Use long division to fine the quotient
(4x^3-6x^2-4x+8) divided by (2x-1)

The expression when factored by the GCF is 6x(x - 2y + 3y^2)

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

6x^2 - 12xy + 18xy^2

The greatest common factor (GCF) of 6x^2, -12xy, and 18xy^2 is 6x, which is the largest number that divides each term evenly.

To factor the polynomial by factoring out the GCF, we can divide each term by 6x to get:

6x^2/6x = x

-12xy/6x = -2y

18xy^2/6x = 3y^2

So the polynomial can be factored as:

6x^2 - 12xy + 18xy^2 = 6x(x - 2y + 3y^2)

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The angle between the vectors "[−12​] and [6−1​]α" is approximately 2.03 radians.

To find the angle between the vectors, we can use the formula:

cos(α) = ([tex]V{1}[/tex] • [tex]V{2}[/tex]) / (||[tex]V{1}[/tex]|| ||[tex]V{2}[/tex]||)

Where v1 and v2 are the two vectors, and ||[tex]V{1}[/tex]|| and ||[tex]V{2}[/tex]|| are the magnitudes of the vectors.

For the given vectors:

[tex]V{1}[/tex] = [−1 2] and [tex]V{2}[/tex] = [6 −1]

[tex]V{1}[/tex]• v2 = (−1)(6) + (2)(−1) = −6 + −2 = −8

||[tex]V{1}[/tex]|| = √((-1)^2 + 2^2) = √(5)

||[tex]V{2}[/tex]|| = √(6^2 + (-1)^2) = √(37)

cos(α) = (−8) / (√(5) √(37)) = −8 / √(185)

α = cos^-1(−8 / √(185))

α ≈ 2.03 radians

Therefore, the angle between the vectors is approximately 2.03 radians.

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we can graph s=s(t),v=v(t), and a=a(t) by plotting the equations for s, v, and a as functions of t. The graph of s=s(t) is a sinusoidal curve with a linear trend. The graph of v=v(t) is a sinusoidal curve with a constant value. The graph of a=a(t) is a sinusoidal curve with a zero value.

The velocity v of the pendulum is the first derivative of the distance s with respect to time t. That is, \[ v=v(t)=\frac{ds}{dt} \] Similarly, the acceleration a of the pendulum is the first derivative of the velocity v with respect to time t. That is, \[ a=a(t)=\frac{dv}{dt} \] We can find the velocity v and acceleration a by taking the derivatives of the given equation for the distance s. \[ s=s(t)=0.04 \sin (2 t)+3 t \] The first derivative of s with respect to t is \[ v=v(t)=\frac{ds}{dt}=0.08 \cos (2 t)+3 \] The second derivative of s with respect to t is \[ a=a(t)=\frac{dv}{dt}=-0.16 \sin (2 t) \] Now, we can find the velocity v and acceleration a at the given times t= 8π ,t= 4π , and t= 2π by plugging in the values of t into the equations for v and a. \[ v( 8π )=0.08 \cos (2( 8π ))+3=3 \quad \text{m/s} \] \[ v( 4π )=0.08 \cos (2( 4π ))+3=3 \quad \text{m/s} \] \[ v( 2π )=0.08 \cos (2( 2π ))+3=3 \quad \text{m/s} \] \[ a( 8π )=-0.16 \sin (2( 8π ))=0 \quad \text{m/s}^2 \] \[ a( 4π )=-0.16 \sin (2( 4π ))=0 \quad \text{m/s}^2 \] \[ a( 2π )=-0.16 \sin (2( 2π ))=0 \quad \text{m/s}^2 \] Finally, we can graph s=s(t),v=v(t), and a=a(t) by plotting the equations for s, v, and a as functions of t. The graph of s=s(t) is a sinusoidal curve with a linear trend. The graph of v=v(t) is a sinusoidal curve with a constant value. The graph of a=a(t) is a sinusoidal curve with a zero value.

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AB is indeed the zero matrix, and we have found a 2 x 2 matrix B such that AB is the zero matrix.

