Use the relationships in the diagrams below to solve for the given variable. Justify your solution with a definition or theorem.

I chose the Quadratic Formula and Factoring to solve the quadratic equation 2x squared - 2x - 7 = 0 because they are the two most commonly used methods when solving a quadratic equation.

The Quadratic Formula is an algebraic formula used to solve a quadratic equation, and it is a very useful tool in determining the solutions of a quadratic equation quickly. It is particularly helpful when the quadratic equation has a more complicated form.

Factoring is also a very useful tool when solving a quadratic equation. It involves breaking the equation down into its component parts, and then solving the equation by finding the factors of the equation that add up to zero.

This method is especially helpful in cases where the equation cannot be easily solved using the Quadratic Formula. Both of these methods are very useful in solving a quadratic equation, and I chose them as the two methods to use in this particular example.

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Step-by-step explanation:

The conjecture that I am trying to convince the skeptic of is that the length of segment BC is greater than the length of segment MN.

First, let us consider the slopes of the lines AC and MN. Since both lines slope slightly down from left to right, we can conclude that the slope of AC is steeper than that of MN. This means that the distance between points A and C is greater than the distance between points M and N.

Next, let us consider the position of point B. Since point B is below the lines, we know that segment BA is shorter than segment BC. This is because segment BC is the hypotenuse of the right triangle BNC, while segment BA is just one of the legs of that triangle.

Finally, we can use the fact that segment BA intersects both lines AC and MN to show that segment MN is shorter than segment BC. This is because segment MN is just one of the legs of the right triangle AMN, while segment BC is the hypotenuse of the larger right triangle ABC. Since the hypotenuse of a triangle is always longer than its legs, we can conclude that segment BC is longer than segment MN.

Therefore, based on the above reasoning, we can conclude that the length of segment BC is indeed greater than the length of segment MN, which confirms our conjecture.

The end behavior is that as x moves toward negative infinity, f(x) moves toward negative infinity and as x moves toward positive infinity, f(x) moves toward positive infinity. F(x) only has one x-intercept, which is x = 3.

The given function is f(x) = -x³ + 2x² + 3

1. End behavior:

As the leading coefficient of f(x) is negative, -x^3 dominates the behavior of the function as x moves towards -∞ or +∞. Therefore, the end behavior of the function is:

2. X-intercepts:

To determine the x-intercepts of the function, we have to calculate the equation f(x) = -x^3 + 2x^2 + 3 = 0. We can factor out a common factor of -1 to simplify the equation:

-x^3 + 2x^2 + 3 = 0

x^3 - 2x^2 - 3 = 0

Applying synthetic division or long division, one of the roots of this equation is x = 3. We can then factor out (x - 3) using polynomial division to find the other two roots:

x^3 - 2x^2 - 3 = (x - 3)(x^2 + x + 1)

The factor of x^2 + x + 1 has no real roots, so the only x-intercept of f(x) is x = 3.

3. Positive/negative:

To determine the sign of f(x) for different values of x, we can look at the sign of each factor in the polynomial:

Putting these together, we can make the following sign chart for f(x):

x                         f(x)

x < 0                   -

0 < x < 3                  +

x > 3                  -

4. Multiplicity: The single root of f(x) has multiplicity 1 and is given by x = 3. This indicates that at x = 3, the graph of f(x) crosses the x-axis and switches signs.

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Answer: -4

Basically, just plug in m and n as the values you were given:

tell me if you still have questions

Answer: The given vector can be expressed as a linear combination of u, v, and w as (14, 9, 14) = u - v + 3w.

Question: Express the following as a linear combination of u =(3, 1,6), v = (1.-1.4) and w=(8,3,8). (14, 9, 14) = ____ u- _____ v+ _____

Current Progress: To express the given vector as a linear combination of u, v, and w, we need to find scalars a, b, and c such that (14, 9, 14) = a*u + b*v + c*w.

Step 1: Write the equation in component form:
(14, 9, 14) = (3a + b + 8c, a - b + 3c, 6a + 4b + 8c)

Step 2: Equate the corresponding components and solve for a, b, and c:
3a + b + 8c = 14
a - b + 3c = 9
6a + 4b + 8c = 14

Step 3: Solve the system of equations using any method (substitution, elimination, etc.). One possible solution is a = 1, b = -1, and c = 3.

