When x increases from a toa 2, y increases by a difference of 8. For which function is this statement true?

Answer:

Standard form: f(x) = -x^2 + 6x + 16

Step-by-step explanation:

Identifying the function:

General equation of a quadratic's standard form:

The general equation for the standard form of a quadratic function is given by:

f(x) = ax^2 + bx + c

Method to find the standard form of the equation of f(x):

We can find either the vertex form or the intercept form, whose general equations are given by:

Vertex form:  f(x) = a(x - h)^2 + k, where

Intercept form: f(x) = a(x - p)(x - q), where

Connection between the vertex and intercepts forms and the standard form:

Identifying the vertex:

Finding the vertex form:

Thus, we have part of the vertex form:  f(x) = a(x - 3)^2 + 25.

Thus, we can plug in (-2, 0) for (x, f(x)) in the vertex form to find a:

0 = a(-2 - 3)^2 + 25

0 = a(-5)^2 + 25

0 = a * 25 + 25

(0 = 25a + 25) - 25

(-25 = 25a) / 25

-1 = a

Thus, -1 is a in the vertex form.

Therefore, the full vertex form is given by:

f(x) = -(x - 3)^2 + 25

Convert from vertex form the standard form:

Now we can convert to standard for by expanding:

f(x) = -(x - 3)(x - 3) + 25

f(x) =-(x^2 - 3x - 3x + 9) + 25

f(x) = -x^2 + 3x + 3x - 9 + 25

f(x) = -x^2 + 6x + 16

Thus, the standard form of the equation of f(x) is f(x) = -x^2 + 6x + 16

The accurate statement about the graph is: Point D is a y-intercept of the graph.

Based on the given graph of y = -x + 1, the following statements about the graph are accurate:

Point A is a relative minimum of the graph:

False.

Point A is located at (-2, 3), and it is not the lowest point on the graph. Moving to the left or right of point A, the graph continues to decrease.

Point B is a turning point of the graph:

False.

Point B is not a turning point.

It is located at (-1, 2), and the graph does not change its direction or slope at this point.

Point C is a maximum of the graph:

False.

Point C is located at (2, -2), and it is not the highest point on the graph. Moving to the left or right of point C, the graph continues to increase.

Point D is a y-intercept of the graph:

True.

Point D is located at (0, 1), and it lies on the y-axis.

This point represents the y-intercept of the graph, where the value of x is zero.

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There were 14 players on the soccer team.

In the given problem, the members of a soccer team raised $1210.50 to go to a tournament. They rented a bus for $769.50 and budgeted $31.50 per player for meals. We need to find how many players were on the team.To solve this problem, we can start by subtracting the cost of the bus rental from the total amount raised by the team:$1210.50 - $769.50 = $441Now, we can divide this remaining amount by the amount budgeted per player for meals:$441 ÷ $31.50 = 14Therefore, there were 14 players on the soccer team. We can check our answer by multiplying the number of players by the amount budgeted per player for meals:14 x $31.50 = $441This is equal to the remaining amount after the bus rental was paid, which confirms that our answer is correct.

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Answer:

9000

Step-by-step explanation:

The relative change from Ohio to Indiana in terms of new mathematics teachers is approximately 49.08%. This indicates that Indiana has experienced a significant increase compared to Ohio in the number of new mathematics teachers.

To calculate the relative change from Ohio to Indiana in terms of new mathematics teachers, we need to determine the percentage increase or decrease in the number of teachers between the two states.

The relative change can be calculated using the formula:

Relative Change = (New Value - Old Value) / Old Value * 100

Let's apply this formula to the given data:

Ohio: Old Value = 4392

Indiana: New Value = 6546

Relative Change = (6546 - 4392) / 4392 * 100

Simplifying the equation:

Relative Change = 2154 / 4392 * 100

Relative Change ≈ 0.4908 * 100

Relative Change ≈ 49.08%

The relative change from Ohio to Indiana in terms of new mathematics teachers is approximately 49.08%. This indicates that Indiana has experienced a significant increase compared to Ohio in the number of new mathematics teachers.