When multiplying two matrices, we use the dot product of the rows of the first matrix and the columns of the second matrix to obtain the entries of the resulting matrix. In order for the product of two matrices to be the zero matrix, the dot product of each row of the first matrix with each column of the second matrix must be zero.

Let's construct a 2 x 2 matrix B such that AB is the zero matrix. We can use two different nonzero columns for B to obtain the zero matrix. Let's start with the first column of B:

B = [b1 b3]
   [b2 b4]

We need to find values for b1, b2, b3, and b4 such that the dot product of each row of A with each column of B is zero. Let's start with the first row of A and the first column of B:

-12b1 + 4b2 = 0

We can rearrange this equation to solve for b2 in terms of b1:

b2 = 3b1

Now let's look at the second row of A and the first column of B:

15b1 - 5b2 = 0

Substituting the value of b2 from the first equation gives us:

15b1 - 5(3b1) = 0

Simplifying gives us:

0 = 0

This equation is always true, so we can choose any value for b1 and find the corresponding value for b2. Let's choose b1 = 1:

b2 = 3(1) = 3

Now let's look at the first row of A and the second column of B:

-12b3 + 4b4 = 0

We can rearrange this equation to solve for b4 in terms of b3:

b4 = 3b3

Now let's look at the second row of A and the second column of B:

15b3 - 5b4 = 0

Substituting the value of b4 from the first equation gives us:

15b3 - 5(3b3) = 0

Simplifying gives us:

0 = 0

This equation is always true, so we can choose any value for b3 and find the corresponding value for b4. Let's choose b3 = 2:

b4 = 3(2) = 6

Now we have the values for all of the entries of B:

B = [1 2]
   [3 6]

Let's check our work by multiplying A and B:

AB = [-12 4] [1 2]
    [ 15 -5] [3 6]

 = [(-12)(1) + (4)(3)  (-12)(2) + (4)(6)]
    [(15)(1) + (-5)(3)  (15)(2) + (-5)(6)]

 = [0 0]
    [0 0]

So AB is indeed the zero matrix, and we have found a 2 x 2 matrix B such that AB is the zero matrix.

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The Height of the pool should be 1.6 m.

Every three-dimensional item requires some amount of space. The volume of this space is measured. Volume is defined as the space occupied by an object inside the confines of three-dimensional space. It is also known as the object's capacity.

Given:

length of pool= 12.5 m

width of pool = 5.7 m

and, Maximum volume of pool = 114 m³

So, Volume= l w h

114 = 12.5 x 5.7 x h

114 = 71.25 h

h = 114/ 71.25

h = 1.6 m

Thus, the height can be 1.6 m.

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The product of the inverse of these elementary matrices gives us the original matrix A:
A = E1^(-1)E2^(-1)E3^(-1)E4^(-1) =
[ 1 2 0 ]
[ -4 8 0 6 ]
[ 0 1 0 ]

Consider the given Gauss-Jordan reduction:
A =
[ 1 2 0 ]
[ -4 8 0 6 ]
[ 0 1 0 ]

We need to find the elementary matrices E1, E2, E3, and E4 such that:
A = E1^(-1)E2^(-1)E3^(-1)E4^(-1)

First, we can use an elementary matrix E1 to subtract 4 times the first row from the second row:
E1 =
[ 1 0 0 ]
[ -4 1 0 ]
[ 0 0 1 ]

Next, we can use an elementary matrix E2 to add -2 times the second row to the first row:
E2 =
[ 1 -2 0 ]
[ 0 1 0 ]
[ 0 0 1 ]

Then, we can use an elementary matrix E3 to subtract the third row from the second row:
E3 =
[ 1 0 0 ]
[ 0 1 -1 ]
[ 0 0 1 ]

Finally, we can use an elementary matrix E4 to divide the second row by 8:
E4 =
[ 1 0 0 ]
[ 0 1/8 0 ]
[ 0 0 1 ]

Therefore, the product of the inverse of these elementary matrices gives us the original matrix A:
A = E1^(-1)E2^(-1)E3^(-1)E4^(-1) =
[ 1 2 0 ]
[ -4 8 0 6 ]
[ 0 1 0 ]

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1)Substituting the given values, we get:

xyz = 3(-1)(2) = -6

Therefore, when x=3, y=-1, and z=2, the value of xyz is -6.

