#7Change from standard form to vertex formy= x²-6x+7

Answer:

centered at x = 0.

Step-by-step explanation:

Part A: The power series for g(x) can be obtained by using the formula for a geometric series with a first term of 1 and a common ratio of (x/2). Then we have:

g(x) = 4/((x+2)(x-2)) = 4/(4*(1 + x/2)*(1 - x/2))

= 1/(1 - x/2) - 1/(1 + x/2)

We can then use the power series for 1/(1-x) to find the power series for g(x):

g(x) = 1/(1 - x/2) - 1/(1 + x/2)

= (1/2) * (1 + x/2 + (x/2)^2 + (x/2)^3 + ...) - (1/2) * (1 - x/2 + (x/2)^2 - (x/2)^3 + ...)

= x + (3/4)*x^2 + (5/8)*x^3 + (35/64)*x^4 + ...

To evaluate the infinite sum when x = 1, we can substitute x = 1 into the power series and use the formula for an infinite geometric series:

g(1) = 1 + (3/4) + (5/8) + (35/64) + ...

= 1/(1 - 1/2) - 1/(1 + 1/2)

= 2 - 2/3

= 4/3

Therefore, the infinite sum when x = 1 is 4/3.

Part B: To find an upper bound for the error of the approximation sin(0.3) ≈ 0.3 - 0.3^3/3!, we can use the formula for the remainder term in a Taylor series:

Rn(x) = f^(n+1)(c) * (x-a)^(n+1) / (n+1)!

where f(x) = sin(x), a = 0.3, n = 3, and c is some number between a and x. We want to find an upper bound for |R3(0.3)|.

Taking the fourth derivative of f(x) = sin(x), we get:

f^(4)(x) = -sin(x)

Since |sin(c)| ≤ 1 for any c, we have:

|R3(0.3)| ≤ |f^(4)(c)| * (0.3-0)^4 / 4!

≤ 1 * 0.3^4 / 24

≤ 0.000625

Therefore, an upper bound for the error is 0.000625, rounded to five decimal places.

Part C: We can find the power series for h(x) by differentiating the power series for ln(1+x) term by term. The power series for ln(1+x) is:

ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ...

Taking the derivative, we get:

h(x) = ln(1+x)'

= 1 - x + x^2 - x^3 + ...

which is the power series for (-1)^n x^n. Therefore, the power series for h(x) is:

h(x) = ∑(-1)^n x^n

centered at x = 0.

Answer:

What is the full question?

So the vertex form of the equation is: y = -(x + 3)² + 2.

To change the standard form y = -x² - 6x - 7 to vertex form, we need to complete the square.

First, let's factor out the negative sign:

y = -(x² + 6x + 7)

Next, we need to add and subtract a constant term inside the parentheses that will allow us to complete the square:

y = -(x² + 6x + 9 - 9 - 7)

Now, we can group the first three terms and factor the quadratic trinomial inside the parentheses:

y = -(x + 3)² + 2

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The linear function parallel to the line shown in the graph passing through the point (−2, 3) is given as follows:

y = 2/3x + 13/3. (option A).

The slope-intercept representation of a linear function is given by the equation presented as follows:

y = mx + b

The coefficients of the function and their meaning are described as follows:

On the graphed function, when x increases by 9, y increases by 6, hence the slope is given as follows:

m = 6/9

m = 2/3.

When two lines are parallel, they have the same slope, hence:

y = 2x/3 + b.

When x = -2, y = 3, hence the intercept b is obtained as follows:

3 = -4/3 + b

b = 13/3.

Hence the equation is given as follows:

y = 2/3x + 13/3. (option A).

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For every positive integer n, there are n consecutive composite integers.

We will prove the statement using the fact that every integer greater than 1 is either prime or can be factored into a product of primes.

Consider the n consecutive integers starting with (n+1)!+2, which are:

(n+1)!+2, (n+1)!+3, (n+1)!+4, ..., (n+1)!+n+1

We will show that each of these integers is composite.