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Answer:

Combining the like terms in the expression 9 + a - 5b - 2, we can group the constants together and the variables together:

(9 - 2) + a + (-5b) = 7 + a - 5b

Therefore, the equivalent expression after combining the like terms is 7 + a - 5b.

Step-by-step explanation:

Answer:

0.956

Step-by-step explanation:

The smallest value among the given numbers is 0.956.

Answer:

0.956

Step-by-step explanation:

To make this problem more clear, I will take every number and find what two numbers it's between.

3.11

Between 3 and 4

0.956

Between 0 and 1

4.21

Between 4 and 5

10.1

Between 10 and 11

Since 0.956 is between the smallest numbers, then it too must be the smallest of the four numbers.

If this answer helped you, please leave a thanks!

Have a GREAT day!!!

The values of a and b are a = 6 and b = 2. The values of c and d are c = 576 and d = -384, after solving the square-rooted equations.

Given the equation [tex]\(\sqrt{6 - 4\sqrt{2}} = \sqrt{a} - \sqrt{b}\)[/tex], let's find the values of a and b:

1. Square both sides of the equation to eliminate the square root:

[tex]\[6 - 4\sqrt{2} = a - 2\sqrt{ab} + b\][/tex]

2. Equate the rational and irrational parts on both sides:

[tex]\[6 = a + b\]\\-4\sqrt{2} = -2\sqrt{ab}\][/tex]

3. Square both sides of the second equation to solve for ab:

[tex]\[16 \cdot 2 = 4ab\]\[ab = 8\][/tex]

Thus, the values of a and b are a = 6 and b = 2.

To find the values of c and d in the second equation [tex]\(\sqrt{23} - 6\sqrt{6 - 4\sqrt{2}} = \sqrt{c} + \sqrt{d}\)[/tex], we can follow a similar process:

1. Square both sides of the equation:

[tex]\[23 - 12\sqrt{6 - 4\sqrt{2}} + 36(6 - 4\sqrt{2}) = c + d + 2\sqrt{cd}\][/tex]

2. Simplify and equate the rational and irrational parts:

[tex]\[23 + 216 - 144\sqrt{2} - 72\sqrt{12} = c + d\]\[-12\sqrt{6 - 4\sqrt{2}} = 2\sqrt{cd}\][/tex]

3. Square both sides and solve for cd:

[tex]\[144(6 - 4\sqrt{2}) = 4cd\]\[576 - 384\sqrt{2} = cd\][/tex]

Thus, the values of c and d are c = 576 and d = -384.

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The difference in height between the two trees is 29 inches.

To find the difference in height between the two trees, we need to subtract the height of one tree from the height of the other tree.

Let's convert the measurements to a consistent unit, such as inches, for easier calculation.

The first tree is 12 feet 8 inches tall. In inches, this is calculated as:

12 feet [tex]\times[/tex] 12 inches/foot + 8 inches.

= 144 inches + 8 inches

= 152 inches.

The second tree is 15 feet 1 inch tall.

In inches, this is calculated as:

15 feet [tex]\times[/tex] 12 inches/foot + 1 inch = 180 inches + 1 inch = 181 inches.

Now we can find the difference in height by subtracting the height of the first tree from the height of the second tree:

181 inches - 152 inches = 29 inches.

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Answer:

[tex]y=2x-3[/tex]

Step-by-step explanation:

Find the equation of the line that passes through the points A and B.

[tex]A=(4,5)\\\\B=(2,1)\\\\\\\hrule[/tex]

To find the equation of a line given two points, you can use the point-slope form of a linear equation, which is given by:

[tex]y-y_1=m(x-x_1)[/tex]

Where...

To find the equation, follow these steps:

(1) - Determine the coordinates of the two points on the line. Let's call them (x₁, y₁) and (x₂, y₂).

(2) - Calculate the slope, "m," using the formula:

[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

This calculates the change in y divided by the change in x between the two points.

(3) - Substitute the slope, "m," and the coordinates of one of the points (x₁, y₁) into the point-slope form equation.