2)y^2 = (-1)^2 = 1

Therefore, when x = 3, y = -1, and z = 2, y^2 = 1.

3) 2

The price per can has been lowered by R2, making the standard can cost R7.

A mathematical assertion that establishes the equivalence of two expressions is known as an equation. It is made up of two sides, a left and a right side, separated by the equal sign (=). The equation, which is typically used to determine an unknown variable's value, requires that the values on both sides be equal. Exponents, logarithms, multiplication, division, addition, subtraction, and other mathematical operations can all be used in equations.

given

Let's say that the standard price per can is "x" Rand.

The revised price is (x-2) Rands if the price is decreased by R2.

The issue is that 14 cans are being offered for the customary price of 10 cans. Thus, we can construct the equation:

10x = 14(x-2) (x-2)

After finding x, we obtain:

10x = 14x - 28

-4x = -28

x = 7

The price per can has been lowered by R2, making the standard can cost R7.

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Both statements are false because the sum of two angles does not equal the sum of their cosines or sines.

Statement 53 is false. This is because the sum of two angles does not equal the sum of their cosines. The correct equation for the sum of two angles in terms of their cosines is:

\( \cos \left(30^{\circ}+60^{\circ}\right)=\cos 30^{\circ} \cos 60^{\circ} - \sin 30^{\circ} \sin 60^{\circ} \)

Using this equation, we can calculate the correct value of the cosine of the sum of the two angles:

\( \cos \left(30^{\circ}+60^{\circ}\right)=\cos 30^{\circ} \cos 60^{\circ} - \sin 30^{\circ} \sin 60^{\circ} \)
\( \cos 90^{\circ}=\frac{\sqrt{3}}{2} \cdot \frac{1}{2} - \frac{1}{2} \cdot \frac{\sqrt{3}}{2} \)
\( \cos 90^{\circ}=0 \)

Statement 54 is also false. This is because the sum of two angles does not equal the sum of their sines. The correct equation for the sum of two angles in terms of their sines is:

\( \sin \left(30^{\circ}+60^{\circ}\right)=\sin 30^{\circ} \cos 60^{\circ} + \cos 30^{\circ} \sin 60^{\circ} \)

Using this equation, we can calculate the correct value of the sine of the sum of the two angles:

\( \sin \left(30^{\circ}+60^{\circ}\right)=\sin 30^{\circ} \cos 60^{\circ} + \cos 30^{\circ} \sin 60^{\circ} \)
\( \sin 90^{\circ}=\frac{1}{2} \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} \)
\( \sin 90^{\circ}=1 \)

In conclusion, both statements are false because the sum of two angles does not equal the sum of their cosines or sines. The correct equations for the sum of two angles in terms of their cosines and sines are shown above.

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Answer:

The area of the shaded region can be calculated by subtracting the area of the inner circle from the area of the outer circle. The area of the inner circle is pi multiplied by the square of the radius, or 3.14 * (125.6/2)^2, which equals 19644.8. The area of the outer circle is pi multiplied by the square of the radius, or 3.14 * (125.6)^2, which equals 49766.56. Subtracting the inner circle’s area from the outer circle's area, the area of the shaded region is 30121.76, rounded to the nearest hundredth

Step-by-step explanation:

slope = -1/5

used slope forumula

Answer:

h ≤ 6

Step-by-step:

Let h be the height of the triangular pennant in inches.

The formula for the area of a triangle is:

A = 1/2 * base * height

We know that the base of the triangle is 10 inches, so we can substitute this value into the formula:

A = 1/2 * 10 * h

Simplifying this equation, we get:

A = 5h

We also know that the area of the pennant must be at most 30 square inches. So we can write:

A ≤ 30

Substituting the formula for the area, we get:

5h ≤ 30

Dividing both sides by 5, we get:

h ≤ 6

Therefore, the possible heights of the triangle must be at most 6 inches in order for the area of the pennant to be at most 30 square inches.

