06.02 LC) What best describes the expression y/3?
Group of answer choices
A) 3 less than some number
B) 3 more than some number
C) Some number divided by 3
D) Some number times 3

The dot represents the vertex, and the x represents the point where the parabola touches the x-axis. The parabola is symmetric about the vertical line x = h and does not cross the x-axis anywhere else.

A parabola is a type of curve that is defined by a specific mathematical equation, namely, the quadratic equation. It is a symmetrical curve that can be described as the shape of the graph of a quadratic function.

by the question.

If (h, k) is a zero of the function, then. [tex]f(h) = 0[/tex]. Substituting this into the equation for f(x), we get:

[tex]0 = a(h - h)^2[/tex]

[tex]0 = 0[/tex]

This is a true statement, which tells us that (h, k) is indeed a zero of the function.

Now, let's consider the graph of this function. Since the coefficient a is non-zero, the parabola will be facing either upwards or downwards. If a > 0, then the parabola will be facing upwards, and if a < 0, then the parabola will be facing downwards.

Since the vertex of the parabola is at (h, k), the axis of symmetry is the vertical line x = h. Therefore, the parabola is symmetric about this line.

Finally, since (h, k) is also a zero of the function, the parabola must cross the x-axis at x = h with a single point of tangency. This means that the parabola just touches the x-axis at this point and does not cross it.

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The length QR of the right angle triangle is 26 cm.

A right triangle is a triangle that has one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.

Therefore, the side of a right angle triangle can be found using Pythagoras's theorem as follows:

Hence,

c² = a² + b²

where

Therefore,

24² + 10² = QR²

576 + 100 = QR²

QR² = 676

QR = √676

QR = 26 cm

Therefore,

length of QR = 26 cm

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According to the conditions, in right angled triangle, the value of x is 6.

What is angle ?

In mathematics, an angle is a geometric figure formed by two rays (or line segments) that share a common endpoint, known as the vertex of the angle.

In a right-angled triangle, the sum of the two acute angles is always 90 degrees. Therefore, we can use this fact to find the value of x.

We are given that one of the acute angles is 60 degrees, and the other angle is 7x - 12 degrees. So we can write:

60 + (7x - 12) = 90

Simplifying this equation, we get:

7x + 48 = 90

Subtracting 48 from both sides, we get:

7x = 42

Dividing both sides by 7, we get:

x = 6

Therefore, in right angled triangle, the value of x is 6.

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Answer: The given line has a slope of -1, since its equation is y = -x + 5. The line that is perpendicular to this line will have a slope that is the negative reciprocal of -1, which is 1. So, we know that the equation of the line we're looking for will have a slope of 1.

To find the equation of this line, we need to use the point-slope form of the equation of a line:

y - y1 = m(x - x1)

where m is the slope of the line, and (x1, y1) is a point on the line.

We know that the point (-2, 5) is on the line we're looking for, and we know that the slope of the line is 1. So we can substitute these values into the point-slope form:

y - 5 = 1(x - (-2))

Simplifying, we get:

y - 5 = x + 2

Adding 5 to both sides, we get:

y = x + 7

Therefore, the equation of the line that passes through the point (-2, 5) and is perpendicular to the line y = -x + 5 is y = x + 7.

Step-by-step explanation:

The distance between the two points (-6, 8) and (6, 3) is equal to 13 units.

Mathematically, the distance between two (2) points that are on a coordinate plane can be calculated by using this formula:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where:

x and y represents the data points (coordinates) on a cartesian coordinate.

Substituting the given points into the distance formula, we have the following;

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Distance = √[(6 - (-6))² + (3 - 8)²]

Distance = √[(12)² + (-5)²]

Distance = √(144 + 25)

Distance = √169 units.

Distance = 13 units.

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a. The inequality required is 2x-y≤3 b. We can graph the inequality of x+2y<4. c.  it is not common solution point.

An inequality is a mathematical statement that compares two quantities and indicates whether one is greater than, less than, or equal to the other. Inequalities use special symbols, such as "<" (less than), ">" (greater than), "≤" (less than or equal to), and "≥" (greater than or equal to), to represent the relationship between the two quantities.

For example, the inequality 2x + 3 < 7 means that the quantity 2x + 3 is less than 7. To solve this inequality, we can subtract 3 from both sides to get 2x < 4, and then divide both sides by 2 to get x < 2.