Step 4: Plug the values of a, b, and c back into the linear combination equation:
(14, 9, 14) = 1*u + (-1)*v + 3*w

Step 5: Simplify the equation:
(14, 9, 14) = u - v + 3w

Answer: The given vector can be expressed as a linear combination of u, v, and w as (14, 9, 14) = u - v + 3w.

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Fully  parenthesize form of arithmetic operation are -

1. For the expression x + y * z - x, you can fully parenthesize it as follows: ((x + (y * z)) - x)
2. For the expression y - z * x / w + z, you can fully parenthesize it as follows: (((y - ((z * x) / w)) + z)
3. For the expression (z - w + x) / z * w, you can fully parenthesize it as follows: (((z - w) + x) / z) * w
4. For the expression (x * w / z) - (x - y) * y, you can fully parenthesize it as follows: (((x * w) / z) - ((x - y) * y))
5. For the expression z - (w + x ) - y * z - y, you can fully parenthesize it as follows: (((z - (w + x)) - (y * z)) - y)
6. For the expression (x - y + z) * z / z, you can fully parenthesize it as follows: (((x - y) + z) * z) / z

Given arithmetic expressions: 1. x + y * z - x2. y - z * x / w + z3. (z - w + x) /z * w4. (x * w / z) - (x - y) * y5. z - (w + x ) - y * z - y6. (x - y + z) * z / z. To apply fully parenthesize form in solving the arithmetic operation, the given arithmetic expressions are:1. ((x) + ((y) * (z))) - (x)2. ((y) - (((z) * (x)) / (w))) + (z)3. ((((z) - (w)) + (x)) / (z)) * (w)4. ((((x) * (w)) / (z)) - ((x) - (y))) * (y)5. (z) - (((w) + (x)) - ((y) * (z))) - (y)6. (((x) - (y) + (z)) * (z)) / (z)

Therefore, the fully parenthesized form in solving the arithmetic operation for the given expressions is shown as above.

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Answer: B

Step-by-step explanation:

A perfect square is in the name: square!

Any number squared means that it can represent a square, since the side lengths are equal. In this example, 6^2 is a perfect square since a square with equal side lengths of 6*6 would be perfect; any other length measures would not make it a square. Hope this helps explain it!

Answer:

6^2

Step-by-step explanation:

62 = (6 ×  6) = 36. Here, 36 is a perfect square because it is the product of two equal integers, 6 × 6 = 36.

A total of 16 customers order the classic cheesecake.

To find out how many customers order the classic cheesecake, c, we need to subtract the number of customers who order ice cream from the total number of customers who order dessert.

This is because the question tells us that there are only two dessert options, ice cream and classic cheesecake, c.

Here's the equation we can use to solve the problem:

c = total number of customers who order dessert - number of customers who order ice cream

We can plug in the given values from the question:

c = 38 - 22

Simplifying the equation:

c = 16

Therefore, 16 customers order the classic cheesecake, c.

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Using coordinate geometry, we can find that Theo is 5.61 units closer to the tube slide than the wave pool.

Using the formula obtained from Pythagoras' theorem, distance can be computed. In coordinate geometry, the formula for distance is:

√ [(x2 – x1) ² + (y2 – y1) ²].

Given,

Theo is at point (5,0)

Wave pool is at point (-8, -12)

Tube Slide is at point (10,11)

We must first figure out how far Theo is from each ride in order to gauge how near he is to any of them. The distance formula can be used to achieve this.

By distance formula to find distance between two points (x1, y1) and (x2, y2), d= √[(x2-x1) ² + (y2-y1) ²]

Distance between Theo and wave pool is=

d= √ [(-13) ² + (-12) ²]

d= √313

d= 17.69 units.

Distance between Theo and tube slide is=

d= √ [5² + 11²]

d= √146

d = 12.08 units.

Difference between the two distances = 17.69 - 12.08

= 5.61 units.

