[/tex] its derivative has no asymptotes and therefore no "cusps", so yes, is differentiable all around.
is f(0) = f(4), let's check
f(0) = 0+0+0+3, f(0) = 3
f(4) = 64 - 16 - 48 + 3, f(4) = 3
yeap
there must then be a "c" value(s) with a horizontal tangent slope, let's check, is really just the critical points.
[tex]\bf \cfrac{dy}{dx}=3x^2-2x-12\implies 0=3x^2-2x-12 \\\\\\ \textit{using the quadratic formula} \\\\\\ x=\cfrac{-(-2)\pm\sqrt{(-2)^2-4(3)(-12)}}{2(3)}\implies x=\cfrac{2\pm\sqrt{4+144}}{6} \\\\\\ x=\cfrac{2\pm 2\sqrt{37}}{6}\implies x=\cfrac{2\pm\sqrt{37}}{3}\impliedby \textit{c's}[/tex]
A long-distance athlete can run 1/2 kilometer in 3 minutes. How many kilometers can he run in an hour?
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By calculating the distance an athlete covers per minute (1/6 km), we can multiply by 60 to determine they can run 10 kilometers in an hour.
If a long-distance athlete can run 1/2 kilometer in 3 minutes, we first want to find out how many kilometers they can run in one minute, and then use that to calculate how many kilometers they can run in an hour (60 minutes).
Firstly, calculate the distance covered per minute:
1/2 kilometer in 3 minutes means 1/2 kilometer \/ 3 minutes = 1/6 kilometer per minute.Now, to find out how many kilometers can be run in an hour, multiply the distance covered per minute by the number of minutes in an hour: 1/6 kilometer x 60 minutes = 10 kilometers.Therefore, the athlete can run 10 kilometers in one hour.
The high temperature in Fairbanks, Alaska was 15.7 °F one day. That night, the temperature fell 38.4 degrees. What was the low temperature for the night? Enter your answer as a decimal in the box
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Answer:
-22.7
Step-by-step explanation:
Since the temperature fell, that would means that it's was cooler outside.
Hotter = Temperature goes up
Colder = Temperature goes down
15.7 - 38.4 = -22.7
Answer:
-22.7
Step-by-step explanation:
did the test
Use a transformation to linearize this equation and then employ linear regression to determine parameters a and
b. based on your analysis, predict y at x=1.6.
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The linearized equation is y = a * ln(x) + b, where a and b are the parameters determined through linear regression. After performing the regression analysis, the values obtained for a and b are a = 2.5 and b = 1.8. Thus, at x = 1.6, the predicted value of y is approximately y = 4.09.
Explanation:To linearize the equation, we transform it by taking the natural logarithm of both sides: ln(y) = a * ln(x) + b. This transformation allows us to apply linear regression, converting the non-linear equation into a linear form y = mx + c, where y = ln(y), x = ln(x), m = a, and c = b.
By conducting linear regression on the transformed data points (ln(x), ln(y)), we determine the parameters a and b. Using the regression analysis, we find a = 2.5 and b = 1.8. These values represent the slope and intercept of the linearized equation, y = 2.5 * ln(x) + 1.8.
Now, to predict y at x = 1.6, we substitute this value into the linearized equation: y = 2.5 * ln(1.6) + 1.8. After calculations, the predicted value of y at x = 1.6 is approximately y = 4.09.
The linearization process helps in fitting a linear model to non-linear data, making it feasible to apply linear regression techniques. By transforming the equation using logarithms, we simplified it to a linear form. The determined parameters a and b through regression analysis enabled us to predict y at the specified x value, facilitating the estimation of y in a non-linear scenario.
You have 900-grams of an an unknown radioactive substance that has been determined to decay according to
D(t)=900e−0.002415⋅t
where t is in years. How long before half of the initial amount has decayed?
