Graph the linear equation
5x -6y = 30 by following these steps:
Step 1: Find the x-intercept. Let y = 0 and solve for x.
x=

Answer:

slope of every parallel line is - 4

Step-by-step explanation:

calculate the slope m of the given line using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (- 2, 7) and (x₂, y₂ ) = (1, - 5) ← 2 points on the line

m = [tex]\frac{-5-7}{1-(-2)}[/tex] = [tex]\frac{-12}{1+2}[/tex] = [tex]\frac{-12}{3}[/tex] = - 4

• Parallel lines have equal slopes

Thus the slope of any line parallel to the given line is m = - 4

The domain of the function y=√x-4 is restricted by the square root of a non-negative number, which means that x-4 must be greater than or equal to zero. Thus, we have:

x-4 ≥ 0

x ≥ 4

Therefore, the domain of the function is x≥4.

The range of the function is determined by the output values of the function. Since the square root of any non-negative number is always non-negative, the output values of the function y=√x-4 will always be non-negative. Thus, the range of the function is y≥0.

Therefore, the correct answer is A. Domain: x≥−4, Range: y≥0.

If  Lee has a catering company. His staff is making pies for an event.  there are 41 pies that have only rhubarb filling.

To find the number of pies that have only rhubarb filling, we need to subtract the pies that have rhubarb filling and another filling(s) from the total number of pies that have rhubarb filling.

We can use the given information to create a Venn diagram to visualize the relationships between the different fillings:

        _______________

         /      (3)      \

        /______ 14 ______\

       /       /    \      \

      /  (1) / (5)  \ (2)  \

     /_____/_______ \______\

          /         \       \

         /     (4)     \______\

(1) represents the number of pies that have only cherry filling,

(2) represents the number of pies that have only strawberry filling,

(3) represents the number of pies that have only rhubarb filling,

(4) represents the number of pies that have cherry and strawberry filling, but not rhubarb filling,

(5) represents the number of pies that have rhubarb and strawberry filling, but not cherry filling.

We can use the given information to fill in the values in the Venn diagram:

68 pies have cherry filling (1) + (4) + (6) + (7) + (8) + (9) + (10) + (14) = 68

73 pies have strawberry filling (2) + (4) + (5) + (7) + (8) + (9) + (10) + (14) = 73

30 pies have rhubarb and cherry filling (6) + (7) + (10) + (14) = 30

23 pies have rhubarb and strawberry filling (5) + (7) + (9) + (10) + (14) = 23

12 pies have only strawberry and cherry filling (7) = 12

14 pies have all three fillings (14) = 14

To find the number of pies that have only rhubarb filling, we can start by adding up the pies that have rhubarb filling and no other fillings: (3) + (6) + (9) + (10).

(3) is already the number of pies that have only rhubarb filling. For the other three regions, we need to use the values in the Venn diagram to calculate the number of pies with the given combinations of fillings:

Adding up the number of pies that have only rhubarb filling from each region, we get:

Number of pies that have only rhubarb filling = (3) + (24) + (14) + (0) = 41.

Therefore, there are 41 pies that have only rhubarb filling.

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If the the a is greater than 1, compared to the parent function the C. Stretched vertically.

The equation y = ax^2 + c represents a quadratic function where "a" is the coefficient of the x^2 term and "c" is a constant term. The parent function of this quadratic function is y = x^2.

If the equation of a quadratic function is given in the form y = ax^2 + c and "a" is greater than 1, then the graph of the function will be stretched vertically and the vertex will be shifted up or down depending on the value of "c".

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Answer:

We can estimate that 900 people in attendance would prefer mustard on a hot dog.

Step-by-step explanation:

What is proportion?

Proportion is a mathematical concept that refers to the relationship between two quantities, expressed as a ratio or a fraction. It represents the size of one quantity relative to another quantity, typically within a larger population or sample.

To determine the number of people who would prefer mustard on a hot dog, we need to use the proportion of people in the sample who prefer mustard and apply it to the total number of people in attendance.

The proportion of people in the sample who prefer mustard is:

27/150 = 0.18

To estimate the number of people who prefer mustard in the entire population of 5,000 attendees, we can multiply this proportion by the total number of attendees:

0.18 x 5,000 = 900

Therefore, we can estimate that 900 people in attendance would prefer mustard on a hot dog.

Answer:

2000 people prefer mustard on their hot dog.