First, note that (n+1)!+2 is composite, since it is greater than 2 and can be factored as 2*(n+1)!/2 + 1.

Next, for each i between 2 and n+1, inclusive, we have:

(n+1)!+i = i*((n+1)!/i) + i

Since i divides (n+1)!, we have (n+1)!/i as an integer, and since i is between 2 and n+1, we have (n+1)!/i greater than 1. Thus, (n+1)!+i can be factored into a product of at least two integers, i and (n+1)!/i + 1. Since i is between 2 and n+1, we have (n+1)!/i + 1 between 3 and (n+1), inclusive. Therefore, (n+1)!+i is composite.

Thus, each of the n integers (n+1)!+2, (n+1)!+3, ..., (n+1)!+n+1 is composite, as desired. Therefore, for every positive integer n, there are n consecutive composite integers.

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The weight of food packed in certain containers is a random variable with a mean of 16 0z. and a standard deviation of 0.6 oz. If 36 packages are randomly selected, the probability that the mean will be greater than 16.2 oz is 2.28%.

Given that the weight of food packed in containers is a random variable with a mean (µ) of 16 oz. and a standard deviation (σ) of 0.6 oz, we want to find the probability that the mean weight of a sample of 36 packages (n) will be greater than 16.2 oz.
Step 1: Calculate the standard error (SE) of the sample mean.
SE = σ / √n = 0.6 / √36 = 0.6 / 6 = 0.1
Step 2: Calculate the z-score for 16.2 oz.
z = (sample mean - population mean) / SE = (16.2 - 16) / 0.1 = 2
Step 3: Find the probability that the mean weight will be greater than 16.2 oz.
To find the probability for a z-score of 2, we can look it up in a standard normal (z) table or use a calculator that provides the area to the left of the z-score.
Using the table or calculator, we find the area to the left of the z-score 2 is approximately 0.9772. Since we are looking for the probability of the mean weight being greater than 16.2 oz, we want the area to the right of the z-score. To find this, subtract the area to the left from 1:
1 - 0.9772 = 0.0228
So, the probability that the mean weight of the 36 packages will be greater than 16.2 oz is approximately 0.0228 or 2.28%.

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Based on the information provided, we have a random sample of 60 measurements taken from a population with a mean (µ) of 25 and a variance (σ²) of 200.

The mean of the sample (X) will be equal to the population mean, so X = 25.

To calculate the standard error (SE) of the sample, we use the formula SE = σ / √n, where σ is the population standard deviation and n is the sample size. Since the variance is given as 200, the standard deviation (σ) is the square root of 200, which is approximately 14.14.

Now, we can calculate the standard error: SE = 14.14 / √60 ≈ 1.82.

So, the pair for the mean and standard error of x is (25, 1.82).

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The standard form is 3x² + 8x -9= 0.

We have,

-8x = 3x² - 9

We know that the standard form of Quadratic Equation is

y = ax² + bx + c

Now, writing the given equation in standard form as

0 = 3x² - 9 + 8x

3x² + 8x -9= 0

Thus, the standard form is 3x² + 8x -9= 0.

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Answer:

  8√23 ≈ 38.37

Step-by-step explanation:

You want the length of a chord 10 units from the center of a circle when a chord 12 units from the center has length 36 units.

The radius OA completes the right triangle OBA. The hypotenuse (OA) is found using the Pythagorean theorem:

  OA² = OB² +BA²

  OA² = 12² +18² = 468

Radius OF is the same length, so we can use the Pythagorean theorem to find FE.

  OF² = OE² +FE²

  FE² = OF² -OE² = 468 -10² = 368

  FD = 2·FE = 2√368 = 8√23

  FD ≈ 38.37

Answer:

$76.00

Step-by-step explanation:

149 + 22 - 95 = 76

Answer: $54

Step-by-step explanation:

Take $149 and minus it with $22

Than he refunded it so add back $22 $22 + $127 = $149

Than he bought the TV which costed $95
$149 - $95 = $54

By solving a positive number over a variable (or vice versa) both raised to a negative power  ,The simplified expression is [tex]\frac{8}{x^{3} }[/tex].