(4) - Rearrange the equation to get it into slope-intercept form. Which is written as:

[tex]y=mx+b[/tex]

Where...

[tex]\hrulefill[/tex]

Step (1) -

[tex](x_1,y_1) \rightarrow (4,5)\\\\(x_2,y_2) \rightarrow (2,1)[/tex]

Step (2) -

[tex]m=\dfrac{y_2-y_1}{x_2-x_1}\\\\\\\\\Longrightarrow m=\dfrac{1-5}{2-4} \\\\\\\\\Longrightarrow m=\dfrac{-4}{-2} \\\\\\\\\therefore \boxed{m=2}[/tex]

Step (3) -

[tex]y-y_1=m(x-x_1); \ (x_1,y_1) \rightarrow (4,5) \ and \ m=2\\\\\\\\\Longrightarrow \boxed{y-5=2(x-4)}[/tex]

Step (4) -

[tex]y-5=2(x-4)\\\\\\\\\Longrightarrow y-5=2x-8 \ \text{(Applied the distributive property)}\\\\\\\\\Longrightarrow y-5+5=2x-8+5\\\\\\\\\therefore \boxed{\boxed{y=2x-3}}[/tex]

Thus, the problem is solved.

Hello !

Answer:

[tex]\Large\boxed{\sf y=2x-3}[/tex]

Step-by-step explanation:

The slope-intercept form of a line is of the form [tex]\sf y=mx+b[/tex] where m is the slope and b is the y-intercept.

The line passes through two points :

- A(4,5)

- B(2,1)

Let's replace x and y with their values in the equation :

[tex]\begin{cases}\sf 5=4m+b \\\sf 1=2m+b\end{cases}[/tex]

We get a system of two linear equations to solve.

Let's subtract the second line from the first one and solve for m :

[tex]\sf 5-1=4m+b-2m-b\\\iff 4=2m\\\iff m=\frac{4}{2}\\\iff \boxed{\sf m=2}[/tex]

Now let's substitute 2 for m in the second equation :

[tex]\sf 1=2\times2+b\\b=1-4\\\boxed{\sf b=-3}[/tex]

The slope-intercept form of the line is :

\boxed{\sf y=2x-3}

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Have a nice day

Of course! To solve this problem, we can use a system of linear equations. I'll guide you step-by-step.

Let's denote the amount of Kenyan French Roast coffee used in the blend by the variable K (in pounds), and the amount of Sumatran coffee by the variable S (also in pounds).

We know that the total weight of the blend is 14 pounds. This information gives us our first equation:

K + S = 14   [Equation 1] (This equation tells us that the sum of the amounts of Kenyan and Sumatran coffee must equal 14 pounds.)

Next, let's think about the cost. We're told that Kenyan French Roast coffee costs $0.700 per pound and Sumatran coffee costs $9.00 per pound. We also know that the blend sells for $8.00 per pound. We can use this information to write another equation based on the total cost of the coffee in the blend:

0.700K + 9.00S = 8.00(14) [Equation 2] (This equation represents the total cost of the coffee in the blend.)

Now, we can solve this system of linear equations. Let's manipulate Equation 1 to solve for one of the variables, say K. From Equation 1, we can write:

K = 14 - S   [Equation 3] (We simply subtracted S from both sides of Equation 1.)

Now, let's substitute this expression for K into Equation 2:

0.700(14 - S) + 9.00S = 8.00(14) (We replaced K with 14 - S in Equation 2.)

Now, distribute:

9.8 - 0.700S + 9.00S = 112 (Multiply 0.700 through the parentheses.)

Combine like terms:

9.8 + 8.30S = 112 (Combine the S terms.)

Subtract 9.8 from both sides:

8.30S = 102.2

Now, divide by 8.30 to solve for S:

S ≈ 12.31

Now that we have the approximate value of S, let's plug this back into Equation 3 to find K:

K = 14 - 12.31

K ≈ 1.69

Therefore, to make a 14-pound blend that sells for $8.00 per pound, you should use approximately 1.69 pounds of Kenyan French Roast coffee and approximately 12.31 pounds of Sumatran coffee.

