There is an upcoming election for student council president at a high school. Candidate A must get over 50% of the vote against Candidate B to be elected. A poll was taken of a random sample of 80 students from the high school and 44 students said they would vote for Candidate A. Simulations were done with an assumption that the population is split 50-50 using a sample size of 80 to see how likely a sample of 80 would have 44 who preferred Candidate A. The results of 200 simulations are shown below. Create an interval containing the middle 95% of the data based on the data from the simulation, to the nearest hundredth, and state whether the observed proportion is within the margin of error of the simulation results.

The correct classification for this triangle is an acute triangle.

The angle measures given are 51, 89, and 40 degrees.

There are no angles that are either equal to or greater than 90 degrees among those mentioned. Consequently, the triangle does not contain any angles that are either right or obtuse.

To categorize a triangle according to its angles, the total of the angles within the triangle, which is invariably 180 degrees, is taken into account.

51 + 89 + 40 = 180

Given that the total of the angles is 180 degrees, we can deduce that this particular triangle is acute in nature. An acute-angled triangle is a type of triangle that has three angles which are each smaller than 90 degrees.

Therefore, the correct classification for this triangle is an acute triangle.

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The required reverse Euclidean algorithm is the solution to the modular equation (5x) mod 91 is

x = 6(mod 91).

Given that (5*x) mod 91 =32.

To solve the modular equation (5*x) mod 91 =32 using reverse Euclidean algorithm is to find the modular inverse of 5 modulo 91.

Consider  (5*x) mod 91 =32.

5x = 32(mod 91)

Apply the Euclidean algorithm to find GCD of 5 and 91 is

91 = 18 * 5 + 1.

Rewrite it in congruence form,

1 = 91 - 18 *5

On simplifying the equation,

1 = 91 (mod 5)

The modular inverse of 5 modulo 91 is 18.

Multiply equation by 18 on both sides,

90x = 576 (mod91)

To obtain the smallest positive  solution,

91:576 = 6 (mod 91)

Divide both sides by the coefficient of x:

x = 6 * 90^(-1).

Apply the Euclidean algorithm,

91 = 1*90 + 1.

Simplify the equation,

1 + 1 mod (90)

The modular inverse of 90 modulo 91 is 1.

Substitute the modular inverse in the given question gives,

x = 6*1(mod 91)

x= 6 (mod91)

Therefore, the solution to the modular equation (5x) mod 91 is

x = 6(mod 91).

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The Volume of Trapezoidal prism is 420 cm².

From the given figure we can write the dimension of the prism as

a = 5, b=15, c= 15, d= 15

h= 7 and l = 6 cm

Now, Volume of Trapezoidal prism

= 1/2 (a+b) x h x l

= 1/2 (5+15) x 7 x 6

= 1/2 x 20 x 42

= 10 x 42

= 420 cm²

Thus, the Volume of Trapezoidal prism is 420 cm².

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Answer: i'm kind of just guessing, but i think x = 13

Step-by-step explanation:

please don't ask me how i don't know

Let's solve the inequality to determine the number of outfits Tariq can purchase while staying within his budget.

Given:

Amount Tariq has to spend: $640

Cost of a new bicycle: $291.24

Cost of 4 bicycle reflectors: $19.56 each

Cost of bike gloves: $16.52

Cost of each biking outfit: $50.80

Let's assume the number of outfits Tariq can purchase is represented by x.

The total cost of the items he has already purchased is:

Cost of bicycle = $291.24

Cost of 4 bicycle reflectors = $19.56 * 4 = $78.24

Cost of bike gloves = $16.52

The remaining amount Tariq has to spend can be calculated as:

Remaining amount = Total amount - (Cost of bicycle + Cost of reflectors + Cost of gloves)

Remaining amount = $640 - ($291.24 + $78.24 + $16.52)

Now, we need to determine the maximum number of outfits Tariq can purchase with the remaining amount. Each outfit costs $50.80.

Inequality: x * $50.80 ≤ Remaining amount

Substituting the values:

x * $50.80 ≤ $640 - ($291.24 + $78.24 + $16.52)

Simplifying further:

x * $50.80 ≤ $640 - $385

x * $50.80 ≤ $255

To solve for x, we divide both sides of the inequality by $50.80:

x ≤ $255 / $50.80

x ≤ 5

Therefore, the maximum number of outfits Tariq can purchase while staying within his budget is 5.

