Answer:
The new function is [tex]g(x)=48x^2+16[/tex]
Step-by-step explanation:
We are given,
The function [tex]f(x)=12x^2+4[/tex] is vertically stretched by a factor of 4.
Now, we know that,
Vertical stretch changes the function f(x) to [tex]kf(x)[/tex] where k is the scale factor.
So, we get,
The new function is [tex]g(x)=4f(x)[/tex]
i.e. [tex]g(x)=4(12x^2+4)[/tex]
i.e. [tex]g(x)=48x^2+16[/tex]
Thus, the equation of the new function is [tex]g(x)=48x^2+16[/tex].
suppose Marcus delivers papers at a rate of 9 papers in 18 minutes. How much longer would it take him to deliver 100 papers and 72 papers? justify your response.
Is the difference of 1.48-0.25 less than or greater than one explain
12x−3y=, when x=−1/4 and y=3
Use multiplication and distributive property to find the quotient of 75 divided by 3
find the discriminant & describe the nature of the roots 2x2=3x+5
Ordering numbers from greatest to least is called putting them in _____ order.
Ordering numbers from greatest to least is known as putting them in descending order. This organization method is a form of emphatic order, which could also be the answer if the context relates to emphasizing the importance of the items being ordered rather than their numerical value.
Ordering numbers from greatest to least is called putting them in descending order. When we arrange objects based on the level of importance or magnitude, this is often referred to as emphatic order, which prioritizes items from least to most important or vice versa. In mathematics, specifically, when numbers are organized from the largest to the smallest, they are said to be in descending order, which is the opposite of ascending order, where numbers are arranged from smallest to greatest.
There are various ways in which ideas, objects, or numbers can be ordered. For example, chronological order refers to presenting events in the sequence they happened in time, whereas sequential order is similar but can apply to any sequence of actions or items, not just those related to time. Conversely, spatial order involves the arrangement of items as they are placed in space.
In the context of your question, when asked to rank objects by their distance from the origin, a descending order would start with the object that is farthest away (largest distance) and end with the object that is closest to the origin (smallest distance).
The rate in which a function increases or decreases between its points are called the slope, or the rate of change. For any function, the rate of change is calculated by the slope formula. Slope formula: where m = slope (a, f(a)) and (b, f(b)) are two points on the function. Here is an example: For the function, f(x) = 2x - 1, calculate the rate of change between the points, (-1, f(-1)) and (4, f(4)). Let(-1,f(-1))=(a,f(a)) and (4,f(4))=(b,f(b)). Use the given functin, f(x)=2x-1, to complete the points. For (-1, f(-1): For (4,f(4)) f(x)=2x-1 f(x)=2x-1 f(-1)=2(-1)-1 f(4)=2(4)-1 f(-1)=-3 f(4)=7 (-1,f(-1))=(-1,-3) (4,f(4))=(4,7) Next, calculate the slope for the function, using the points (-1, -3) and (4, 7). For the formula, let(-1,-3) =(a,f(a)) and (4,7) = (b,f(b)): The slope of the function, f(x)=2x-1, between the points (-1, -3) and (4,7) is 2. For each function, use the slope formula to calculate the rate of change between the points. In your final answer, include all of your calculations. f(x): (a, f(a) and (b,f(b) 1.) f(x)=x - 3 (0,f(0)) and (6,f(6)) 2.) f(x) = -x (-4,f(-4)) and(2,f(2)) 3.) f(x)=x2 (-2,f(-2)) and (0,f(0)) 4.) f(x)=x3 (-1,f(-1)) and (1,f(1)) 5.) f(x)=2x (0,f(0)) and (4,f(4))
1. f(x)=1/2x-3 f(x)=1/2x-3
f(0)=1/2(0)-3 f(6)=1/2(6)-3
f(0)=-3 f(6)=0
m=f(b) - f(a)/b - a = 0- (-3)/6 - 0 = 3/6 = 1/2
2. f(x) = -x f(x) = -x
f(-4) = -(-4) f(2) = -2
f(-4) = 4 f(2) = -2
m=f(b) - f(a)/b - a = -2 - 4/2 - (-4) = -6/6 = -1
3. f(x) = x^2 f(x) = x^2
f(-2) = -2^2 f(0) = 0^2
f(-2) = -4 f(0) = 0
m=f(b) - f(a)/b - a = 0 - (-2)/0 - (-2) = 2/2 = 1
4. f(x) = x^3 f(x) = x^3
f(-1) = -1^3 f(1) = 1^3
f(-1) = -1 f(1) = 1
m=f(b) - f(a)/b - a = 1 - (-1)/1 - (-1) = 2/2 = 1
5. f(x) = 2^x f(x) = 2^x
f(0) = 2^0 f(4) = 2^4
f(0) = 1 f(4) = 16
m=f(b) -f(a)/b - a = 4 - 0/ 4 - 0 = 4/4 = 1
i hope this is right, have a good day
25 pints of apple juice are poured into 1/2 pint bottles. how many bottles can be filled with apple juice
The number of bottles to fill 25 pints of apple juice will be equal to 50.