Inequalities can also involve variables, such as x or y, and they can be used to represent real-world situations, such as the amount of money in a bank account, the temperature of a room, or the speed of a car.

Inequalities are important in mathematics and other fields because they allow us to compare quantities and make decisions based on those comparisons. They are used in a variety of applications, including economics, physics, engineering, and statistics.

a. Here the inequality, represented by graph

Take two points lying in the line

(x1,y1)=(0,-3) and (x2,y2)= (2,1)

So,

equation of line is

y+3= [tex]\frac{1-(-3)}{2} (x-0)=[/tex] 2x-y-3=0

Thus, the inequality required is 2x-y≤3.

b. We can graph the inequality of x+2y<4

c. Here, we are not agree with Oscar
Reason:- Since (2,1) point lies on inequality 2x-y≤3 but it did not lie on inequality x+2y<4

So it is not common solution point.

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In response to the given question, we can state that we know that sum of all angles in a triangle is 180. m∠C = 4*11.67+43 = 89.68 = =90

A triangle is a polygon because it contains four or more parts. It features a simple rectangular shape. A triangle ABC is a rectangle with the edges A, B, and C. When the sides are not collinear, Euclidean geometry produces a single plane and cube. If a triangle contains three components and three angles, it is a polygon. The corners are the points where the three edges of a triangle meet. The sides of a triangle sum up to 180 degrees.

we know that sum of all angles in a triangle is 180.

2x - 12 + 4x + 43 + 9x - 26 = 180

15x  + 5 = 180

15x = 175

x = 11.67

m∠C = 4*11.67+43 = 89.68 = =90

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Answer:

m∠C = 119°

Step-by-step explanation:

According to the Exterior Angle Theorem, the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles of the triangle.

From inspection of the given triangle, the exterior angle is (9x - 26)° and the two non-adjacent interior angles are ∠B and ∠C.

Equate the sum of the two non-adjacent angles to the exterior angle and solve for x:

⇒ (2x - 12)° + (4x + 43)° = (9x - 26)°

⇒ 2x - 12 + 4x + 43 = 9x - 26

⇒ 6x + 31 = 9x - 26

⇒ 57 = 3x

⇒ x = 19

To calculate the measure of angle C, substitute the found value of x into the expression for the angle:

⇒ m∠C = (4x + 43)°

⇒ m∠C = (4(19) + 43)°

⇒ m∠C = (76 + 43)°

⇒ m∠C = 119°

The value of p is 2 and 4.

What is a quadratic equation?

Any equation that can be written in the standard form where x is an unknown value, a, b, and c are known quantities, and a 0 is a quadratic equation. Any equation containing one term in which the unknown is squared and no term in which it is raised to a higher power.

Here, we have

Given: p² - 6p + 8

we have to solve the quadratic formula or complete the square.

= p² - 6p + 8

= p² -4p - 2p + 8

= p(p-4) -2(p-4)

= (p-4)(p-2)

(p-4)(p-2) = 0

p-4 = 0,

p-2 = 0

p = 4, 2

Hence, the value of p is 2 and 4,

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After answering the presented question, we can conclude that As a cylinder result, the depth of the water in the tank rises by 6.33 cm.

A cylinder is a three-dimensional geometric shape made up of two parallel congruent circular bases and a curving surface connecting the two bases. The bases of a cylinder are always perpendicular to its axis, which is an imaginary straight line passing through the centre of both bases. The volume of a cylinder is equal to the product of its base area and height. A cylinder's volume is computed as V = r2h, where "V" represents the volume, "r" represents the radius of the base, and "h" represents the height of the cylinder.

This cylinder has the following volume:

V_cylinder = π × r² × h

= π × (40 cm)² × (50 cm)

= 251,327.41 cm³

Hence the volume of water displaced by the cube is 125,000 cm, and the volume of water displaced by the cylinder with the same height and radius as the tank is 251,327.41 cm3. As a result, the depth of the water rises by:

Δh = V_cube / (π × r²) - V_cylinder / (π × r²)

= (125,000 cm³) / (π × (40 cm)²) - (251,327.41 cm³) / (π × (40 cm)²)

= 6.33 cm

As a result, the depth of the water in the tank rises by 6.33 cm.

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Therefore, the surface area of the cylinder is approximately 301m².

In mathematics, area refers to the measurement of the size of a two-dimensional surface or region. It is a measure of the amount of space inside a flat figure, such as a square, circle, or triangle. The area of a figure is usually expressed in square units, such as square centimeters, square meters, or square inches.