Answer:  if the length of the second rectangular field is 200 meters, the width should be 35 meters to have the same perimeter but a larger area.

Step-by-step explanation:

STEP1:- Let's denote the length of the second rectangular field as L2 and the width as W2.

The perimeter of a rectangle is given by the formula:

Perimeter = 2(length + width).

For the first rectangular field with length L1 = 135 meters and width W1 = 100 meters, the perimeter is:

Perimeter1 = 2(135 + 100) = 470 meters.

STEP 2:- To find the length and width of the second rectangular field with the same perimeter but a larger area, we need to consider that the perimeters of both rectangles are equal.

Perimeter1 = Perimeter2

470 = 2(L2 + W2)

STEP 3 :- To determine the larger area, we need to find the corresponding length and width. However, there are multiple solutions for this problem. We can set an arbitrary value for one of the dimensions and calculate the other.

For example, let's assume the length of the second rectangular field as L2 = 200 meters:

470 = 2(200 + W2)

470 = 400 + 2W2

2W2 = 470 - 400

2W2 = 70

W2 = 35 meters

HENCE L2 = 200 meters and W2 = 35 meters

The total number of possible outcomes is the number of rows in the table, which depends on the size of the table.

To determine the experimental probability of correctly guessing exactly one correct answer out of five choices, we can utilize the random number table provided, where 0's represent incorrect answers and 1's represent correct answers.

Since we have five choices for each answer, we will focus on a single row of the random number table, considering five consecutive values.

Let's assume we have randomly selected a row from the table, and the numbers in that row are as follows:

0 1 0 1 0

In this case, the second and fourth answers are correct (represented by 1's), while the remaining three choices are incorrect (represented by 0's).

To calculate the experimental probability of exactly one correct answer, we need to determine the number of favorable outcomes (i.e., rows with exactly one 1) and divide it by the total number of possible outcomes (which is equal to the number of rows in the table).

Looking at the table, we can see that there are several possible rows with exactly one 1, such as:

0 1 0 0 0

0 0 0 1 0

0 0 0 0 1

Let's assume there are 'n' favorable outcomes. In this case, 'n' is equal to 3.

The total number of possible outcomes is the number of rows in the table, which depends on the size of the table. Without the specific size of the table, we cannot provide an accurate value.

To calculate the experimental probability, we divide the number of favorable outcomes by the total number of possible outcomes:

Experimental probability = n / Total number of possible outcomes

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Answer:

Refer to the step-by-step, follow along carefully.

Step-by-step explanation:

Verify the given identity.

[tex]\frac{\sin(x)}{1-\cos(x)} -\frac{\sin(x)\cos(x)}{1+\cos(x)} =\csc(x)(1+\cos^2(x))[/tex]

Pick the more complicated side to manipulate, so the L.H.S.

(1) - Combine the fractions with a common denominator

[tex]\frac{\sin(x)}{1-\cos(x)} -\frac{\sin(x)\cos(x)}{1+\cos(x)}\\\\\Longrightarrow \frac{\sin(x)(1+\cos(x))}{(1-\cos(x))(1+\cos(x))} -\frac{\sin(x)\cos(x)(1-\cos(x))}{(1+\cos(x))(1-\cos(x))} \\\\\Longrightarrow \frac{\sin(x)+\sin(x)\cos(x)-\sin(x)\cos(x)+\sin(x)\cos^2(x)}{(1+\cos(x))(1-\cos(x))} \\\\\Longrightarrow \boxed{\frac{\sin(x)+\sin(x)\cos^2(x)}{(1+\cos(x))(1-\cos(x))}} \\\\[/tex]

(2) - Simplify the denominator

[tex]\frac{\sin(x)+\sin(x)\cos^2(x)}{(1+\cos(x))(1-\cos(x))}\\\\\Longrightarrow \frac{\sin(x)+\sin(x)\cos^2(x)}{1-\cos(x)+\cos(x)-\cos^2(x)}\\\\\Longrightarrow \boxed{\frac{\sin(x)+\sin(x)\cos^2(x)}{1-\cos^2(x)}}[/tex]

(3) - Apply the following Pythagorean identity to the denominator

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Pythagorean Identity:}}\\\\1-\cos^2(\theta)=\sin^2(\theta)\end{array}\right}[/tex]

[tex]\frac{\sin(x)+\sin(x)\cos^2(x)}{1-\cos^2(x)}\\\\\Longrightarrow \boxed{\frac{\sin(x)+\sin(x)\cos^2(x)}{\sin^2(x)}}[/tex]