To solve an expression with a positive number over a variable, both raised to a negative power, you should follow these steps:

1. Identify the base and exponent: In your example, the base is (x/2) and the exponent is -3.

2. Apply the negative exponent rule: When an expression with a negative exponent is raised to a power, you can rewrite it with a positive exponent by taking the reciprocal of the base.

The negative exponent rule states that [tex]a^{-n}[/tex] = 1/([tex]a^{n}[/tex]), where 'a' is the base and 'n' is the exponent.

3. In your example, apply the negative exponent rule to [tex](x/2)^{-3}[/tex] This becomes 1/([tex](x/2)^{-3}[/tex]).

4. Simplify the expression: Raise the base (x/2) to the power of 3. Remember that when you raise a fraction to an exponent, you should raise both the numerator and denominator to that exponent. So, [tex](x/2)^{-3}[/tex] = ([tex]X^{3}[/tex])/([tex]2^{3}[/tex]) =[tex]\frac{x^{3}}{8 }[/tex]

5. Substitute the simplified expression back into the original equation: 1/(([tex](x/2)^{3}[/tex]) = 1/([tex]\frac{x^{3}}{8 }[/tex]).

6. To further simplify, remember that dividing by a fraction is equivalent to multiplying by its reciprocal. So, 1/([tex]\frac{x^{3}}{8 }[/tex]) = 1 * ([tex]\frac{8}{x^{3} }[/tex]) = [tex]\frac{8}{x^{3} }[/tex].

The simplified expression is [tex]\frac{8}{x^{3} }[/tex]. This is how you solve an expression with a positive number over a variable, both raised to a negative power.

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Answer: 0

Step-by-step explanation:

8-8=0

For instance, in Europe, €5,99 would represent 5 euros and 99 cents.

The symbol used to separate whole numbers from parts of numbers in decimal notation, such as in money with dollars and cents, is a period (also known as a decimal point).

For example, $5.99 represents 5 dollars and 99 cents. The period serves as a visual indicator to show where the whole number ends and the fractional part begins.

It's worth noting that in some parts of the world, such as in Europe, the comma is used instead of the period to separate the whole number and the fractional part. For instance, in Europe, €5,99 would represent 5 euros and 99 cents.

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Answer:

Step-by-step explanation:

A table that displays the number of individuals who fall into each combination of categorical variables is called a contingency table.

Contingency tables, also known as cross-tabulation tables or crosstabs, are a useful tool for analyzing the relationship between two or more categorical variables.

In a contingency table, each row represents a category of one variable, and each column represents a category of another variable. The intersections of the rows and columns, called cells, display the count or frequency of observations that fall into the specific combination of categories. This allows researchers to identify patterns, trends, and possible associations between the variables being studied.

One common application of contingency tables is in hypothesis testing, particularly the chi-square test of independence. This test evaluates whether there is a significant association between the categorical variables, or if the observed frequencies are simply due to chance.

In conclusion, a contingency table is a valuable tool for organizing and analyzing categorical data, enabling researchers to identify patterns and relationships among variables and aiding in the decision-making process.

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Using unit conversions, which are based on division and multiplication operations, Reagan's frog can jump farther.

Unit conversions refer to the expressions of the same properties or values using different units of measurement.

For instance miles can be converted to kilometers and feet can be converted to inches, and vice versa.

All unit conversions operate on division and multiplication operations, two of the four basic mathematical operations.

1 meter = 100 centimeters

The length Libby's bullfrog can jump = 2 meters

2 meters = 200 centimeters (2 x 100)

The length Reagan's bullfrog can jump = 220 centimeters

220 centimeters = 2.2 meters (220 ÷ 100)

Thus, we can conclude that Reagan's frog can jump farther than Libby's.