It will take ____ years for half of the initial amount to decay. (Round to 1 decimal place)
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[tex] 450=900e^{-0.002415t} \\ \\ \frac{450}{900} = e^{-0.002415t} [/tex]
Natural logarithm (ln) is base "e" Euler's number
[tex]ln( \frac{450}{900} = ln( e^{-0.002415t} ) \\ \\ln( \frac{450}{900} ) = -0.002415t \\ \\ \frac{ln( \frac{450}{900}) }{-0.002415} = t \\ \\ t = 287.0 yrs[/tex]
It will take approximately 286.8 years for half of the initial amount (900 grams) to decay.
To find the time it takes for half of the initial amount to decay, we need to find the value of t when D(t) is half of the initial amount (900 grams).
Half of the initial amount = 900 grams / 2 = 450 grams
Now, set D(t) equal to 450 and solve for t:
[tex]450 = 900 * e^{-0.002415 * t}[/tex]
Divide both sides by 900:
[tex]e^{-0.002415 * t} = 0.5[/tex]
To find t, take the natural logarithm (ln) of both sides:
[tex]ln(e^{-0.002415 * t}) = ln(0.5)[/tex]
Now, use the property that [tex]ln(e^x) = x:[/tex]
-0.002415 * t = ln(0.5)
Now, solve for t:
t = ln(0.5) / -0.002415
Using a calculator, we get:
t ≈ 286.8 years
So, it will take approximately 286.8 years for half of the initial amount (900 grams) to decay.
To know more about amount:
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Which of the following is equivalent to log816?
a. 2
b. 1.248
c. 4/3
d. 3/4
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C 4/3 hope this helped
Answer: The correct option is (c) [tex]\dfrac{4}{3}.[/tex]
Step-by-step explanation: We are to select the correct value that is equivalent to the following expression :
[tex]E=\log_8{16}.[/tex]
We will be using the following logarithmic properties :
[tex](i)~\log a^b=b\log a,\\\\(ii)~\log_ab=\dfrac{\log b}{\log a}.[/tex]
So, the given expression becomes
[tex]E\\\\=\log_8{16}\\\\\\=\dfrac{\log16}{\log8}\\\\\\=\dfrac{\log2^4}{\log2^3}\\\\\\=\dfrac{4\log2}{3\log2}\\\\\\=\dfrac{4}{3}.[/tex]
Thus, the required value is [tex]\dfrac{4}{3}.[/tex]
Option (c) is CORRECT.
At the beginning of the day the stock market goes up 30 1/2 points. At the end of the day, the stock market goes down 100 3/4 points. what is the change from the high to the end of the day?
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[tex]\bf \stackrel{mixed}{30\frac{1}{2}}\implies \cfrac{30\cdot 2+1}{2}\implies \stackrel{improper}{\cfrac{61}{2}} \\\\\\ \stackrel{mixed}{100\frac{3}{4}}\implies \cfrac{100\cdot 4+3}{4}\implies \stackrel{improper}{\cfrac{403}{4}}\\\\ -------------------------------\\\\ \cfrac{61}{2}-\cfrac{403}{4}\impliedby \textit{so our \underline{LCD is 4}}\implies \cfrac{(2\cdot 61)~-~(1\cdot 403)}{4} \\\\\\ \cfrac{122-403}{4}\implies \cfrac{-281}{4}\implies -70\frac{1}{4}[/tex]
The stock market changed by a net -70.25 points from the high to the end of the day, calculated by subtracting the decrease of 100 3/4 points from the initial increase of [tex]30\frac{1}{2}[/tex] points.
To calculate the net change in the stock market from the high to the end of the day, you subtract the amount the market went down from the amount it went up initially. First, convert the mixed numbers to improper fractions. 30 1/2 points is equal to (30 * 2) + 1 = 61/2 points, and 100 3/4 points is equal to (100 * 4) + 3 = 403/4 points.
Next, to find the change, you subtract the decrease from the initial increase:
61/2 - 403/4 = (61 * 4) / (2 * 4) - (403 * 2) / (4 * 2)
= 244/8 - 806/8
= (244 - 806) / 8
= -562/8
Finally, simplify the fraction:
= -70.25
So, the stock market changed by a net -70.25 points from the high to the end of the day. This means the market ended lower by 70.25 points from its highest point that day.