The inequality that describes the possible heights of the triangle is:

h ≤ 6

The given permutations can be expressed as the product of the following cyclic permutations: (1 4 5 10)(2 6)(3 9 7 8).

The given permutations can be expressed as products of cyclic permutations as follows:

S10 = ([1,2,3,4,5,6,7,8,9,10],[4,6,9,5,10,2,8,3,7,1])

= (1 4 5 10)(2 6)(3 9 7 8)(7 8 3 9)

= (1 4)(4 5)(5 10)(2 6)(3 9)(9 7)(7 8)(8 3)

= (1 4 5 10)(2 6)(3 9 7 8)

Therefore, the given permutations can be expressed as the product of the following cyclic permutations: (1 4 5 10)(2 6)(3 9 7 8).

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Answer:

see below

Step-by-step explanation:

net worth = assets- liabilities

assets = what u have (good stuff)/own eg money

liabilities = what u owe/spending

so work backwards or put numbers in place

672 = assets- 314

672+314 = assets (ans)

The exponential function showing the relationship between y and t is y = 41,000 x (1.07)^t

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

Initial value, a = 41,000

Rate = 7% increment

The exponential function for the college enrollment y of the college, in dollars, after t years can be expressed as:

y = a(1 + r)^t

Substitute the known values in the above equation, so, we have the following representation

y = 41,000 x (1 + 7%)^t

Evaluate

y = 41,000 x (1.07)^t

Where 1.07 is the factor by which the college enrolment increases

Hence, the function is y = 41,000 x (1.07)^t

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The relative frequency for the total number of boys in this survey, rounded to the nearest hundredth, is 0.43.

What is frequency ?

In statistics, frequency refers to the number of times a particular observation or value occurs in a given dataset or sample. It is often represented in a frequency table, which organizes the data by category or value and shows the number of occurrences for each.

To find the relative frequency for the total number of boys in this survey, we need to divide the number of boys who prefer math or social studies by the total number of students in the survey, which is 120.

Relative frequency of boys who prefer math = 43/52 ≈ 0.83

Relative frequency of boys who prefer social studies = 9/52 ≈ 0.17

Relative frequency of total number of boys = 52/120 ≈ 0.43

Therefore, the relative frequency for the total number of boys in this survey, rounded to the nearest hundredth, is 0.43.

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To solve this Cyberchallenge, you will need to use browser dev tools and curl web requests.

Here are the steps to do it:
1. Open the browser dev tools by pressing F12 on your keyboard or right-clicking on the page and selecting "Inspect Element".
2. Go to the "Network" tab in the dev tools.
3. Start a curl web request by typing "curl" followed by the URL of the security gateway in the command line.
4. Add the "-v" option to the curl command to see the headers and response body of the request.
5. Look for the question asked by the gateway in the response body.
6. Use a calculator to quickly calculate the answer to the question.
7. Add the "-d" option to the curl command followed by the answer to the question to send the answer as a POST request.
8. Look for the flag in the response body of the POST request.
I hope this helps you understand how to solve the Cyberchallenge.

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The line L(t) = (2 - 4t, 3 – 2t, -5 – 4t) is neither parallel nor perpendicular to any of the given planes.

The line L(t) = (2 - 4t, 3 – 2t, -5 – 4t) can be written in parametric form as: x = 2 - 4t, y = 3 - 2t, and z = -5 - 4t. The

direction vector of the line is (-4, -2, -4).

To determine if the line is parallel or perpendicular to a given plane, we can take the dot product of the direction vector of the line and the normal vector of the plane. If the dot product is zero, the line is parallel to the plane. If the dot product is nonzero, the line is neither parallel nor perpendicular to the plane.

For the plane 6x + 3y + 6z = -39, the normal vector is (6, 3, 6). The dot product of the direction vector of the line and the normal vector of the plane is (-4)(6) + (-2)(3) + (-4)(6) = -42. Since the dot product is nonzero, the line is neither parallel nor perpendicular to the plane.