What is an expression?Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
The Number of the bottles = 25 / ( 1 / 2 )
= 25 x 2
= 50
Therefore, the number of bottles to fill 25 pints of apple juice will be equal to 50.
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The population and areas of four states are shown.
The answer is B) The state with the second lowest population has the lowest population density.
You have a 5" by 7" photo that you would like to have enlarged to fit an 8" by 10" frame. Would the two photographs be similar? Explain
5/8 = 0.625
7/10 = 0.7
the 2 ratios are different so it would not be similar
If you were to spin the spinner below, what would be the probability of landing on green or purple?
you have 1/4 one is purple another orange and the other two green
Final answer:
The probability of landing on green or purple on the spinner is 3/4, or 75%.
Explanation:
To determine the probability of landing on green or purple when spinning a spinner that has 1/4 of its sections as purple, another 1/4 as orange, and the remaining two quarters green, you would simply add the probability of landing on each of those colors since they are mutually exclusive events. The probability of landing on purple is 1/4 and the probability of landing on green is 2/4 since there are two quarters of the spinner that are green.
We calculate the probability as follows:
Probability of landing on purple = 1/4Probability of landing on green = 2/4 or 1/2To find the combined probability of landing on green or purple, you add the two probabilities together:
Probability of landing on green or purple = Probability of purple + Probability of green
Probability of landing on green or purple = 1/4 + 1/2
Probability of landing on green or purple = 1/4 + 2/4
Probability of landing on green or purple = 3/4
Thus, the probability of landing on green or purple is 3/4, which means there is a 75% chance of the spinner landing on either of these colors.
A random sample is drawn from a population with mean μ = 66 and standard deviation σ = 5.5. use table 1.
a. is the sampling distribution of the sample mean with n = 16 and n = 36 normally distributed? yes, both the sample means will have a normal distribution. no, both the sample means will not have a normal distribution. no, only the sample mean with n = 16 will have a normal distribution. no, only the sample mean with n = 36 will have a normal distribution.
b. can you use the standard normal distribution to calculate the probability that the sample mean falls between 66 and 68 for both sample sizes? yes, for both the sample sizes, standard normal distribution could be used. no, for both the sample sizes, standard normal distribution could not be used. no, only for the sample size with n = 16, standard normal distribution could be used. no, only for the sample size with n = 36, standard normal distribution could be used.
c. calculate the probability that the sample mean falls between 66 and 68 for n = 36. (round intermediate calculations to 4 decimal places, "z" value to 2 decimal places, and final answer to 4 decimal places.)
The sampling distributions of both sample sizes are normally distributed due to the Central Limit Theorem. The standard normal distribution can be used to calculate the probability for both sample sizes. The probability that the sample mean falls in a particular range for a given sample size can be found by calculating the z-scores for that range and looking at the area under the standard normal curve.
Explanation:The subject of this question relates to the concepts of sampling distribution, normal distribution, and probability in the field of statistics. To answer this:
a. The sampling distribution of the sample means for both n = 16 and n = 36 will be normally distributed. According to the Central Limit Theorem, if the sample size is large enough (generally n > 30), the sampling distribution of the mean tends to be normal regardless of the shape of the population distribution.
b. Yes, a standard normal distribution could be used for both sample sizes to calculate the probability that the sample mean falls between 66 and 68. Since we are dealing with standard normal distribution, we first need to convert the samples into z-scores.
c. To calculate the probability for n = 36, we need to calculate the z-score for both 66 and 68 using the formula z = (X - μ) / (σ / sqrt(n)), where X is the sample mean, μ is the population mean, σ is the standard deviation, and n is the sample size. The probability that the sample mean falls between 66 and 68 for n = 36 is then simply the area under the standard normal curve bounded by these two z-scores.