Answer:

There are 20 presents in total, and 3 of them contain gems.

The probability of randomly selecting a present that contains a gem is 3/20 = 0.15 or 15%.

Answer:

y=12.5x+25

Step-by-step explanation:

y intercept where x =0 is 25

gradient between two points: (0,25) , (8,125)

(y2-y1)/(x2-x1) = 100/8 = 12.5

The distributive property and subtraction were used to prove the equivalence of the expressions.After applying the distributive property and combining like terms, we end up with the expression 6x - 2, which is equivalent to the original expression 5x + x - 2.

To prove that the expressions 5x + x - 2 and 6x - 2 are equivalent, the distributive property and subtraction were used.

Let's break down the steps:Starting with the expression 5x + x - 2, we can combine the like terms (5x and x) to simplify it as 6x.

Then, we have 6x - 2.

To further simplify the expression, we can apply the distributive property:

We can rewrite 6x as (5 + 1)x, using the fact that 5 + 1 equals 6.Applying the distributive property, we distribute (5 + 1) to both terms inside the parentheses: (5 + 1)x - 2.

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Answer:150?

Step-by-step explanation:

Answer:

Step-by-step explanation:

Letter B is the correct answer

Initial population is 200, tripled every hour

200 x 3 = 600

First hour 200

Second hour = 600 + 200 = 800

Third hour 800x3 = 2400 + 800 = 3200

and so on.

The sample variance is approximately 44.79 and the sample standard deviation is approximately 6.692.

To find the sample variance and standard deviation, follow these steps:

1. Calculate the mean of the data set:

  Mean (μ) = (22 + 11 + 4 + 9 + 10) / 5 = 56 / 5 = 11.2

2. Calculate the difference between each data point and the mean, and square each difference:

  [tex](22 - 11.2)^2 = 121.00 \\(11 - 11.2)^2 = 0.04 \\(4 - 11.2)^2 = 51.84 \\(9 - 11.2)^2 = 4.84 \\(10 - 11.2)^2 = 1.44[/tex]

3. Calculate the sum of all the squared differences:

  121.00 + 0.04 + 51.84 + 4.84 + 1.44 = 179.16

4. Calculate the sample variance:

  Variance [tex](s^2)[/tex] = sum of squared differences / (n - 1)

                 = 179.16 / (5 - 1)

                 = 179.16 / 4

                 = 44.79

5. Calculate the sample standard deviation:

  Standard Deviation (s) = square root of the variance

                         = √44.79

                         ≈ 6.692

Therefore, the sample standard deviation is about 6.692, while the sample variance is about 44.79.

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The slope of the line passing through points: (2,4) and (1,3) is 1

By definition, the slope or gradient of a line describes its steepness, incline, or grade. To calculate the slope of the line passing through the points we use the below formula,

[tex]m=\frac{Y_{2}-Y_{1} }{X_{2} -X_{1} }[/tex]

[tex]Here ~Y_{2} = 3, Y_{1} =4 , X_{2}=1 ~ and~ X_{1} =2.[/tex]

[tex]Hence, m=\frac{3-4}{2-1} =\frac{-1}{-1} \\~~~~~~~~~~~ m= 1.[/tex]

In mathematics, a slope is a number that quantifies the steepness and direction of a line, or a portion of a line connecting two points and is commonly symbolized by m. In general, the absolute magnitude of a line's slope, m, is used to determine its steepness. The steeper the line, the greater the value. Based on the sign and value of m, it is feasible to establish the direction of the line that m describes.

The slope is defined as the change in height divided by the change in horizontal distance, and it is also known as "rise over run." It has applications in both geography and civil engineering, such as road construction. In the case of a road, the "rise" is the change in height, while the "run" is the difference in distance between two fixed places, as long as the distance for measurement is not so great that the earth's curvature is taken into account.