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The calculated perimeter of the rectangle is 28 units

From the question, we have the following coordinates that can be used in our computation:

A (-2, 2) B (-2, 6) C (8, 6) D (8, 2)

The distance between the coordinates are

√[(-2 + 2)^2 + (2 - 6)^2] = 4

√[(-2 - 8)^2 + (6 - 6)^2] = 10

The perimeter is then calculated as

Perimeter = 2 * sum of dimensions

So, we have

Perimeter = 2 * (4 + 10)

Perimeter = 28

Hence, the perimeter is 28 units

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Based on the results, the comparison of the overall performance of the portfolios, from best to worst, is: from "Portfolio 1, Portfolio 2, Portfolio 3". Therefore, the Option C is correct.

We have,

To calculate the weighted mean of the RORs for each portfolio, we need to multiply each ROR by its corresponding investment amount, sum the products, and divide by the total investment amount:

Weighted mean ROR for Portfolio 1:

= ((3.9% x $1,250) + (1.7% x $575) + (10.6% x $895) + (-3.2% x $800) + (8.1% x $1,775)) / ($1,250 + $575 + $895 + $800 + $1,775)

= 5.12%

Weighted mean ROR for Portfolio 2:

= ((3.9% x $950) + (1.7% x $2,025) + (10.6% x $1,185) + (-3.2% x $445) + (8.1% x $625)) / ($950 + $2,025 + $1,185 + $445 + $625)

= 0.04464053537

= 4.46%

Weighted mean ROR for Portfolio 3:

= ((3.9% x $900) + (1.7% x $2,350) + (10.6% x $310) + (-3.2% x $1,600) + (8.1% x $2,780)) / ($900 + $2,350 + $310 + $1,600 + $2,780)

= 0.03550251889

= 3.55%

Based on the results, the comparison of the overall performance of the portfolios, from best to worst, is: from "Portfolio 1, Portfolio 2, Portfolio 3". Therefore, the Option C is correct.

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complete question:

Calculate the weighted mean of the RORs for each portfolio. Based on the results, which list shows a comparison of the overall performance of the portfolios, from best to worst?

A)Portfolio 1, Portfolio 3, Portfolio 2

B) Portfolio 2, Portfolio 3, Portfolio 1

C) Portfolio 1, Portfolio 2, Portfolio 3

D) Portfolio 3, Portfolio 2, Portfolio 1

Answer: To find the domain of a function, you need to identify all the possible input values (x) for which the function produces a valid output (y). In other words, you need to find the set of all values of x for which the function is defined and produces real outputs.

Step-by-step explanation: Here are the general steps to find the domain of a function:

Look for any values of x that could lead to undefined results. For example, if the function involves a square root, the value inside the square root cannot be negative, so you need to ensure that the expression inside the square root is non-negative.

Look for any values of x that could lead to division by zero. For example, if the function involves a fraction, the denominator cannot be zero, so you need to ensure that the denominator is not equal to zero.

Look for any other restrictions on the input values based on the definition of the function. For example, some functions may require that x be a certain type of number, such as an integer or a positive real number.

Write the domain of the function as a set of possible input values. For example, you might write the domain as an interval of real numbers or a set of discrete values.

It is important to note that some functions may have restricted domains due to the nature of the function, while other functions may have unrestricted domains that encompass all real numbers.

An interval in which the actual sum of the series, obtained using Lagrange erroe bound is; S₅ - 1/13! ≤ S ≤ S₅ + 1/13!

The Lagrange error bound can be used to find an interval in which the actual sum exists as follows;

The Lagrange error bound, with regards to alternating series, states that the error in approximating the sum of an alternating series using a partial sum is less than or equivalent to the absolute value of the next or first omitted term, aₙ ₊ ₁

The absolute value of the next omitted term therefore is |a₆| and when x = 1, |a₆| = |(-1)⁶ × 1⁽⁽²⁾⁽⁶⁾⁺¹⁾ ÷ (2×6 + 1)!| = |1/13!|

An interval for which the actual sum exists is therefore;

|S - S₅| ≤ |R₅| = 1/13!