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You have deposited $800 in a savings account, which pays three percent interest compounded quarterly. Find the amount in the account for 5 years
Find 1/4+13/20
. Write your answer as a fraction in simplest form.
A fraction is a way to describe a part of a whole. The sum of the two of the given fractions is equal to 9/10.
What is a Fraction?A fraction is a way to describe a part of a whole. such as the fraction ¼ can be described as 0.25.
The given fractions can be added as shown below.
(1/4) + (13/20)
Taking the LCM of the denominator which is equal to 20,
= (1×5 / 4×5) + (13/20)
= (5/20) + (13/20)
= (5 + 13)/20
= 18/20
Divide both the numerator and the denominator by 2,
= 9/10
Hence, the sum of the two of the given fractions is equal to 9/10.
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Calculate the quotient and remainder of –111 divided by 11. (quotient, remainder)
Final answer:
The quotient of –111 divided by 11 is –10, and the remainder is –1. We obtain this by dividing –111 by 11 and then finding the difference between the product of the quotient and 11 and –111.
Explanation:
To calculate the quotient and remainder of –111 divided by 11, we perform the division as follows:
Divide –111 by 11 to find the quotient. Since 111 is divisible by 11, the quotient is –10.To find the remainder, we calculate –10 multiplied by 11, which is –110, and subtract it from –111. The difference is –1, so the remainder is –1.Therefore, the quotient is –10 and the remainder is –1.
What is 800+60+4 in standard form
Ellen Saver went to her bank. She had a balance of $2,447.67 in her savings account. She withdrew $231.49 and the teller credited her account with $36.61. What is her new balance?
$2,179.57
$2,252.79
$2,642.77
$2,715.77
Find the particular solution of the differential equation \frac{dy}{dx} + y\cos(x) = 5\cos(x)
What is the product of 5 + 2i and 7 + 3i?
(1) 41 + 29i
(2)29 + 29i
(3)41 + i
(4)35 + 6i
(5)29 + 41i
Given:
A = O, WA = NO, AS = OT.
SSS
SAS
ASA
AAS
"15. describe the three data fragmentation strategies. give some examples of each."
Charlotte embroidered a piece of cloth with a length of 4 meters and a width of 2 meters in 412 hours.
What is the approximate speed, in square meters per hour, at which Charlotte can embroider? Round your answer to the nearest whole number.
If 9:7 is the ratio, there are here 116 more boys than girls how many total students are there
The Sugar Sweet Company will choose from two companies to transport its sugar to market. The first company charges $3157 to rent trucks plus an additional fee of $50.25 for each ton of sugar. The second company does not charge to rent trucks but charges $275.75 for each ton of sugar.
For what amount of sugar do the two companies charge the same?
What is the cost when the two companies charge the same?
Both companies will charge the same $3,860.50 for 14 tons of sugar.
CalculusGiven that The Sugar Sweet Company will choose from two companies to transport its sugar to market, and the first company charges $3157 to rent trucks plus an additional fee of $50.25 for each ton of sugar, while the second company does not charge to rent trucks but charges $275.75 for each ton of sugar, to determine for what amount of sugar do the two companies charge the same, and what is the cost when the two companies charge the same, the following calculation must be made:
Company 1 =
3157 + 50.25 x 10 = 3157 + 502.5 = 3659.503157 + 50.25 x 20 = 3157 + 1005 = 41623157 + 50.25 x 15 = 3157 + 753.75 = 3910.753157 + 50.25 x 14 = 3157 + 703.50 = 3860.50Company 2 =
275.75 x 10 = 2757.50275.75 x 20 = 5515275.75 x 15 = 4136.25275.75 x 14 = 3860.50Therefore, both companies will charge the same $3,860.50 for 14 tons of sugar.
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Write the steps for how to use a number line to multiply a 2 digit number by 20. Give an example
To multiply a two-digit number by 20 using a number line, first choose a two-digit number. Each step on your number line will represent this number. Then count 20 steps on your number line from 0. The number you land on is the product. Alternatively, you can multiply the number by 2 and then move the decimal point one place to the right.
Explanation:To use a number line to multiply a two-digit number by 20, follow these steps with an example:
Choose a two-digit number. Let's use 15 as an example.On your number line, make a mark at zero and at your two-digit number (in this case, 15).Each step on your number line will represent an increment of 15. This means the step after 15 will be 30, then 45, and so on.Count 20 steps on your number line from 0. The number you land on will be the product of your two-digit number (15) and 20.Alternatively, as per the shortcuts mentioned, multiplying by 10 involves moving the decimal point one place to the right. Multiplying a number by 20 is thus equivalent to multiplying the number by 2 and then by 10. As an example, 15 x 20 can also be calculated as (15 x 2) x 10 = 30 x 10 = 300.