(4) - Simplify the fraction and split it up

[tex]\frac{\sin(x)+\sin(x)\cos^2(x)}{\sin^2(x)}\\\\\Longrightarrow \frac{1+\cos^2(x)}{\sin(x)}\\\\\Longrightarrow \boxed{\frac{1}{\sin(x)}+\frac{\cos^2(x)}{\sin(x)}}[/tex]

(5) - Apply the following reciprocal identity

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Reciprocal Identitiy:}}\\\\\csc(\theta)=\frac{1}{\sin(\theta)} \end{array}\right}[/tex]

[tex]\frac{1}{\sin(x)}+\frac{\cos^2(x)}{\sin(x)}\\\\\Longrightarrow \csc(x)+\frac{1}{\sin(x)}\cos^2(x) \\\\\Longrightarrow \csc(x)+\csc(x)\cos^2(x) \\\\\therefore \boxed{\boxed{\csc(x)(1+\cos^2(x))}}[/tex]

Thus, the identity is verified.

Prove of the expression sin x / (1 - cos x) - [sin x cos x ] / (1 + cos x) by using trigonometry formula is shown below.

We have to given that,

Expression to verify is,

⇒ sin x / (1 - cos x) - [sin x cos x ] / (1 + cos x)

Now, We can simplify as,

⇒ sin x / (1 - cos x) - [sin x cos x ] / (1 + cos x)

⇒ sin x [ 1 / (1 - cos x) - cos x / (1 + cos x)]

⇒ sin x [1 + cos x - cos x (1 - cos x )] / (1 - cos²x)

⇒ sin x [1 + cos x - cos x + cos²x] / sin²x

⇒ (1 + cos²x) / sin x

⇒ cosec x (1 + cos²x)

Thus, Prove of the expression sin x / (1 - cos x) - [sin x cos x ] / (1 + cos x) by using trigonometry formula is shown above.

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Answer:

Step-by-step explanation:

z=6

Answer:

Step-by-step explanation:

Answer:

For month 1, the current balance is $2750.00, the interest is $45.38, and the payment is $75.00. The amount applied to principal is $29.62.

For the remaining months, the interest and payment amount will stay the same, but the current balance and amount applied to principal will change based on the previous month's numbers.

Point of view:

Here's your answer but I prefer you to focus and study hard because school isn't that easy. But i'm glad I could help you!

:)

The simplified expression in the context of this problem is given as follows:

[tex]5\sqrt{5}(\sqrt{10} - 3) = 25\sqrt{2} - 15\sqrt{5}[/tex]

The expression in the context of this problem is given as follows:

[tex]5\sqrt{5}(\sqrt{10} - 3)[/tex]

Applying the distributive property, we multiply the outer term by each of the inner terms, hence:

[tex]5\sqrt{50} - 15\sqrt{5}[/tex]

The number 50 can be written as follows:

50 = 2 x 25.

Hence the square root is simplified as follows:

[tex]\sqrt{50} = \sqrt{2 \times 25} = 5\sqrt{2}[/tex]

Hence the simplified expression is given as follows:

[tex]25\sqrt{2} - 15\sqrt{5}[/tex]

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The other options provided, -3x - 1, -x - 12, and -x - 3, do not match the simplified form of the given expression. Only -3x - 4 corresponds to the original expression after simplification. It is important to carefully distribute and combine like terms to simplify expressions correctly.

The expression shown is 2 + 2(x – 3) – 5x. To find an equivalent expression, we need to distribute the 2 to both terms inside the parentheses, resulting in 2x - 6. Now we can simplify the expression further:

2 + 2x - 6 - 5x

Combining like terms, we have:

(2x - 5x) + (2 - 6)

This simplifies to:

-3x - 4

Hence, the expression -3x - 4 is equivalent to 2 + 2(x – 3) – 5x.