Here,

The net of the right circular cylinder consists of three rectangles: one for the lateral surface, and two for the top and bottom faces.

The lateral surface of a cylinder can be found using the formula:

Lateral surface area = 2πrh

where π is the value of pi (approximately 3.14), r is the radius of the base, and h is the height of the cylinder. In this case, the height of the cylinder is given as 8m, and the radius is given as 4m (half of the width of one of the rectangles). Therefore, we have:

Lateral surface area = 2πrh = 2(3.14)(4m)(8m) ≈ 201m²

The area of each of the circular faces is given by the formula:

Circular face area = πr²

Since the radius is 4m, we have:

Circular face area = πr² = 3.14(4m)² ≈ 50m²

The total surface area of the cylinder is the sum of the lateral surface area and the areas of the two circular faces:

Total surface area = Lateral surface area + 2 × Circular face area

Total surface area = 201m² + 2(50m²) = 301m² (rounded to the nearest whole number)

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Answer:%40.50

Step-by-step explanation: i tired

The OLS assumption that is violated when predicted errors have a kurtosis of 5 is normality. The correct option is (d). Kurtosis is a statistical measure of the peak of a probability distribution curve. It measures how the tails of the distribution compare to a normal distribution.

Oridinary Least Squares (OLS) is a regression technique that assumes that the response variable has a linear relationship with the explanatory variable(s) and that the response variable has normal distribution error terms. However, in some cases, such as when the predicted errors have a kurtosis of 5, this assumption of normality is violated. If the distribution has more of its observations in the tails than a normal distribution, it is said to be leptokurtic. If it has fewer of its observations in the tails than a normal distribution, it is said to be platykurtic.

Kurtosis of 5 means that the distribution is leptokurtic and has fatter tails than the normal distribution.Assuming normality of the errors means that the residuals or errors are normally distributed. If the errors are not normally distributed, then the residuals will not be normally distributed either. This will affect the accuracy of the confidence intervals and hypothesis tests. The coefficient estimates may be biased and the confidence intervals may be too wide or too narrow. Therefore, normality of the errors is an important assumption of OLS regression and in this case it has been violated.

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a) Sample space = {36}

b) Probability of the sum of the dice is a 7 = P(E) = 6/36 = 1/6

c) Probability of the sum of the dice is at least a 7 = P(F) = 21/36 = 7/12.

We need to find, What is the size of the sample space? Probability of the sum of the dice is a 7 ?Probability of the sum of the dice is at least a 7? Solution a)Sample space is defined as the set of all possible outcomes. Suppose that you roll two 6 sided dice. So, The possible outcomes of each die is {1,2,3,4,5,6}.The total outcomes for rolling two dice are {1,1}, {1,2}, {1,3}, {1,4}, {1,5}, {1,6}, {2,1}, {2,2}, {2,3}, {2,4}, {2,5}, {2,6}, {3,1}, {3,2}, {3,3}, {3,4}, {3,5}, {3,6}, {4,1}, {4,2}, {4,3}, {4,4}, {4,5}, {4,6}, {5,1}, {5,2}, {5,3}, {5,4}, {5,5}, {5,6}, {6,1}, {6,2}, {6,3}, {6,4}, {6,5}, and {6,6}.Therefore, Sample space = {36}.b)Let E be the event that the sum of the dice is a 7.The events where the sum of the dice is a 7 are {1,6}, {2,5}, {3,4}, {4,3}, {5,2}, and {6,1}.The number of events where the sum of the dice is a 7 is 6.Therefore, Probability of the sum of the dice is a 7 = P(E) = 6/36 = 1/6.c)Let F be the event that the sum of the dice is at least a 7.Therefore, F = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1), (2,6), (3,5), (4,4), (5,3), (6,2), (3,6), (4,5), (5,4), (6,3), (4,6), (5,5), (6,4), (5,6), and (6,5)}The number of events where the sum of the dice is at least a 7 is 21.Therefore, Probability of the sum of the dice is at least a 7 = P(F) = 21/36 = 7/12.a) Sample space = {36}.b) Probability of the sum of the dice is a 7 = P(E) = 6/36 = 1/6.c) Probability of the sum of the dice is at least a 7 = P(F) = 21/36 = 7/12.