Therefore, Theo is 5.61 units closer to the tube slide than the wave pool.

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The model that represents the answer is a rectangle divided into three pieces with one shaded, which indicates that each friend will receive 1/3 of the candy bar.

What is rectangle?

A rectangle is a four-sided flat shape that has four right angles, where opposite sides are parallel and congruent (i.e., of equal length).

The model that represents the answer to the given problem is a rectangle divided into three pieces with one shaded.

Each friend will receive an equal portion of the candy bar, so the candy bar needs to be divided into three equal parts.

Each part represents the fraction of the candy bar that each kid will receive.

Therefore, the model that represents the answer is a rectangle divided into three pieces with one shaded, which indicates that each friend will receive 1/3 of the candy bar.

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The mode of the set of variables "17,7,6,3,17,17,4,3,9,5,17,17" is 17 .

The Mode of a data set is defined as the value that appears most frequently in the dataset. The mode is one of the measures of central tendency, along with the mean and median.

The mode is useful in describing the most common value in a dataset, and can be specially helpful when dealing with categorical or discrete data.

The mode can be more than one value or may not exist at all in some datasets.

From the given set of variables "17,7,6,3,17,17,4,3,9,5,17,17",

The number 17 appears 6times, which is more than any other number in the dataset.

Therefore, the mode is 17.

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Answer: 85.41 percent

Step-by-step explanation:

2882.45 / 3374.60 = 0.8541 or 85.41%

Answer:

85.416%

Step-by-step explanation:

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The required probabilities are,

1. P(A and B) = 7/100.

2. P(AIB) = 16/25

3. P(AIB) = 0.495

4. P(A and B) = 0.7.

Conditional probability is a term used in probability theory to describe the likelihood that one event will follow another given the occurrence of another event.

1. Given:

The probabilities are,

P(A) = 1/5, P(BIA) = 7/20.

P(A and B) = P(A ∩ B) = P(BIA) x P(A)

Now, P(A and B) = 7/20 x 1/5 = 7/100.

2. Given:

P(B) = 3/4, P(A and B) = P(A ∩ B) = 12/25

P(AIB) = P(A ∩ B) /P(B) = 12/25 x 4/3 = 16/25

3. Given:

P(B) = 0.5, P(A and B) = P(A ∩ B) = 0.2475

P(AIB) = P(A ∩ B) /P(B) = 0.2475/0.5 = 0.495

4.  Given:

P(A) = 0.5, P(BIA) = 0.35.

P(A and B) = P(A ∩ B) = P(BIA) x P(A)

Now, P(A and B) = 0.35/0.5 = 0.7.

Therefore, all the required values are given above.

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$630,463.17

The revenue function in dollars from the sale of x campers is given by:

R(x) = 66,000x + 46,000(10+x)^(-1) - 8000

We can simplify this function as follows:

R(x) = 66,000x + 46,000/(10+x) - 8000

= 66,000x + 4600/(1+x/10) - 8000

To find the x-value that maximizes revenue, we need to find the critical points of the function R(x) and then determine whether each critical point corresponds to a local maximum, local minimum, or neither.

Taking the derivative of R(x) with respect to x, we get:

R'(x) = 66,000 - 4600/(1+x/10)^2

Setting R'(x) equal to zero, we get:

66,000 - 4600/(1+x/10)^2 = 0

Multiplying both sides by (1+x/10)^2, we get:

66,000(1+x/10)^2 - 4600 = 0

Expanding and simplifying, we get:

(x+10)^2 = 44/3

Taking the square root of both sides, we get:

x+10 = ±(2*sqrt(11))/3

Subtracting 10 from both sides, we get:

x = -10 ± (2*sqrt(11))/3

So the critical points of R(x) are:

x = -10 + (2sqrt(11))/3 and x = -10 - (2sqrt(11))/3

To determine whether each critical point corresponds to a local maximum, local minimum, or neither, we can use the second derivative test. Taking the second derivative of R(x), we get:

R''(x) = 9200/(1+x/10)^3

At the critical point x = -10 + (2*sqrt(11))/3, we have:

R''(-10 + (2sqrt(11))/3) = 9200/(1+(-10+(2sqrt(11))/3)/10)^3

= 9200/(1+2*sqrt(11)/30)^3

≈ -318.68

Since R''(-10 + (2sqrt(11))/3) is negative, the critical point x = -10 + (2sqrt(11))/3 corresponds to a local maximum of R(x).