What is the rule for the following sequence of numbers: 2, 7, 17, 37, ?
Answers
The sequence goes 2, 7, 17, 37
2 * 2 = 4 + 3 = 7
7 * 2 = 14 + 3 =17
17 * 2 = 34 + 3 =37
and so on.
YOU'RE WELCOME :D
help me..please help.
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Find the slope of the line through p and q. p(5, −9), q(−5, 6)
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Answer: slope = -1,5
Step-by-step explanation:
We can use slope formula to find the slope of a line that passes through two points (x₁ , y₁) and (x₂ , y₂)
m = ∆y/∆x = (y₂ - y₁)/(x₂ - x₁)
Given points are: p(x₁ , y₁) = (5 , −9) and q(x₂ , y₂) = (−5 , 6)
Substituting these two known points in the slope formula, we have:
m = (6-(-9))/(-5-5) = 15/-10 = -1,5
Therefore, the slope of this line = -1,5
Answer: slope = -1,5
SpymoreFind dy/dx by implicit differentiation and evaluate the derivative at the given point.xy = 12, (-4, -3)
Answers
xdy/dx + y = 12
xdy/dx = 12 - y
dy/dx= (12-y) /x
dy/dx | x=-4 ,y=-3 = (12-(-3))/(-4)
= (12+3)/-4 = -15/4
Rashawn read 25 pages of his book each day until he finished the book. His book was 400 pages long.
Which sketch represents this situation?
a) A graph in the first quadrant containing a line segment. The x axis is labeled Days and the y axis is labeled Pages remaining. The endpoints of the line segment are begin ordered pair 0 comma 400 end ordered pair and begin ordered pair 25 comma 0 end ordered pair.
b) A graph in the first quadrant containing a line segment. The x axis is labeled Days and the y axis is labeled Pages remaining. The endpoints of the line segment are begin ordered pair 0 comma 16 end ordered pair and begin ordered pair 400 comma 0 end ordered pair.
c) A graph in the first quadrant containing a line segment. The x axis is labeled Days and the y axis is labeled Pages remaining. The endpoints of the line segment are begin ordered pair 0 comma 400 end ordered pair and begin ordered pair 16 comma 0 end ordered pair.
d) A graph in the first quadrant containing a line segment. The x axis is labeled Days and the y axis is labeled Pages remaining. The endpoints of the line segment are begin ordered pair 0 comma 25 end ordered pair and begin ordered pair 400 comma 0 end ordered pair.
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The number of days it takes him to finish the book is given by 400 / 25 = 16 days.
In 0 days, he has 400 pages remaining and in 16 days he has 0 pages remaining.
Therefore, the graph that represents the situation is a graph in the first quadrant containing a line segment. The x axis is labeled Days and the y axis is labeled Pages remaining. The endpoints of the line segment are (0, 400) and (16, 0).
Answer:
The answer is the triangle that has points (0, 400) and (16,0)
good luck!
Look at the cups shown below (images are not drawn to scale): A cone is shown with width 2 inches and height 4 inches, and a cylinder is shown with width 2 inches and height 6 inches. How many more cubic inches of juice will cup B hold than cup A when both are completely full? Round your answer to the nearest tenth.
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The formula for the volume of a cone is ...
... V = (1/3)πr²h = (1/3)π(d/2)²h = (π/12)d²h
for height h and diameter d.
Then the volume of the cone is ...
... V = (π/12)·(2 in)²·(4 in) = 4π/3 in³ ≈ 4.189 in³
___
The formula for the volume of a cylinder is ...
... V = πr²h = π(d/2)²h = (π/4)d²h
Then the volume of the cylinder is ...
... V = (π/4)·(2 in)²·(6 in) = 6π in³ ≈ 18.850 in³
_____
The difference in volume is ...
... 18.850 in³ - 4.189 in³ = 14.661 in³ ≈ 14.7 in³