Hence slope of the line passing through points: (2,4) and (1,3) is 1

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Answer:

slope = 1

Step-by-step explanation:

slope = change in x /change in y

slope = 3 - 4 / 1 - 2

slope = -1 / -1

slope = 1

So slope = 1

Step-by-step explanation: We may determine the range of potential further miles Charlie will run by deducting his present mileage from the maximum mileage if he has already run 17 miles and will only cover 35 miles this week.

Maximum mileage - current mileage = potential extra miles.

17 miles minus 35 miles equals 18 miles.

As a result, Charlie may run somewhere between 0 and 18 additional miles.

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Answer:

18 possible numbers

Step-by-step explanation:

If Charlie has already run 17 miles and he will run at most 35 miles this week.

We can calculate the possible numbers of additional miles he will run by subtracting the distance he has already run from the maximum distance he can run.

Maximum distance Charlie can run = 35 miles

Distance Charlie has already run = 17 miles

Possible additional miles he will run = Maximum distance - Distance already run

                              = 35 miles - 17 miles

                              = 18 miles

Therefore, there are 18 possible numbers of additional miles Charlie will run.

The probabilities are

b) 0.2231 ( for a blood pressure of 110.4 mmHg or more.)

c) 0.8531. ( for a blood pressure of 152.1 mmHg or less.)

e)0.0058. ( for a blood pressure that is at most 70.5 mmHg)

f) A blood pressure of 70.5 mmHg is unusually low for a randomly selected person in China because the probability is very small (0.0058).

g) The blood pressure value can be calculated using the formula:

blood pressure = (z-score * standard deviation) + mean

b) To find the probability that a randomly selected person in China has a blood pressure of 110.4 mmHg or more, we need to calculate the z-score and then use a standard normal distribution table. The z-score is calculated as (110.4 - mean) / standard deviation.
Z = (110.4 - 128) / 23
Z ≈ -0.7565
Using the standard normal distribution table, the probability corresponding to a z-score of -0.7565 is approximately 0.2231.
c) To find the probability that a randomly selected person in China has a blood pressure of 152.1 mmHg or less, we again need to calculate the z-score.
Z = (152.1 - mean) / standard deviation
Z = (152.1 - 128) / 23
Z ≈ 1.0522
Using the standard normal distribution table, the probability corresponding to a z-score of 1.0522 is approximately 0.8531.
d) To find the probability that a randomly selected person in China has a blood pressure between 110.4 and 152.1 mmHg, we need to find the difference in probabilities.
P(110.4 ≤ X ≤ 152.1) = P(X ≤ 152.1) - P(X ≤ 110.4)
Using the standard normal distribution table, we can find these probabilities and subtract them.
e) To find the probability that a randomly selected person in China has a blood pressure that is at most 70.5 mmHg, we need to calculate the z-score.
Z = (70.5 - mean) / standard deviation
Z = (70.5 - 128) / 23
Z ≈ -2.5217
Using the standard normal distribution table, the probability corresponding to a z-score of -2.5217 is approximately 0.0058.
f) A blood pressure of 70.5 mmHg is unusually low for a randomly selected person in China. This is because the probability of having a blood pressure of 70.5 mmHg or lower is very small (0.0058), indicating that it is a rare occurrence.
g) To find the blood pressure that 75% of all people in China have less than, we need to find the z-score corresponding to a cumulative probability of 0.75 using the standard normal distribution table.

Once we have the z-score, we can use it to find the blood pressure value using the formula: blood pressure = (z-score * standard deviation) + mean. Round the answer to two decimal places.

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The hight of the rectangle is equal to be 29.3 inches to one decimal place using the Pythagoras rule.

The Pythagoras rule states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 

Let the height of the rectangle be x so that by Pythagoras rule considering the right triangle formed with the diagonal of 40 inches;

x² + (x + 2)² = (40in)²

x² + x² + 4x + 4 = 1600

2x² + 4x + 4 = 1600

x² + 2x + 2 = 800

x² + 2x - 798 = 0

x = 1 + √799 or x = 1 - √799 {by quadratic formula}

x = 29.2666

Therefore, the hight of the rectangle is equal to be 29.3 inches to one decimal place using the Pythagoras rule.