Which indicates that the interval is; S₅ - 1/13! ≤ S ≤ S₅ + 1/13!

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Answer:

Boys equal 189 and girls equal 108

Step-by-step explanation:

Let b = the number of boys

Let g = the number of girls

b + g = 297

Rewrite as g = 297 - b

[tex]\frac{4}{9}[/tex]b = [tex]\frac{7}{9}[/tex]g Multiply both sides by 9

4b = 7g  Substitute 297 - b for g

4b = 7(297 -b)  Distribute the 7

4b = 2079 - 7b   Add 7 b to both sides

11b = 2097  Divide both sides by 11

b = 189

There are 189 boys.

g = 297 - b  Substitute 189 for b to solve for g

g = 297 - 189

g = 180

The number of girls is 108.

Helping in  the name of Jesus.

Jamie has 33 nickels in the jar.

Let's solve the problem with the given information: Jamie has a jar of coins containing the same number of nickels, dimes, and quarters, and the total value is $13.20.

Let's use N for the number of nickels, D for dimes, and Q for quarters. Since there's an equal number of each coin, we can say N = D = Q.

The value of these coins can be represented as:
0.05N + 0.10D + 0.25Q = 13.20

Now, substitute N for D and Q since N = D = Q:
0.05N + 0.10N + 0.25N = 13.20

Combine the terms:
0.40N = 13.20

Now, divide by 0.40 to find the number of nickels:
N = 13.20 / 0.40
N = 33

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The equation that represents the line is y = 9x + 6

This is calculated as

Slope = Change in cost/DVDs

So, we have

Slope = (24 - 15)/(2 - 1)

Slope = 9

In this problem, the slope represents the cost per number of DVD

So, the slope is $9 per DVD

We have

Slope = 9

So, the y-intercept is

y-intercept = 15 - 9

y-intercept = 6

In this context, the y‐intercept represents the initial cost

So, the y‐intercept (i.e. the initial cost) is $6

This is calculated as

y = mx + c

So, we have

y = 9x + 6

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To solve the equation √(2x) + 5 = 1 using mental math, we can follow these steps:

   Subtract 5 from both sides of the equation: √(2x) = -4

   Square both sides of the equation to eliminate the square root: 2x = 16

   Divide both sides by 2 to solve for x: x = 8

Therefore, the solution to the equation √(2x) + 5 = 1 is x = 8.

Regarding the problem m +4-√3m = 0, we can solve for m algebraically by following these steps:

   Move the constant term (4) to the other side of the equation: √(3m) = -m + 4

   Square both sides to eliminate the square root: 3m = (4 - m)^2

   Simplify the right-hand side: 3m = 16 - 8m + m^2

   Rearrange the terms and set equal to zero: m^2 - 11m + 16 = 0

   Factor the quadratic equation: (m - 1)(m - 16) = 0

   Solve for m by setting each factor equal to zero: m - 1 = 0 or m - 16 = 0

   Solve for m in each equation: m = 1 or m = 16

Therefore, the solutions to the equation m +4-√3m = 0 are m = 1 and m = 16.

The average value of a function f(x) on an interval [0, 6] is represented by 'a'. To determine if there is a value 'c' for which the average value of f(x) on the interval [0, c] is greater than 'a', we need to consider the properties of the function and the Mean Value Theorem.

The Mean Value Theorem states that if a function f(x) is continuous on the interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point 'c' in the interval (a, b) such that the average rate of change equals the instantaneous rate of change or f'(c) = (f(b) - f(a)) / (b - a).

Without more information about the function f(x), we cannot definitively say whether there is a value 'c' for which the average value of f(x) on the interval [0, c] is greater than the average value on the interval [0, 6]. However, if the function meets the conditions of the Mean Value Theorem, it is possible that such a value 'c' exists.