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The measure of length of the triangle is solved and

a) x = 4.9 units

b) x = 14 units

c) x = 4.8 cm

d) b = 68.5 units

Given data ,

Let the triangle be represented as ΔABC

where the measure of the lengths of the sides are given as

a)

The measure of hypotenuse AC = 12

The measure of angle ∠BAC = 66°

So , from the trigonometric relations , we get

cos θ = adjacent / hypotenuse

cos 66° = x / 12

So , x = 12 cos ( 66 )°

x = 4.9 units

b)

The measure of base of triangle BC = 20 units

And , the angle ∠BAC = 55°

So , from the trigonometric relations , we get

tan θ = opposite / adjacent

tan 55° = 20/x

x = 20 / tan55°

x = 14 units

c)

The measure of base of triangle BC = 4 cm

And , the angle ∠BAC = 57°

So , from the trigonometric relations , we get

sin θ = opposite / hypotenuse

sin 57° = 4/x

x = 4 / sin 57°

x = 4.8 cm

d)

The measure of base of triangle BC = 38 units

And , the angle ∠BAC = 61°

So , from the trigonometric relations , we get

tan θ = opposite / adjacent

tan 61° = b/38

b = 38 x tan 61°

b = 68.5 units

Hence , the trigonometric relations are solved.

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The product of (7x - 5)(6x - 4) is 42x^2 - 58x + 20.

First, distribute the first term of the first polynomial (7x) to each term in the second polynomial (6x - 4):

7x × 6x = 42x²

7x × (-4) = -28x

Next, distribute the second term of the first polynomial (-5) to each term in the second polynomial (6x - 4):

-5 × 6x = -30x

-5 × (-4) = 20

Now, combine the like terms:

42x² - 28x - 30x + 20

Simplify the expression:

42x² - 58x + 20

Therefore, the product of (7x - 5)(6x - 4) is 42x^2 - 58x + 20.

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Answer:

The change in volume is -5.5 mL (a decrease in volume of 5.5 mL) when 60.0 mL of gas is cooled from 33.0°C to 5.00°C.

Step-by-step explanation:

To calculate the change in volume, we need to use the ideal gas law equation:

V1/T1 = V2/T2

where V1 and T1 are the initial volume and temperature, and V2 and T2 are the final volume and temperature.

Given:

V1 = 60.0 mL

T1 = 33.0°C = 33.0 + 273.15 = 306.15 K

T2 = 5.00°C = 5.00 + 273.15 = 278.15 K

Now we can calculate V2, the final volume:

V1/T1 = V2/T2

(60.0 mL) / (306.15 K) = V2 / (278.15 K)

Cross-multiplying and solving for V2:

V2 = (60.0 mL) * (278.15 K) / (306.15 K)

V2 = 54.5 mL

The final volume, V2, is 54.5 mL.

To find the change in volume, we subtract the initial volume from the final volume:

Change in volume = V2 - V1

Change in volume = 54.5 mL - 60.0 mL

Change in volume = -5.5 mL

Therefore, the change in volume is -5.5 mL (a decrease in volume of 5.5 mL) when 60.0 mL of gas is cooled from 33.0°C to 5.00°C.

The y-intercept of f(x + 4) + 8 is given as follows:

10.

A translation happens when either a figure or a function is moved horizontally or vertically on the coordinate plane.

The four translation rules for functions are defined as follows:

The parent function for this problem is given as follows:

[tex]f(x) = \log_2{x}[/tex]

The translated function is then given as follows:

[tex]g(x) = \log_2{x + 4} + 8[/tex]

The y-intercept of the function is the numeric value at x = 0, hence:

[tex]g(0) = \log_2{0 + 4} + 8[/tex]

g(0) = 2 + 8 = 10.

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The marked price (M.P.) is 1,72,000 when the selling price is 1,61,680 with discount of 6%.

To find the marked price (M.P.), we can use the formula:

M.P. = S.P. / (1 - D%)

Given:

S.P. = 1,61,680

D% = 6%

First, we need to convert the discount percentage to decimal form by dividing it by 100:

D% = 6/100 = 0.06

Now, we can substitute the values into the formula:

M.P. = 1,61,680 / (1 - 0.06)

M.P. = 1,61,680 / 0.94

M.P. = 1,72,000

Therefore, the marked price (M.P.) is approximately 1,72,000.

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Answer: There are several expressions that are equivalent to the given expression 3x^2 - 5a^3 + 2y^4. Here are a few examples:

These expressions have the same terms but may differ in the order in which the terms are written. It's important to note that the coefficients and exponents of the variables remain unchanged in each expression.

The correct option is the second one, the order is:

g(x), f(x), h(x).