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The two compositions between the functions are:

To do so, just evaluate one of the functions into the other one.

We start with:

g(x) = 8/x

The composition of g and g will be:

g(g(x)) = 8/(g(x)) = 8/(8/x) = x

And for the case of h(x) = x² + 9 we will get:

h(h(x)) = h(x)² + 9 = (x² + 9)² + 9

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the inverse of F(x)=2x-4 is f^-1(x) = (x + 4) / 2.

In simple terms, the inverse function takes the output of the original function and gives you the original input back. So, if you input 2 into the original function, you get 6 as the output. If you input 6 into the inverse function, you get 2 as the output.

I hope this helps!

Answer:

f^-1(x)=( x + 4 ) / 2  

Step-by-step explanation:

by putting f(x) = y , we got the x in y i.e. x = (y+4)/2.

then replace y with x, we get the inverse of function f(x).

Each person will receive X/32 peanuts if Joe has 1/2 of a bag of peanuts and wants to share them equally among his 3 friends.

If Joe has 1/2 of a bag of peanuts, we need to determine the total number of peanuts in the bag. Since the bag is divided equally among Joe and his 3 friends, there are a total of 1 + 3 = 4 equal portions to distribute.

To find out how many peanuts are in each portion, we divide 1/2 by 4. Dividing 1/2 by 4 gives us 1/8. Therefore, each portion contains 1/8 of the bag.

Next, we need to determine the number of peanuts in 1/8 of the bag. To do this, we multiply the number of peanuts in the whole bag by 1/8. Let's assume there are X peanuts in the whole bag. So, we have X * 1/8 peanuts in each portion.

Finally, to find out the number of peanuts each person gets, we divide the number of peanuts in each portion by the number of people (Joe and his 3 friends). Therefore, each person receives (X * 1/8) / 4 = X/32 peanuts.

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For the following piecewise-defined function the answers are:

(a) f(-2) = -6

(b) f(2) = 4

(c) f(4) = 16

(d) f(9) = 1

To find the values of the piecewise-defined function at different points, we evaluate each case separately.

(a) For f(-2), we need to evaluate the function at x = -2. Since -2 is less than -1, we use the first case: f(x) = 3x. Plugging in x = -2, we have f(-2) = 3(-2) = -6.

(b) For f(2), we again evaluate the function at x = 2. Since 2 falls within the range -1 ≤ x ≤ 8, we use the second case: f(x) = x^2. Plugging in x = 2, we have f(2) = 2^2 = 4.

(c) For f(4), we evaluate the function at x = 4. Again, 4 falls within the range -1 ≤ x ≤ 8, so we use the second case: f(x) = x^2. Plugging in x = 4, we have f(4) = 4^2 = 16.

(d) For f(9), we need to evaluate the function at x = 9. Since 9 is greater than 8, we use the third case: f(x) = x - 8. Plugging in x = 9, we have f(9) = 9 - 8 = 1.

The answers are:

(a) f(-2) = -6

(b) f(2) = 4

(c) f(4) = 16

(d) f(9) = 1

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The distance covered by the car using Simpson's rule is 135 km.

Simpson's rule is a numerical integration method that uses quadratic approximations to estimate the value of a definite integral.

It is based on dividing the integral into small sections, fitting a quadratic function to each section, and then summing the areas of the sections.

The table below shows the velocity of a car at various intervals of time.

Time (min)Velocity (km/hour)0 22273027187 00 2 4 6 8 10 12

We can use Simpson’s rule to find the distance covered by the car.

The formula for Simpson’s rule is:

[tex]∫_a^b[f(x)dx\approx((b-a)/6)(f(a)+4f((a+b)/2)+f(b))][/tex], where a and b are the lower and upper limits of integration, respectively, and f(x) is the function to be integrated.

Let's begin by calculating the distance covered by the car during the first interval, which is from t = 0 to t = 2.

We'll need to use the formula above to estimate the integral of velocity with respect to time.