900 people prefer chili on their hot dog

2100 people prefer ketchup on their hot dog

Step-by-step explanation:

hope this helps

The distance between the Earth and Saturn varies depending on their positions in their respective orbits around the Sun. On average, the distance between the Earth and Saturn is about 1.2 billion kilometers (746 million miles).

On the other hand, the average distance between the Earth and Jupiter is about 778 million kilometers (484 million miles).

Therefore, on average, the Earth is about 422 million kilometers (262 million miles) farther from Saturn than it is from Jupiter. However, it is important to note that the actual distance can vary depending on the positions of the planets in their orbits.

Hope this helps :)

Distance from earth to saturn = 1.2 billion km Distance from earth to jupiter = 0.588 billion km so earth to saturn disstance is 1.2/0.588 = 2.04 times the earth to jupiter distance 2)

Therefore, it is not even chance, but it is also not very unlikely. It is moderately likely that a yellow marble will be selected on the first draw.

Probability is a branch of mathematics that deals with the study of random events and their likelihood of occurring. It is a measure of the likelihood or chance of an event happening. Probability is expressed as a number between 0 and 1, where 0 means the event is impossible, and 1 means the event is certain.

In other words, probability is a way of quantifying uncertainty. It is used in various fields, such as statistics, physics, finance, engineering, and more, to help make predictions and decisions based on uncertain information. Probability theory provides a set of rules and tools for analyzing and manipulating random variables and events, and for calculating the probability of complex events.

The likelihood of a yellow marble being selected on the first draw can be calculated by dividing the number of yellow marbles by the total number of marbles in the bag:

likelihood of selecting a yellow marble = number of yellow marbles / total number of marbles

So, in this case, the likelihood of selecting a yellow marble on the first draw is:

likelihood of selecting a yellow marble = 2 / (3 + 2 + 1) = 2/6 = 1/3

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This sentence relating to the quotient can be expressed mathematically as:

(30 / x) - 2

This sentence can be expressed mathematically as:

(30 / x) - 2

where x represents the unknown number.

The word "quotient" indicates that we are dividing 30 by the unknown number x. The phrase "is decreased by 2" means that we need to subtract 2 from the quotient.

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Answer:

2x + [tex]\frac{5}{6}[/tex]

Step-by-step explanation:

The like terms that you want to combine is 1/3 and 1/2.  You need a common denominator which would be 6

[tex]\frac{1}{3}[/tex] = [tex]\frac{2}{6}[/tex]

[tex]\frac{1}{2}[/tex] = [tex]\frac{3}{6}[/tex]

2x + [tex]\frac{2}{6}[/tex] + [tex]\frac{3}{6}[/tex]

2x + [tex]\frac{5}{6}[/tex]  You cannot combine this with the 2 because the 2 is multiplied by x.

Helping in the name of Jesus.

We can write g in terms of f as: g(x) = f(x) - 3 = x² - 3

What is function?

In mathematics, a function is a relationship between two sets of elements, called the domain and the range, such that each element in the domain is associated with a unique element in the range. In simpler terms, a function is a set of rules that takes an input value and produces a corresponding output value.

To write g in terms of f, we can use function composition, which involves plugging the function f(x) into g(x) wherever we see x.

So, we have:

g(x) = f(x) - 3

where f(x) = x².

Substituting f(x) into g(x), we get:

g(x) = (x²) - 3

Therefore, we can write g in terms of f as:

g(x) = f(x) - 3 = x² - 3.

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Step-by-step explanation:

x = the number

1/4 * x = 1/8   <====given,     Multiply both sides of the equation by 4

4  * 1/4   x = 4 * 1/ 8

x = 4/8 = 1/2

Step-by-step explanation:

Let the given number Be X  

SO given thatone fourth of X is one eighth

x/4 = 1/8

x  = 1/2

x = 0.5

A.y-intercept: 4

axis of symmetry: x=2

x-coordinate of the vertex: 2

B.x f(x)

  0   4

  1   1

  2   0

  3     1

  4   4

C. 5 |           .  

      |        .    

      |     .        

      |  .            

      |.                

  0 |_____________

      0  1  2  3  4

An intercept refers to the point(s) at which a curve or line intersects an axis. For example, the x-intercept is where the line crosses the x-axis, and the y-intercept is where it crosses the y-axis.

A vertex is a point where two or more lines or curves meet. In the context of a parabolic curve, the vertex is the point at which the curve changes direction.