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The distance from Georgetown to Brett's house, measured in a straight line, is approximately 64.75 miles, which has been calculated through Pythagorean Theorem.

You may determine the right angled triangle's missing length using the Pythagorean Theorem. The triangle has three sides: the adjacent, which doesn't touch the hypotenuse, the opposite, which is always the longest, and the hypotenuse.

We can solve this question through the Pythagorean theorem. Let's assume that Brett's house is at point B, Springfield is at point S, and Georgetown is at point G  

We want to find the length of the line segment BG, which is the distance from Brett's house to Georgetown.

We can say that the length of the line segment BS is x miles (we don't know the value of x yet), and the length of the line segment SG is 17 miles. We can also say that the line segments BS and SG are perpendicular to each other, since Brett's house is due west of Springfield and due south of Georgetown.

Using the Pythagorean theorem, we can write:

[tex]BG^2 = BS^2 + SG^2[/tex]

Substituting the known values, we get:

[tex]BG^2 = x^2 + 17^2[/tex]

Simplifying and solving for BG, we get:

[tex]BG = sqrt(x^2 + 17^2)[/tex]

We also know that the line segments BS and SG form a right triangle, so we can use the Pythagorean theorem again to write:

[tex]x^2 + BG^2 = (17 + BG)^2[/tex]

Expanding and simplifying, we get:

[tex]x^2 + BG^2 = 289 + 34BG + BG^2[/tex]

Substituting BG^2 with its value from the first equation, we get:

[tex]x^2 + x^2 + 17^2 = 289 + 34BG + x^2 + 17^2[/tex]

Simplifying, we get:

[tex]2x^2 = 289 + 34BG[/tex]

Substituting BG with its value from the first equation, we get:

[tex]2x^2 = 289 + 34sqrt(x^2 + 17^2)[/tex]

Simplifying and solving for x, we get:

[tex]x = sqrt((289/2)^2 - 17^2) = sqrt(4196.25) = 64.75[/tex]

Therefore, the distance from Georgetown to Brett's house, measured in a straight line, is approximately 64.75 miles.

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Answer:

1 game, for $5.75

Step-by-step explanation:

To find this answer, all you need to do is create an equation for both Rock and Bowl and the Super Bowling and then set them equal to each other to find out and solve which value of x games makes them equal.

The first equation is 2.75x +3

The second one is 2.25x + 3.5

2.75x + 3 = 2.25x + 3.5

2.75x = 2.25x + .5

From here, you can either solve for x or realize that x being equal to 1 makes both values equal.

So, the answer is 1 game now to plug it into one of our equations, we get

2.75(1) + 3 which we know is equal to the other equation too which is $5.75 dollars

Answer: i believe its 5

Step-by-step explanation:

The first three terms of the sequence Tₙ = n^2 - 2n - 6 are -7, -8, and 0, and the sequence is a quadratic sequence with a parabolic graph that opens upward.

To find the first three terms of the sequence Tₙ = n^2 - 2n - 6, we simply need to substitute the first three positive integers for n, which gives us:

T₁ = 1^2 - 2(1) - 6 = -7

T₂ = 2^2 - 2(2) - 6 = -8

T₃ = 3^2 - 2(3) - 6 = 0

Therefore, the first three terms of the sequence are -7, -8, and 0.

The sequence Tₙ is a quadratic sequence, which means that it has a second-order difference. In other words, the differences between the terms of the sequence form a linear sequence.

Specifically, the first differences are 2, 4, 6, 8, and so on, which form an arithmetic sequence with a common difference of 2. The second differences are all equal to 2, which confirms that the sequence is quadratic.

The graph of the sequence Tₙ is a parabola that opens upward, with a vertex at (1, -7). This means that the sequence starts with a negative term, then decreases until it reaches a minimum at n = 1, and then increases indefinitely.

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(A) The interquartile range for numbers of employees at Mall A is 65 and at Mall B is 7.5. (B) Mall B would we expect to have greater variability in regard to numbers of employees.

Part A: To determine the interquartile range (IQR) for numbers of employees at each mall, we first need to find the median for each mall. The median is the middle value of a set of data when arranged in order.

For Mall A:

Arrange the numbers in ascending order: 18, 20, 21, 22, 25, 26, 27, 28, 28, 29

The median is the average of the two middle numbers, which are 25 and 26.

Median = (25 + 26) / 2

             = 25.5

Next, we need to find the first quartile (Q1) and the third quartile (Q3). The first quartile is the median of the lower half of the data, and the third quartile is the median of the upper half of the data.