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In the regression equation, what does the letter "x" represent?
which is the solution set of the compound inequality 3.5x - 10 > -3 and 8x - 9 < 39
What are the steps for using a compass and straightedge to construct a square? Drag and drop the steps in order from start to finish.
1.Construct a line perpendicular to line m through point F. Label a point on this line as point G.
2.Use a straightedge to draw line m and label a point on the line as point F.
3.Without changing the compass width, place the compass point on point H and draw an arc in the interior of ∠HFK .
4.Use the straightedge to draw JH¯¯¯¯¯ and JK¯¯¯¯¯.
5.With the compass open to the desired side length of the square, place the compass point on point F and draw an arc on line m and an arc on FG←→ . Label the points of intersection as points H and K.
6.Keeping the same compass width, place the compass on point K and draw an arc in the interior of ∠HFK to intersect the previously drawn arc. Label the point of intersection as point J.
These are the right answers
Use the chain rule to find ∂z/∂s and ∂z/∂t. z = sin(θ) cos(ϕ), θ = st5, ϕ = s7t
We use the chain rule to find the partial derivatives ∂z/∂s and ∂z/∂t, considering z as a function of θ and ϕ, treating θ and ϕ as intermediate variables. We then differentiate θ with respect to s and t, and similarly for ϕ.
Explanation:To find the partial derivatives ∂z/∂s and ∂z/∂t, we first need to make use of the chain rule in calculus. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Here, z is a function of θ and ϕ, which are themselves functions of s and t - θ = st^5 and ϕ = s^7t.
First, find the partial derivative ∂z/∂s. We differentiate z = sin(θ)cos(ϕ) with respect to s, treating θ and ϕ as intermediates, and then differentiate θ and ϕ with respect to s.
Similarly, to find the partial derivative ∂z/∂t, we differentiate z = sin(θ)cos(ϕ) with respect to t treating θ and ϕ as intermediates, and then differentiate θ and ϕ with respect to t.
Please note that further calculation will require the exact differentiation of trigonometric functions and power functions.
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When certain kinds of chemicals are combined, the rate at which the new compound is formed is modeled by the autonomous differential equation dX/dt = k(a-X)(B-X) where k > 0 is a constant of proportionality and B > a > 0. Here X(t) denotes the number of grams of the new compound formed in time t. (a) Use a phase portrait of the differential equation to predict the behavior of X(t) as t -> infinity. (b) Consider the case when a = B. Use a phase portrait of the differential equation to predict the behavior of X(t) as t -> infinity when X(0) < a. When X(0) > a. (c) Verify that an explicit solution of the DE in the case when k=1 and a=B is X(t)=a-1/(t+c). Find a solution that satisfies X(0) = a/2. Then find a solution that satisfies X(0)=2a. Graph these two solutions. Does the behavior of the solutions as t->infinity agree with your answers to part (b)?
To predict the behavior of the autonomous differential equation as t -> infinity, analyze the phase portrait of the system. The behavior of X(t) as t approaches infinity depends on the initial conditions. Verify the explicit solution of the DE when k=1 and a=B. Graph the solutions and observe their behavior as t-> infinity.
Explanation:To predict the behavior of the autonomous differential equation dX/dt = k(a-X)(B-X) as t -> infinity, we can analyze the phase portrait of the system. The phase portrait will show the equilibrium points and the direction of the solutions as time increases. In this case, there will be two equilibrium points, one at X = a and one at X = B, assuming a < B. The behavior of X(t) as t approaches infinity depends on the initial conditions. If X(0) < a, the solution will approach X = a, and if X(0) > a, the solution will approach X = B.
To verify the explicit solution of the differential equation when k=1 and a=B, we substitute these values into the equation: X(t) = a - 1/(t+c). For the solution that satisfies X(0) = a/2, we substitute t=0 and X(0) = a/2 into the equation and solve for c. Similarly, for the solution that satisfies X(0) = 2a, we substitute t=0 and X(0) = 2a into the equation and solve for c. We can then graph these two solutions and observe that as t approaches infinity, X(t) approaches a for both solutions, which confirms our analysis from part (b).
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