Similarly, at the critical point x = -10 - (2*sqrt(11))/3, we have:

R''(-10 - (2sqrt(11))/3) = 9200/(1+(-10-(2sqrt(11))/3)/10)^3

= 9200/(1-2*sqrt(11)/30)^3

≈ 318.68

Since R''(-10 - (2sqrt(11))/3) is positive, the critical point x = -10 - (2sqrt(11))/3 corresponds to a local minimum of R(x).

Therefore, the x-value that maximizes revenue is x = -10 + (2*sqrt(11))/3, and the maximum revenue is:

R(-10 + (2sqrt(11))/3) = 66,000(-10 + (2sqrt(11))/3) + 46,000/(10-10+(2*sqrt(11))/3) - 8000

≈ $630,463.17

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Rounding up to the nearest integer, the sample size is approximately 803.

To find the sample size, we need to use the formula for margin of error:

The margin of error = Z * √[(p * (1 - p))/n]

Where Z is the Z-score, p is the proportion of the population with the characteristic of interest (in this case, favoring the death penalty), and n is the sample size.

We are given the margin of error (3.4 percentage points, or 0.034) and the proportion (0.52), and we need to find the Z-score and the sample size.

The Z-score corresponds to the confidence level, which is 95% in this case. Using the standard normal cumulative probabilities table, we can find that the Z-score for a 95% confidence level is 1.96

Plugging in the values we have into the formula:

0.034 = 1.96 * √[(0.52 * (1 - 0.52))/n]

Squaring both sides and rearranging:

n = (1.96^2 * 0.52 * (1 - 0.52))/(0.034^2)

n = 802.33

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Answer:

1. 1hour 10minutes = 70 minutes

2. 190 miles

Step-by-step explanation:

1.
*Fraction*

Feet/minutes

- Divide

4/1 - 4feet over 1minute

4/1 = 280/x

4x70=280

1x70= 70

70 minutes = 1 hour 10 minutes

2.
1 hour = 65 miles

65x2hours=130miles

2hours= 130 miles

65x3hours= 195miles

3hours = 195miles

This function 8a-27k+m is equal to -(71/27).

The exponential function f(x)=((2)/(3))^(2-3x)-2 can be rewritten in the form f(x)=ka^(x)+m by following these steps:

1. Start with the original equation: f(x)=((2)/(3))^(2-3x)-2
2. Rewrite the exponent as a product: f(x)=((2)/(3))^(2*(1-3/2)*x)-2
3. Use the property of exponents that a^(b*c)= (a^b)^c: f(x)=(((2)/(3))^2)^(1-3/2*x)-2
4. Simplify the exponent: f(x)=((4)/(9))^(1-3/2*x)-2
5. Rewrite the equation in the form f(x)=ka^(x)+m, where k=((4)/(9))^1, a=((4)/(9))^(-3/2), and m=-2: f(x)=((4)/(9))*((4)/(9))^(-3/2*x)-2

Now, we can plug in the values for k, a, and m into the expression 8a-27k+m to find the answer:

8a-27k+m = 8*((4)/(9))^(-3/2)-27*((4)/(9))-2
= (8/(4^(3/2)*9^(3/2)))-(27*(4/9))-2
= (8/(8*27))-(27*(4/9))-2
= (1/27)-(12/9)-2
= (1/27)-(4/3)-2
= -(71/27)

Therefore, 8a-27k+m is equal to -(71/27).

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The solutions for A if 5A−|[1 0], [2 3]|=3A−|[5 2] ,[ 6 1]|  is 5  and for equation 3A−[2],[1]=5A−2[3],[0] is 2.

To find A in the given equations, we need to use the properties of matrices and solve for A.

For equation a:
5A−|[1 0], [2 3]|=3A−|[5 2], [6 1]|

First, we need to find the determinants of the matrices. The determinant of a 2x2 matrix is found by multiplying the elements on the main diagonal and subtracting the product of the elements on the other diagonal.