For Mall A:

Lower half: 18, 20, 21, 22, 25

Upper half: 26, 27, 28, 28, 29

Q₁ = median of the lower half

    = (21 + 22) / 2

    = 21.5

Q₃ = median of the upper half

     = (28 + 28) / 2

     = 28

The interquartile range for Mall A is:

IQR = Q3 - Q1

      = 28 - 21.5

      = 6.5

For Mall B:

Arrange the numbers in ascending order: 19, 21, 23, 24, 25, 26, 27, 29, 30, 33

The median is the average of the two middle numbers, which are 25 and 26.

Median = (25 + 26) / 2

             = 25.5

Lower half: 19, 21, 23, 24, 25

Upper half: 26, 27, 29, 30, 33

Q₁ = median of the lower half

    = (21 + 23) / 2

    = 22

Q3 = median of the upper half

     = (29 + 30) / 2

     = 29.5

The interquartile range for Mall B is:

IQR = Q3 - Q1

      = 29.5 - 22

      = 7.5

Part B: To determine which mall would have greater variability in regard to numbers of employees, we can compare the interquartile ranges for each mall. The interquartile range measures the spread of the middle 50% of the data. The larger the interquartile range, the greater the variability.

In this case, Mall B has a larger interquartile range than Mall A (7.5 vs. 6.5). This suggests that there is greater variability in the number of employees at Mall B.

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The angle R is  53.1° by the use of the cosine which is one of the trigonometric ratios that we have.

In trigonometry, the three most important trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions relate the angles of a triangle to the ratios of its sides. For example, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse (the longest side of the right-angled triangle).

Cos R = 3/5

R = Cos-1(3/5)

R = 53.1°

Hence, we can see that the required angle R is obtained as  53.1°

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Step-by-step explanation:

*change density to kg per cubic meter

*change volume to cubic meter

*rearrange the formula to solve for mass

The area of the deck is 816 square feet.

We need to subtract the area of the pool from the area of the deck plus pool combination.

The dimensions of the deck plus pool combination can be found by adding twice the deck width to the pool dimensions.

The length of the deck plus pool combination is:

40ft + 2(6ft) = 52ft

The width of the deck plus pool combination is:

16ft + 2(6ft) = 28ft

Therefore, the area of the deck plus pool combination is:

52ft * 28ft = 1456ft²

The area of the pool is:

16ft * 40ft = 640ft²

So the area of the deck is:

1456ft² - 640ft² = 816ft²

Therefore, the area of the deck is 816 square feet.

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To the nearest foot, the car travels approximately 21 feet over an angle of π/3 radians.

In this problem, we are given an angle of π/3 radians. To find out how far the car travels, we need to calculate the length of the arc that the car travels along. The formula for the length of an arc is given by:

arc length = radius x angle in radians

So, in this case, the arc length that the car travels is:

arc length = 20 x π/3

arc length ≈ 20 x 1.05

arc length ≈ 21

Therefore, the solution of arc length is 21.

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According to the given conditions of volume,[tex]$V_{cone} = \frac{1}{3} \pi \frac{120}{\pi h} h = \frac{1}{3} \cdot 120 = 40}$[/tex] cubic inches.

Volume is the amount of space that a three-dimensional object occupies or contains. It is a measure of the total amount of enclosed space inside a solid figure, such as a cube, cylinder, sphere, or any other three-dimensional shape.

According to given information :

Since the cylinder and cone have the same height and congruent circular bases, their volumes are proportional to the squares of their radii.

Let the radius of the base of the cylinder and cone be denoted as r.

The volume of a cylinder is given by:

[tex]$V_{cylinder} = \pi r^2 h$[/tex]

where h is the height of the cylinder.

We are given that the volume of the cylinder is 120 in, so we can plug this into the formula and solve for r:

[tex]$120 = \pi r^2 h$[/tex][tex]$r^2 = \frac{120}{\pi h}$[/tex]

The volume of a cone is given by:

[tex]$V_{cone} = \frac{1}{3} \pi r^2 h$[/tex]

We know that the cone and cylinder have the same height, so we can substitute the expression we found for [tex]$r^2$[/tex] into the formula for the volume of the cone:

According to the given conditions,

[tex]$V_{cone} = \frac{1}{3} \pi \frac{120}{\pi h} h = \frac{1}{3} \cdot 120 = 40}$[/tex] cubic inches.