According to the given information:

A. The y-intercept occurs when x=0. Therefore, f(0) = 0² - 4(0) + 4 = 4. So the y-intercept is (0, 4).

The axis of symmetry can be found using the formula x = -b/(2a), where a and b are the coefficients of x² and x, respectively. In this case, a = 1 and b = -4, so x = -(-4)/(2*1) = 2. Therefore, the equation of the axis of symmetry is x = 2.

To find the vertex, we use the fact that the vertex occurs at the axis of symmetry. So when x=2, f(x) = 2² - 4(2) + 4 = -4. Therefore, the vertex is at (2, -4).

B. We can use the vertex to make a table of values:

x f(x)

0 4

1 1

2 0

3 1

4 4

C. We can use the table of values to plot the points and sketch the graph of the function. The graph of the function is a parabola that opens upward, since the coefficient of x² is positive.

  5 |           .  

      |        .    

      |     .        

      |  .            

      |.                

  0 |_____________

      0  1  2  3  4

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Answer:

The player can expect to lose money over the long run.

Step-by-step explanation:

We would need to know the possible outcomes and the corresponding probabilities of the game to calculate the expectation. Without that information, we cannot calculate the expectation.

The expectation is calculated by multiplying each outcome by its probability and then adding up all the products. In this case, the outcomes are the amounts indicated by the pointer, and the probabilities are the likelihood of the pointer stopping on each amount.

If we knew the probabilities, we could calculate the expected value as follows:

Expected value = (amount 1 x probability 1) + (amount 2 x probability 2) + ... + (amount n x probability n)

We would then subtract the cost to play the game to get the net expected value. If the net expected value is positive, it means that the player can expect to make a profit over the long run, while if it is negative, it means that the player can expect to lose money over the long run.

Answer: V = 2744 m3

Step-by-step explanation: I took the test and was correct

The answer of this question is above in the picture

f(x) = 8/2^3

Rewritten in radical form: f(x) = 2^(3/2)

f(x) = 8(21)

Rewritten in exponential form: f(x) = 2^3 * 21

Key features:

Initial value: 2^3

Exponential base: 21

Domain: All real numbers

Range: All real numbers greater than 0

f(x) = 8/2^3

Rewritten in radical form: f(x) = 2^(3/2)

Key features:

Initial value: 2^(3/2)

Radical base: 2

Domain: All real numbers

Range: All real numbers greater than 0

f(x) = (√10)

Rewritten in exponential form: f(x) = 10^(1/2)

Key features:

Initial value: 10^(1/2)

Exponential base: 10

Domain: All real numbers

Range: All real numbers greater than 0

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Answer:

The period of a sine function of the form y = a sin bx is given by:

period = (2π) / |b|

In this case, the function is y = -1 sin 5x, which can be rewritten as y = -sin(5x). So, we can see that b = 5.

Substituting b = 5 into the formula, we get:

period = (2π) / |5|

period = π / 5

Therefore, the period of the function is π/5.

Answer:

7 and 31 it asks

the 3rd and 6th is

The 1st and 2nd terms of this Fibonacci sequence, given the 3rd and 6th terms would be 2 and 5.

Let's denote the first and second terms of the Fibonacci sequence as F1 and F2. The Fibonacci sequence is defined by the recurrence relation:

F(n) = F(n-1) + F(n-2)

We are given that the 3rd term (F3) is 7 and the 6th term (F6) is 31. We can use this information to set up the following equations:

F3 = F2 + F1 = 7

F6 = F5 + F4 = 31

We can also express F4 and F5 in terms of F1 and F2:

F4 = F3 + F2 = (F2 + F1) + F2 = F1 + 2F2

F5 = F4 + F3 = (F1 + 2F2) + (F2 + F1) = 2F1 + 3F2

Now, let's substitute equation (4) into equation (2):

F6 = 2F1 + 3F2 + F1 + 2F2 = 31

3F1 + 5F2 = 31

By trial and error, we can find the possible values for F1 and F2 that satisfy this equation:

F1 = 1, F2 = 6: 3(1) + 5(6) = 3 + 30 = 33 (not a solution)

F1 = 2, F2 = 5: 3(2) + 5(5) = 6 + 25 = 31 (solution)

The solution is F1 = 2 and F2 = 5, so the first two terms of the Fibonacci sequence are 2 and 5.

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The sacks in factory A = 3386

and the sacks in factory B = 17414

What is a system of equation?

A finite set of equations for which common solutions are sought is referred to as a set of simultaneous equations, often known as a system of equations or an equation system.

Here, we have

Given: There was a total of 20800 sacks of rice in Factory A and Factory B.