So, |[1 0], [2 3]| = (1*3) - (0*2) = 3
And, |[5 2], [6 1]| = (5*1) - (2*6) = -7

Now we can substitute these values back into the equation and solve for A.

5A - 3 = 3A - (-7)
5A - 3 = 3A + 7
2A = 10
A = 5

For equation b:
3A−[2],[1]=5A−2[3],[0]

First, we need to find the determinant of the matrix [2],[1]. The determinant of a 1x1 matrix is simply the value of the single element.

So, |[2],[1]| = 2

Now we can substitute this value back into the equation and solve for A.

3A - 2 = 5A - 2(3)
3A - 2 = 5A - 6
2A = 4
A = 2

Therefore, the solutions for A are 5 for equation a and for equation b the value A is 2

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The worth of the equipment after 7 years will be $3145.7

A geometric sequence is a special type of sequence where the ratio of every two successive terms is a constant.

Given is table with values of product for 3 years, we need to find its worth after 7 years, using the formula given,

aₙ = a₁rⁿ⁻¹

The given formula is of geometric sequence,

r = 9600 / 12000 = 0.8

a₁ = 12000

a₇ = 12000(0.8)⁷⁻¹

a₇ = 12000(0.8)⁶

a₇ = 3145.7

Hence, the worth of the equipment after 7 years will be $3145.7

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The angles of the diagram of a triangle is given by the statements ,

a) m∠5 + m∠6 = 180°

b) m∠2 + m∠3 = m∠6

c) m∠2 + m∠3 + m∠5 = 180°

A triangle is a plane figure or polygon with three sides and three angles.

A Triangle has three vertices and the sum of the interior angles add up to 180°

Let the Triangle be ΔABC , such that

∠A + ∠B + ∠C = 180°

The area of the triangle = ( 1/2 ) x Length x Base

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

if a² + b² = c² , it is a right triangle

if a² + b² < c² , it is an obtuse triangle

if a² + b² > c² , it is an acute triangle

Given data ,

Let the triangle be represented as ΔABC

Now , the measures of angles of a triangle are

Let the Triangle be ΔABC , such that

∠A + ∠B + ∠C = 180°

So , the measure of m∠2 + m∠3 + m∠5 = 180°

And , for a straight line = 180° ( angles in a straight line is 180° )

So , m∠5 + m∠6 = 180°

And , the exterior angle of a triangle = sum of the interior opposite angles

So , m∠2 + m∠3 = m∠6

Hence , the angles of a triangle are solved

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The complete question is attached below :

Which statements are always true regarding the diagram? Select three options.

m∠5 + m∠3 = m∠4

m∠3 + m∠4 + m∠5 = 180°

m∠5 + m∠6 =180°

m∠2 + m∠3 = m∠6

m∠2 + m∠3 + m∠5 = 180°

Answer: He has 80 pages in his book.

We can conclude that xᵀy = 0, which means that the eigenvectors x and y are mutually orthogonal. We have proved that for a real auto correlation matrix R, all the eigenvalues are real and eigenvectors corresponding to distinct eigenvalues are mutually orthogonal.

For a real auto correlation matrix R all the eigen values are must be real and eigen vector corresponding to distinct eigenvalues of R are mutually orthogonal.Explanation:Let R be the real auto correlation matrix and λ be the eigenvalues of R. Since R is a real matrix, we know that λ must also be real. Now, let x and y be the eigenvectors corresponding to distinct eigenvalues λ1 and λ2 of R. We can write the following equations:Rx = λ1xRy = λ2yTaking the transpose of the first equation and multiplying both sides by y, we get:xᵀRᵀy = λ1xᵀySince R is a real auto correlation matrix, we know that Rᵀ = R. Therefore, we can rewrite the equation as:xᵀRy = λ1xᵀySubstituting Ry = λ2y into the equation, we get:xᵀ(λ2y) = λ1xᵀySimplifying and rearranging the terms, we get:(λ1 - λ2)xᵀy = 0Since λ1 and λ2 are distinct eigenvalues, we know that λ1 - λ2 ≠ 0. Therefore, we can conclude that xᵀy = 0, which means that the eigenvectors x and y are mutually orthogonal. Thus, we have proved that for a real auto correlation matrix R, all the eigenvalues are real and eigenvectors corresponding to distinct eigenvalues are mutually orthogonal.