To find the rate of change for a function f(x) on an interval [a, b] we need to get:

R = (f(b) - f(a))/(b - a)

Here the interval is [-1, 3]

The first function is:

f(x)=  (x + 3)² - 2

Evaluating we get:

f(-1) = (-1 + 3)² - 2

f(-1) = 2² - 2 = 4 -2 = 2

and f(3) = (3 + 3)² - 2 = 34

Then the rate is:

R = (34 - 2)/(3 + 1) = 8

For g(x) we can use the graph, we have:

R = (0 + 2)/4 = 1/2

For the last function we need to use the table, then we will get:

R = (62 - 14)/(3 + 1) = 12

Then the order, from least to greatest is:

g(x), f(x), h(x).

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Based on the information, A. There is a 99% chance that the proportion of students who eat cauliflower on Jane's campus is between 5.03% and 17.83%.

Calculate the sample proportion:

= x / n = 4 / 35

= 0.1143

Calculate the Agresti and Coull's adjustment factor:

zα/2 = z(1 - α/2) = z(1 - 0.99/2)

= 2.576

Calculate the margin of error:

= 2.576 √(0.1143(1 - 0.1143) / 35)

= 0.064

Calculate the confidence interval:

= 0.1143 ± 0.064

= (0.0503, 0.1783)

We are 99% confident that the true proportion of students who eat cauliflower on Jane's campus is between 5.03% and 17.83%.

In other words, if we were to repeat this study many times, we would expect to obtain a confidence interval that includes the true proportion of students who eat cauliflower on Jane's campus 99% of the time.

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Answer:

Decrease in temp.

Step-by-step explanation:

Here is the reason:

Initially, at an altitude of 3,000 meters above sea level, the temperature was 12 degrees Celsius. As the altitude increased, the temperature dropped to 4 degrees Celsius. Since the temperature decreased from 12 degrees Celsius to 4 degrees Celsius, there was a decrease in the temperature

The volume of the sphere, given that the sphere has a diameter of 14.6 mm is 1628.7 mm³

The following data were obtained from the question:

The volume of a sphere is giving by the following formula

Volume of sphere = 4/3πr³

Inputting the given parameters, we can obtain the volume of the sphere as follow:

Volume of sphere = (4/3) × 3.14 × 7.3³

Volume of sphere = (4/3) × 3.14 × 389.017

Volume of sphere = 1628.7 mm³

Thus, we can conclude from the above calculation that the volume of the sphere is 1628.7 mm³

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Step-by-step explanation:

prism

v=1/2 X 12cm X 8cm

V= 48

rectangular prism

v=80cm+48cm

v=128cm

The volume of the largest piece is 10, 597. 5 cm³

The largest part of the cone takes the shape of a cylinder.

Now, the formula for calculating the volume of a cylinder is expressed as;

V = πr²h

The parameters of the formula are enumerated as;

Now, substitute the values, we get;

Diameter = 2 radius

Radius = 30/2

Radius = 15cm

Height = 15cm

Now, substitute the values, we get;

Volume = 3.14 × 15² ×15

Find the square value and substitute, we have;

Volume = 10, 597. 5 cm³

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Answer:

The remaining is 75%, the length of the stick would be 1.2

Step-by-step explanation:

According to the information given,

We know that Brian cut off 25% of a stick which was 1.6 meters long

and 1.6 is the 100% of the stick:

It is fairly easy to calculate this, subtract 25 from 100 ( 100 - 25 ), which is equal to 75.

Hence, the answer is 75% and 1.2 for the remaining length of the stick

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Answer:

-8 and 4

Step-by-step explanation:

P= -2

-2+6 = 4

-2-8= -8

The required answer are :

6. The transformation from the graph of f to the graph of r in equation (6) involves compressing the graph horizontally by a factor of 5/2.

7. The transformation from the graph of f to the graph of r in equation (7) involves stretching the graph vertically by a factor of 6.

8.  The transformation from the graph of f to the graph of r in equation (8) involves shifting the graph horizontally to the right by 3 units.

9. The transformation from the graph of f to the graph of r in equation (9) involves compressing the graph horizontally by a factor of 4/3.

10.  The transformation from the graph of f to the graph of r in equation (10) involves shrinking the graph vertically by a factor of 1/2.

11.  The transformation from the graph of f to the graph of r in equation (11) involves shifting the graph vertically upward by 3 units.

In formula form: r(x) = f(2/5x)

This transformation causes the graph of r to become narrower compared to the graph of f, as it is compressed horizontally. The rate at which x-values change is increased, resulting in a steeper slope. The overall shape and direction of the graph remain the same, but it is narrower and more compact.