For the plane 8x + 24y - 20z = -8, the normal vector is (8, 24, -20). The dot product of the direction vector of the line and the normal vector of the plane is (-4)(8) + (-2)(24) + (-4)(-20) = 8. Since the dot product is nonzero, the line is neither parallel nor perpendicular to the plane.

For the plane 8x + 4y + 8z = 48, the normal vector is (8, 4, 8). The dot product of the direction vector of the line and the normal vector of the plane is (-4)(8) + (-2)(4) + (-4)(8) = -56. Since the dot product is nonzero, the line is neither parallel nor perpendicular to the plane.

For the plane 52 5y + 3z = -17, the normal vector is (0, 52, 3). The dot product of the direction vector of the line and the normal vector of the plane is (-4)(0) + (-2)(52) + (-4)(3) = -112. Since the dot product is nonzero, the line is neither parallel nor perpendicular to the plane.

Therefore, the line L(t) = (2 - 4t, 3 – 2t, -5 – 4t) is neither parallel nor perpendicular to any of the given planes.

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Answer:

Step-by-step explanation: I don’t know

Step-by-step explanation:

these 2 parallel lines create similar triangles.

that mags they're is one common scaling factor for the side lengths bergen the 2 triangles.

and that means they must have the same ratio.

the large triangle has the side lengths

24 + 9 = 33

32 + x

the smaller triangle has the corresponding side lengths

24

32

so, the equal ratios are

24/33 = 32/(32+x)

24(32 + x) = 32×33

32 + x = 32×33/24 = 4×11 = 44

x = 44 - 32 = 12

[tex](\stackrel{x_1}{4}~,~\stackrel{y_1}{-16})\qquad (\stackrel{x_2}{8}~,~\stackrel{y_2}{-12}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-12}-\stackrel{y1}{(-16)}}}{\underset{\textit{\large run}} {\underset{x_2}{8}-\underset{x_1}{4}}} \implies \cfrac{-12 +16}{4} \implies \cfrac{ 4 }{ 4 } \implies 1[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-16)}=\stackrel{m}{ 1}(x-\stackrel{x_1}{4}) \implies y +16 = 1 ( x -4) \\\\\\ y+16=x-4\implies y=x-20\impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}[/tex]

a) Yes, I am skeptical of my friend's claim. The mean score of the class is 80, which means that the average score of all the students is 80.

b) I am also skeptical of my friend's claim that Class 2022 did better than Class 2023 in the Final Test. While the mean score of Class 2022 is higher than the mean score of Class 2023, this does not necessarily mean that Class 2022 did better.

If 50% of the students had a score higher than 80, then the other 50% of the students would have to have a score lower than 80 in order for the mean to be 80. However, my friend's estimation error of 0.07 means that there is a possibility that the true percentage of students with a score higher than 80 is actually between 43% and 57%. Therefore, it is not certain that 50% of the students had a score higher than 80.

It is possible that the distribution of scores for Class 2022 is more spread out, with some students scoring very high and some students scoring very low, while the distribution of scores for Class 2023 is more consistent. Without knowing the standard deviation or the range of the scores for each class, it is not possible to accurately compare the performance of the two classes.

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The variance of X, the total number of students to whom problem 1 is assigned, is 12.5.

We have,

To solve this problem, we can use the concept of a binomial distribution.

The number of students to whom problem 1 is assigned can be modeled as a binomial random variable.

Let's define X as the random variable representing the number of students to whom problem 1 is assigned.

We know that each student has a 3/6 = 1/2 probability of being assigned problem 1.

In a class of 50 students, the probability of a single student being assigned problem 1 is p = 1/2.

The number of students to whom problem 1 is assigned follows a binomial distribution with parameters n = 50 (number of students) and p = 1/2 (probability of success).

The variance of a binomial distribution is given by the formula:

Var(X) = np (1 - p)

Substituting the values, we have:

Var(X) = 50 x (1/2) x (1 - 1/2)

= 50 x (1/2) x (1/2)

= 25 x 1/2

= 25/2

= 12.5

Therefore,

The variance of X, the total number of students to whom problem 1 is assigned, is 12.5.