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The value of w using the equation w + 45 = w + 12 + w - 23 is 56

From the question, we have the following parameters that can be used in our computation:

w+12°

w+45°

w-23°

Assuming an equation to solve w, we have

w + 45 = w + 12 + w - 23

Evaluating the like terms, we have

w + 45 = 2w - 11

Collecting the like terms, we have

-w = -56

Divide by -1

w = 56

Hence, the value of w is 56

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Answer:

Part A:

To find the median of the data, we need to arrange the values in ascending order:

$2.99, $3.05, $3.25, $3.25, $3.43, $3.50, $3.60, $3.65

There are 8 values, so the median is the average of the 4th and 5th values, which are $3.25 and $3.43. Therefore, the median is:

Median = ($3.25 + $3.43) ÷ 2 = $3.34

To find the mode, we need to look for the value that appears most frequently in the data. Here, the value $3.25 appears twice, which is more than any other value. Therefore, the mode is:

Mode = $3.25

Part B:

To find the mean of the data, we need to add up all the values and divide by the total number of values:

Mean = ($2.99 + $3.05 + $3.25 + $3.25 + $3.43 + $3.50 + $3.60 + $3.65) ÷ 8

Mean = $3.33

Part C:

Based on the answers in Part A and Part B, we can make the following generalization about the price of milk: The median and mode of the data are both close to $3.25, while the mean is slightly higher at $3.33. This suggests that there are a few values in the data that are higher than the rest, which is pulling up the mean. Overall, the price of milk seems to be centered around $3.25, with some variation above and below that value.

Step-by-step explanation:

The scale factor for ΔXYZ to ΔUVW include the following: A. 2.

In Mathematics and Geometry, the scale factor of a geometric figure can be calculated by dividing the dimension of the image (new figure) by the dimension of the pre-image (original figure):

Scale factor = Dimension of image (new figure)/Dimension of pre-image (original figure)

By substituting the given parameters into the formula for scale factor, we have the following;

Scale factor = Dimension of image/Dimension of pre-image

Scale factor = 20/10 = 16/8 = 12/6

Scale factor = 2.

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Without knowing the initial population (P0) and the growth rate (r), we cannot provide an interpretation for the population size of 4100.

It seems like some parts of the question are missing, so I'll assume the question is: "A population grows at a rate of r, where P(t) is the population after t months. a) Find a formula for the population size after t months, given that the initial population is P0. Select the correct interpretation of the population size of 4100."

Regarding the interpretation of a population size of 4100, some possible options are:
- It is the actual number of individuals in the population.
- It is an estimate of the population size based on some method (e.g., sampling, census).
- It is a desirable or undesirable population size depending on the context (e.g., for conservation, or urban planning).
- It is a change from a previous population size that may be positive, negative, or neutral.
To solve this problem, we'll use the exponential growth formula:

P(t) = P0 * (1 + r)^t

Here, P(t) represents the population size after t months, P0 is the initial population size, r is the growth rate, and t is the number of months.

To interpret the population size of 4100, you can plug 4100 into the formula as P(t) and solve for t. Without knowing the initial population (P0) and the growth rate (r), we cannot provide an interpretation for the population size of 4100.

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The value of this continuous stream of  compound interest  after 18 years  will be [tex]\$2526[/tex].

The interest  that we earn even on interest is termed as compound interest .

We know that formulae for compound interest when compounded annually will be [tex]A = P(1 + \dfrac{r}{n})^{nt}[/tex]

Where A is the amount,

P is the principal.

r is the interest rate,

t is the time (in years).

On putting   the values in the formulae , we get:

[tex]A = 2000 ( 1 +\dfrac{1.3}{100})^{18}[/tex]

On simplifying, we get:

 A = [tex]2000\times1.263[/tex]

A = [tex]2526.24[/tex]

Rounding to the nearest integer, we get:

A =[tex]\$2526[/tex]

Therefore, the value of the continuous stream after 18 years will be [tex]\$2526.[/tex]

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