Therefore, the transformation from the graph of f to the graph of r in equation (6) involves compressing the graph horizontally by a factor of 5/2. This means that every x-coordinate in the graph of f is multiplied by 2/5 to obtain the corresponding x-coordinate in the graph of r. The vertical positioning of the graph remains unchanged.

In formula form: r(x) = 6f(x)

This transformation causes the graph of r to become taller compared to the graph of f, as it is stretched vertically. The rate at which y-values change is increased, resulting in a steeper slope. The overall shape and direction of the graph remain the same, but it is taller and more elongated.

Therefore, the transformation from the graph of f to the graph of r in equation (7) involves stretching the graph vertically by a factor of 6. This means that every y-coordinate in the graph of f is multiplied by 6 to obtain the corresponding y-coordinate in the graph of r. The horizontal positioning of the graph remains unchanged.

In formula form: g(x) = f(x - 3)

This transformation causes the entire graph of f to shift to the right by 3 units. Every point on the graph of f moves horizontally to the right, maintaining the same vertical position. The overall shape and slope of the graph remain the same, but it is shifted to the right.

Therefore, the transformation from the graph of f to the graph of r in equation (8) involves shifting the graph horizontally to the right by 3 units. This means that each x-coordinate in the graph of f is increased by 3 to obtain the corresponding x-coordinate in the graph of r. The vertical positioning of the graph remains unchanged.

In formula form: g(x) = f(4/3x)

This transformation causes the graph of r to become narrower compared to the graph of f, as it is compressed horizontally. The rate at which x-values change is increased, resulting in a steeper slope. The overall shape and direction of the graph remain the same, but it is narrower and more compact.

Therefore, the transformation from the graph of f to the graph of r in equation (9) involves compressing the graph horizontally by a factor of 4/3. This means that every x-coordinate in the graph of f is multiplied by 4/3 to obtain the corresponding x-coordinate in the graph of r. The vertical positioning of the graph remains unchanged.

In formula form: g(x) = 1/2 f(x)

This transformation causes the graph of r to become shorter compared to the graph of f, as it is vertically shrunk. The rate at which y-values change is decreased, resulting in a flatter slope. The overall shape and direction of the graph remain the same, but it is shorter and more compact.

The transformation from the graph of f to the graph of r in equation (10) involves shrinking the graph vertically by a factor of 1/2. This means that every y-coordinate in the graph of f is multiplied by 1/2 to obtain the corresponding y-coordinate in the graph of r. The horizontal positioning of the graph remains unchanged.

In formula form: g(x) = f(x) + 3

This transformation causes the entire graph of f to shift upward by 3 units. Every point on the graph of f moves vertically upward, maintaining the same horizontal position. The overall shape and slope of the graph remain the same, but it is shifted upward.

The transformation from the graph of f to the graph of r in equation (11) involves shifting the graph vertically upward by 3 units. This means that every y-coordinate in the graph of f is increased by 3 to obtain the corresponding y-coordinate in the graph of r. The horizontal positioning of the graph remains unchanged.

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Answer:

[tex]\textsf{a)} \quad T_n=-n^2+n+20[/tex]

[tex]\textsf{b)} \quad T_{12}=-112[/tex]

[tex]\textsf{c)} \quad \sf 8th\;term[/tex]

a)  Second difference is 2.

b)  First term is 10.

Step-by-step explanation:

The given number pattern is:

To determine the type of sequence, begin by calculating the first differences between consecutive terms:

[tex]20 \underset{-2}{\longrightarrow} 18 \underset{-4}{\longrightarrow} 14 \underset{-6}{\longrightarrow}8[/tex]

As the first differences are not the same, we need to calculate the second differences (the differences between the first differences):

[tex]-2 \underset{-2}{\longrightarrow} -4 \underset{-2}{\longrightarrow} -6[/tex]

As the second differences are the same, the sequence is quadratic and will contain an n² term.

The coefficient of the n² term is half of the second difference.

As the second difference is -2, the coefficient of the n² term is -1.

Now we need to compare -n² with the given sequence (where n is the position of the term in the sequence).

[tex]\begin{array}{|c|c|c|c|c|}\cline{1-5}n&1&2&3&4\\\cline{1-5}-n^2&-1&-4&-9&-16\\\cline{1-5}\sf operation&+21&+22&+23&+24\\\cline{1-5}\sf sequence&20&18&14&8\\\cline{1-5}\end{array}[/tex]

We can see that the algebraic operation that takes -n² to the terms of the sequence is to add (n + 20).