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P(event) represents the probability of an event occurring.

Probability is a branch of mathematics that deals with the study of random events or experiments. It is the measure of the likelihood or chance that a particular event will occur. Probability is expressed as a number between 0 and 1, where 0 means that the event is impossible and 1 means that the event is certain to happen.

An event is a set of outcomes of a random experiment, and the probability of an event is the measure of how likely it is to occur.

Here are some examples:

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As a financial analyst, I should opt for an Initial cost of $125000 and after all the making and labor charges each bike is made under $225.

When a linear function is exposed to various constraints, it is maximized or reduced using the mathematical modeling technique known as linear programming. In corporate planning, industrial engineering, and other fields, this technique has proven helpful for guiding quantitative judgments.

Given, The Bici bicycle company is making a low-price ultra-light bicycle.

They have two plans,

I. Initial cost of $125000 and after all the making and labor charges

each bike is made under $225.

II. Initial cost of $100000 and after all the making and labor charges

each bike is made for under $275.

As they want bicycles to cost less we should opt for the first plan even if the initial cost is more, As the market demand is for a cycle that is low in price so it would sell more and an extra investment of $25000 won't be wasted.

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Answer:

Step-by-step explanation:

1. Solve for

x by simplifying both sides of the equation, then isolating the variable.

x= 3, −5

2. x by simplifying both sides of the equation, then isolating the variable.

x= −7, 1

In response to the aforementioned query, we may say that D is equal to equation sqrt((3 - 3)2 + (-8 - 2)2 = sqrt(0 + (-10)2) = sqrt(100) = 10. As a result, it takes 10 units to get from position X to point Y.

A mathematical equation links two statements and utilises the equals sign (=) to indicate equality. In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.

The distance formula may be used to determine the separation between two points:

Sqrt((x2 - x1) + (y2 - y1)) = sqrt(d)

where d is the distance between the two locations, and (x1, y1) and (x2, y2) are the coordinates of the two points.

We may get the separation between points X at (3, 2) and Y at (3, -8), using the following formula:

D is equal to sqrt((3 - 3)2 + (-8 - 2)2 = sqrt(0 + (-10)2) = sqrt(100) = 10.

As a result, it takes 10 units to get from position X to point Y.

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Answer:

(5,7)



x = 5

y = 7

The equation that quickly reveals the y-intercept is f(x) = 3x²+ 36x + 33 and the y- intercept is 33

The x-intercept is the point where a line crosses the x-axis, and the y-intercept is the point where a line crosses the y-axis.

This shows that the y-intercept is gotten when x is 0

Therefore amongst the equation above equation 1 is the equation that easily shows the y-intercept.

The equation f(x) = 3x²+ 36x + 33 is presented in a standard form of quadratic equation.

when x= 0

f(x) = 3x²+ 36x + 33

f(x) = 3(0)²+ 36(0) + 33

f(x) = 0+ 0+ 33

= 33

therefore the y-intercept is 33

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The equation of the sine function is: y = 4 sin(1/4 x).

What is a sine function?

A sine function is a type of trigonometric function that relates the angle of a right triangle to the ratio of the length of its opposite side to its hypotenuse. It is defined as the ratio of the length of the opposite side of an angle in a right triangle to the length of the hypotenuse.

The general form of a sine function is:

y = A sin(Bx + C) + D

where A is the amplitude, B determines the period (B = 2π/period), C is the phase shift, and D is the vertical shift.

Substituting the given values, we get:

A = 4 (amplitude)

Period = 8π

B = 2π/Period = 2π/(8π) = 1/4

C = 0 (no phase shift)

D = 0 (no vertical shift)

the equation of the sine function is:

y = 4 sin(1/4 x)

Note that the value of B is the reciprocal of the period, since the formula for period is T = 2π/B, which can be rearranged to get B = 2π/T.

Hence, the equation of the sine function is: y = 4 sin(1/4 x).

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