Therefore, the volume of the cone is 40 cubic inches.

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The formula that describes a way to produce each term from the previous term in the given geometric sequence is: aₙ = ¹/₂(2)ⁿ⁻¹

In mathematics, a geometric sequence, is defined as a sequence that consists of non-zero numbers whereby each of the terms after the first is found by multiplying the previous one by a fixed, non-zero number that is referred to as the common ratio.

The formula that is usually utilized in finding the nth term of a geometric sequence is expressed as:

aₙ = arⁿ⁻¹

where:

a is first term

r is common ratio

Thus:

a = 1/2

r = 2/1 = 2

Thus:

aₙ = ¹/₂(2)ⁿ⁻¹

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Answer:

Theoretical probability for each color is 1/4, or 25%.

Experimental probability of blue is 42/200, or 21%.

Experimental probability of purple is 55/200, or 27.5%.

Experimental probability of green is 71/200, or 35.5%.

Experimental probability of red is 32/200, or 16%.

Correct statements:

Experimental property of purple (27.5%) is more than theoretical property of blue (25%).

Theoretical property of blue (25%) is more than experimental property of red (16%).

Answer:

Step-by-step explanation:

Answer:



Step-by-step explanation:

Given: The table of linear function.

x :  -3  , 0 , 3,  6

y :  -6 , -2 , 2,  6

Slope=change in y over change in x



Passing point: (0,-2)

Point slope form:







Slope of the linear function

Linear function is

Hence, The slope of linear function is

The slope of the given linear function which graph represents the data is 3/2.

The slope of a linear function can be calculated by finding the difference between two points on the graph and dividing it by the difference of the corresponding x-values of those points.

In this case, the two points are (2, -2) and (6, 4).

The difference between these y-values = 6,

and the difference between the x-values = 4.

Therefore, the slope of the linear function which graph represents the data = 6/4

= 3/2

This means that the linear function has a slope of 3/2 which summarizes that for every two units that x increases, y increases by three units.

If x increases from 0 to 4, y increases from -5 to 1, which is a difference of 6 units (4 x 3/2 = 6).

The linear function can be written as y = 3/2x -5.

This means that for any given x-value, the corresponding y-value can be calculated by multiplying 3/2 by the x-value and subtracting 5.

If x = 2, the y-value is -2,that is

(2* 3/2)- 5= -2

Therefore, the slope of the linear function which graph represents the data is 3/2.

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If $40.90 divided by 66, the rounded answer to the nearest set of 40 is given as $0.

To round the answer of $40.90 divided by 66 to the nearest set of 40, we need to perform the division and then round the quotient to the nearest multiple of 40.

First, let's perform the division:

$40.90 / 66 = 0.6206...

The quotient is a decimal, but we need to round it to the nearest multiple of 40. To do this, we need to find out how close the quotient is to each of the multiples of 40 and then round to the nearest one.

The nearest multiples of 40 are 0, 40, 80, 120, etc.

To determine how close the quotient is to each of these multiples, we can subtract the quotient from each multiple and take the absolute value of the result:

|0 - 0.6206| = 0.6206

|40 - 0.6206| = 39.3794

|80 - 0.6206| = 79.3794

The smallest absolute difference is between the quotient and 0, so we round down to 0.

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[tex]$\cos 6x$[/tex] can be expressed in terms of the first power of cosine as [tex]$2(\cos^2 x)^3 - 1$[/tex].

Structure-based algebraic expressiοn rewriting is the same as rearranging an expressiοn tο plug it intο anοther expressiοn. Sοlve fοr οne οf the variables in these kinds οf prοblems and then insert the resulting expressiοn fοr that variable intο the οther expressiοn.

We may rewrite [tex]$cos 6x$[/tex]  in terms of [tex]$cos 2 x$[/tex] and [tex]$cos 4 x$[/tex] using the fοrmula for lowering powers as fοllows:

[tex]\begin{aligned}\cos 6x = \cos^2 3x - \sin^2 3x \\= \cos^2 3x - (1 - \cos^2 3x) \\= 2\cos^2 3x - 1 \\= 2(\cos^2 x)^3 - 1. \end{aligned}[/tex]

Hence, [tex]$\cos 6x$[/tex]can be written in terms οf the first pοwer of cosine as [tex]$2(\cos^2 x)^3 - 1$[/tex].

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