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The solution is given by [tex](18r^4+30r^3+25r^2-154r)/(r^6-5r^5+16r^4-45r^3-126r^2+3969r)[/tex]

To perform the indicated operations, we need to find a common denominator for all of the fractions. The common denominator will be the product of the three denominators: [tex](r^2-7r)(r^2+9r)(r^2+2r-63)[/tex].

Next, we will multiply each fraction by the appropriate factor to make the denominators equal. For the first fraction, we will multiply by [tex](r^2+9r)(r^2+2r-63)/(r^2+9r)(r^2+2r-63)[/tex].

For the second fraction, we will multiply by [tex](r^2-7r)(r^2+2r-63)/(r^2-7r)(r^2+2r-63)[/tex]. For the third fraction, we will multiply by [tex](r^2-7r)(r^2+9r)/(r^2-7r)(r^2+9r)[/tex].

After multiplying, we will combine the numerators and keep the common denominator:

[tex](9(r^2+9r)(r^2+2r-63)+10(r^2-7r)(r^2+2r-63)-1(r^2-7r)(r^2+9r))/(r^2-7r)(r^2+9r)(r^2+2r-63)[/tex]

Finally, we will simplify the numerator and denominator as much as possible to get the final answer:

[tex](9r^4+72r^3-567r^2+18r^3+162r^2-1134r+10r^4-70r^(3)+630r^2-20r^3-140r^2+980r-r^4+7r^3-9r^2)/(r^6-7r^5+9r^4-63r^3+2r^5-14r^4+18r^3-126r^2-63r^4+441r^3-567r^2+3969r)[/tex]

[tex](18r^4+30r^3+25r^2-154r)/(r^6-5r^5+16r^4-45r^3-126r^2+3969r)[/tex]

So the final answer is [tex](18r^4+30r^3+25r^2-154r)/(r^6-5r^5+16r^4-45r^3-126r^2+3969r)[/tex]

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Check the picture below.

Answer:

Heterozygous matches up with C

Homozygous dominant is D

Homozygous recessive is E

Genotype is A

and Phenotype is B

The profit function is obtained by subtracting the cost function from the revenue function. In other words, Profit = Revenue - Cost.

In the given question, the profit function is already given as P(x) = -x^(2) + 1230x - 36000. This means that the profit is a function of x, which could represent the number of units sold or produced.

To find the profit for a specific value of x, simply substitute the value of x into the profit function and simplify. For example, if x = 100, then the profit would be:

P(100) = -(100)^(2) + 1230(100) - 36000
P(100) = -10000 + 123000 - 36000
P(100) = 77000

Therefore, the profit when x = 100 is $77,000.

Similarly, you can find the profit for any value of x by substituting the value into the profit function and simplifying.

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The equation of the function is f(x) = 1/2cos(x)

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

The sinusoidal graph

The sinusoidal graph is a cosine function

The parent function of a cosine function is

y = cos(x)

Based on the points on the graph, we can see that the graph is shrinked by 2

Mathematically, this is represented as

f(x) = 1/2y

So, we have

f(x) = 1/2cos(x)

Hence, the function is f(x) = 1/2cos(x)

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By completing the square, the expression x²+10x+140 can be rewritten in the form (x+A)²+B. Where A=5 and B=-115.

To do this, we need to find the values of A and B.

Start with the original expression: x²+10x+140
Move the constant term to the right side of the equation: x²+10x = -140
Take half of the coefficient of the x term and square it: (10/2)²= 25
Add this value to both sides of the equation: x²+10x+25 = -140+25
Simplify the right side of the equation: x²+10x+25 = -115
Factor the left side of the equation: (x+5)² = -115

Therefore, the expression x²+10x+140 can be rewritten as (x+5)²-115, where A=5 and B=-115.

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