[tex]\begin{array}{|c|c|c|c|c|}\cline{1-5}n&1&2&3&4\\\cline{1-5}-n^2&-1&-4&-9&-16\\\cline{1-5}+n&0&-2&-6&-12\\\cline{1-5}+20&20&18&14&8\\\cline{1-5}\sf sequence&20&18&14&8\\\cline{1-5}\end{array}[/tex]

Therefore, the expression to find the the nth term of the given quadratic sequence is:

[tex]\boxed{T_n=-n^2+n+20}[/tex]

To find the value of T₁₂, substitute n = 12 into the nth term equation:

[tex]\begin{aligned}T_{12}&=-(12)^2+(12)+20\\&=-144+12+20\\&=-132+20\\&=-112\end{aligned}[/tex]

Therefore, the 12th term of the number pattern is -112.

To find the position of the term that has a value of -36, substitute Tₙ = -36 into the nth term equation and solve for n:

[tex]\begin{aligned}T_n&=-36\\-n^2+n+20&=-36\\-n^2+n+56&=0\\n^2-n-56&=0\\n^2-8n+7n-56&=0\\n(n-8)+7(n-8)&=0\\(n+7)(n-8)&=0\\\\\implies n&=-7\\\implies n&=8\end{aligned}[/tex]

As the position of the term cannot be negative, the term that has a value of -36 is the 8th term.

[tex]\hrulefill[/tex]

Given terms of a quadratic number pattern:

We know the first differences are negative, since the difference between the second and third terms is -7. Label the unknown differences as -a, -b and -c:

[tex]T_1 \underset{-a}{\longrightarrow} 1 \underset{-7}{\longrightarrow} -6 \underset{-b}{\longrightarrow}T_4 \underset{-c}{\longrightarrow} -14[/tex]

From this we can create three equations:

[tex]T_1-a=1[/tex]

[tex]-6-b=T_4[/tex]

[tex]T_4-c=-14[/tex]

The second differences are the same in a quadratic sequence. Let the second difference be x. (As we don't know the sign of the second difference, keep it as positive for now).

[tex]-a \underset{+x}{\longrightarrow} -7\underset{+x}{\longrightarrow} -b \underset{+x}{\longrightarrow}-c[/tex]

From this we can create three equations:

[tex]-a+x=-7[/tex]

[tex]-7+x=-b[/tex]

[tex]-b+x=-c[/tex]

Substitute the equation for -b into the equation for -c to create an equation for -c in terms of x:

[tex]-c=(-7+x)+x[/tex]

[tex]-c=2x-7[/tex]

Substitute the equations for -b and -c (in terms of x) into the second two equations created from the first differences to create two equations for T₄ in terms of x:

[tex]\begin{aligned}-6-b&=T_4\\-6-7+x&=T_4\\T_4&=x-13\end{aligned}[/tex]

[tex]\begin{aligned}T_4-c&=-14\\T_4+2x-7&=-14\\T_4&=-2x-7\\\end{aligned}[/tex]

Solve for x by equating the two equations for T₄:

[tex]\begin{aligned}T_4&=T_4\\x-13&=-2x-7\\3x&=6\\x&=2\end{aligned}[/tex]

Therefore, the second difference is 2.

Substitute the found value of x into the equations for -a, -b and -c to find the first differences:

[tex]-a+2=-7 \implies -a=-9[/tex]

[tex]-7+2=-b \implies -b=-5[/tex]

[tex]-5+2=-c \implies -c=-3[/tex]

Therefore, the first differences are:

[tex]T_1 \underset{-9}{\longrightarrow} 1 \underset{-7}{\longrightarrow} -6 \underset{-5}{\longrightarrow}T_4 \underset{-3}{\longrightarrow} -14[/tex]

Finally, calculate the first term:

[tex]\begin{aligned}T_1-9&=1\\T_1&=1+9\\T_1&=10\end{aligned}[/tex]

Therefore, the first term in the number pattern is 10.

[tex]10 \underset{-9}{\longrightarrow} 1 \underset{-7}{\longrightarrow} -6 \underset{-5}{\longrightarrow}-11 \underset{-3}{\longrightarrow} -14[/tex]

Note: The equation for the nth term is:

[tex]\boxed{T_n=n^2-12n+21}[/tex]