Using Simpson's rule, we obtain:

[tex]∫_0^2[v(t)dt≈((2-0)/6)(v(0)+4v((0+2)/2)+v(2)]= (2/6)(0+4(22)+30) = 34 km[/tex]

Next, we'll need to do the same thing for the other intervals of time and add them all up to find the total distance covered by the car.

Here are the calculations:

∫_2^4▒〖v(t)dt≈((4-2)/6)(v(2)+4v((2+4)/2)+v(4))〗= (2/6)(30+4(27)+22) = 32 km∫_4^6▒〖v(t)dt≈((6-4)/6)(v(4)+4v((4+6)/2)+v(6))〗= (2/6)(22+4(18)+30) = 26 km∫_6^8▒〖v(t)dt≈((8-6)/6)(v(6)+4v((6+8)/2)+v(8))〗= (2/6)(30+4(7)+27) = 22 km∫_8^10▒〖v(t)dt≈((10-8)/6)(v(8)+4v((8+10)/2)+v(10))〗= (2/6)(27+4(0)+18) = 14 km∫_10^12▒〖v(t)dt≈((12-10)/6)(v(10)+4v((10+12)/2)+v(12))〗= (2/6)(18+4(0)+7) = 7 km

Therefore, the total distance covered by the car is:

34 km + 32 km + 26 km + 22 km + 14 km + 7 km = 135 km

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The simplification of the expression -8 - 10x (-1) + 7x (-1) is 3x - 8

-8 - 10x (-1) + 7x (-1)

Following PEMDAS rule:

P = parenthesis

E = exponents

M = Multiplication

D = Division

A = Addition

S = Subtraction

So,

-8 - 10x (-1) + 7x (-1)

= -8 + 10x - 7x

= -8 + 3x

Therefore, it can also be written as 3x - 8

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a. The data is approximately symmetric about

b. The mean, median and mode will be close to 8 .

c. Mean =8.35 ; Median =8 ; Mode is 8 .

First  The histogram of the data can be constructed using excel steps as follow:

Enter the data in to excel sheet.

Select the data and go to insert>chart>insert statistic chart> histogram.

The histogram thus obtained as: [See in attachment]

Step 2: The histogram has a long tail to its right side; therefore, the shape of the histogram is positively skewed. Since, the distribution is positively skewed, therefore, mean will be greater than median, median can be greater than or equal to mode that is -

Mean > Median > Mode

The sample mean of the data can be obtained as -

Mean (x) = [tex]\frac{\sum_{i=1}^{n} X_{i}}{n}[/tex]

= (5 + ...+8)/20

= 8.35

Hence the sample mean is 8.35

Next

Arranging the data into ascending order  -

5, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11, 12

The data points are in even form.

Thus, the required median of the data can be obtained as -

Median = [(n/2)th term + ((n/2) + 1)th term]/2

= [(20/2)th term + ((20/2) + 1)th term]/2

= [(10)th term + ((11)th term]/2

= (8+8)/2

= 8

Tus, the mode =  8

Next

recall that

Mode is the most occurring value of a data set.

8 is occurring the most ( 7 times), therefore, the mode of the data is 8 .

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Let's denote:

- The number of lower box tickets as x.

- The number of upper box tickets as y.

From the problem, we have the following two equations:

1. x + y = 442 (because the total number of tickets purchased is 442)

2. 12.50x + 10y = 4750 (because the total amount of money spent is $4750)

We can solve this system of equations to find the values of x and y. There are a few methods to solve this system, but I'll use substitution here.

First, let's solve the first equation for x:

x = 442 - y

Now, let's substitute x in the second equation:

12.50(442 - y) + 10y = 4750

5510 - 12.50y + 10y = 4750

5510 - 2.50y = 4750

2.50y = 5510 - 4750

2.50y = 760

y = 760 / 2.50

y = 304

Substitute y = 304 in the first equation:

x + 304 = 442

x = 442 - 304

x = 138

So, 138 lower box tickets and 304 upper box tickets were purchased.