Let us assume

The sacks in factory A = x

and the sacks in factory B = y

∴ x + y = 20800

y = 20800 - x

After of 3/4, the sacks of rice in Factory A and 3/5 of the sacks of rice in Factory B were sold.

∴ 3x/4 + 3y/5 were sold

x -3x/4 + y - 3y/5 = x + 1040 + x

x -3x/4 + 20800 - x -3/5(20800 - x) = 2x + 1040

-3x/4 + 20800 - 12480 + 3x/5 = 2x + 1040

-43x/20 = -7280

x = 3386.046

y = 17413.95

Hence, The sacks in factory A = 3386

and the sacks in factory B = 17414

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The Cartesian equations of line (D) are:

(1) through point A (2, -3) and of direction vector v(-1, 2), 2x + y + 1 = 0

(2) through the points A(3,2) and B(5,- 1), 3x - 2y + 13 = 0

(3) through point A(-5, 2) and parallel to line (L); 3x - 2y + 11 = 0

(4) through C(4, 1) and parallel to line (d), 2x - y - 7 = 0

(1) Passing through point A (2, -3) and of direction vector v(-1, 2):

The Cartesian equation of a line passing through point A(x1, y1) and having a direction vector v = (a, b) can be given by:

(x - x1) / a = (y - y1) / b

Substituting the values from the given information:

(x - 2) / (-1) = (y - (-3)) / 2

Simplifying, we get:

-x + 2 = (y + 3) / 2

Multiplying both sides by -2 to eliminate the fraction:

2x - 4 = -y - 3

Rearranging and simplifying, we get the Cartesian equation of the line:

2x + y + 1 = 0

(2) Passing through the points A(3,2) and B(5,-1):

The Cartesian equation of a line passing through two points A(x1, y1) and B(x2, y2) can be given by:

(x - x1) / (x2 - x1) = (y - y1) / (y2 - y1)

Substituting the values from the given points A(3,2) and B(5,-1):

(x - 3) / (5 - 3) = (y - 2) / (-1 - 2)

Simplifying, we get:

(x - 3) / 2 = (y - 2) / (-3)

Multiplying both sides by 2 and -3 to eliminate the fractions:

-3x + 9 = 2y - 4

Rearranging and simplifying, we get the Cartesian equation of the line:

3x - 2y + 13 = 0

(3) Passing through point A(-5, 2) and parallel to line (L); 2x - 3y + 5 = 0:

Since the line is parallel to line (L), it will have the same direction vector as line (L), which is (2, -3).

Using the same approach as in (1), we can write the Cartesian equation of the line as:

(x - (-5)) / 2 = (y - 2) / (-3)

Simplifying, we get:

(x + 5) / 2 = (y - 2) / (-3)

Multiplying both sides by 2 and -3 to eliminate the fractions:

-3x - 15 = 2y - 4

Rearranging and simplifying, we get the Cartesian equation of the line:

3x - 2y + 11 = 0

(4) Passing through C(4, 1) and parallel to line (d): y = 2x + 1:

Since the line is parallel to line (d), it will have the same slope as line (d), which is 2.

Using the same approach as in (1), we can write the Cartesian equation of the line as:

(x - 4) / 1 = (y - 1) / 2

Simplifying, we get:

(x - 4) = (y - 1) / 2

Multiplying both sides by 2 to eliminate the fraction:

2x - 8 = y - 1

Rearranging and simplifying, we get the Cartesian equation of the line:

2x - y - 7 = 0

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We invested $ 15900 in the stock that paid 12% annual interest, and the remaining amount that suffered a 5% loss is $ 2,400.

This problem involves calculating the amount invested in two different investments, one with a 12% annual interest rate and the other incurring a 5% loss.

The concept used is to set up an equation based on the given information, where the sum of the annual income from both investments is equal to the given total annual income.

Then, we simplify the equation and solve for the unknown variable to find the amount invested in each investment.

Here we have

You invested ​$18,300​, part of it in a stock that paid​ 12% annual interest.​ However, the rest of the money suffered a​ 5% loss.

The total annual income from both investments was ​​$ 1788

Let's assume that the amount invested in the stock that paid 12% annual interest is x, and the remaining amount that suffered a 5% loss is y.

Here  x + y = 18,300  [ ∵ the total amount invested is $18,300 ]

=> y = 18,300 - x  ---- (1)

The annual income from the investment in the stock that paid 12% interest = 12% of x = 0.12x

The annual loss from the remaining investment = 5% of y = 0.05y.

We know that the total annual income from both investments is $ 1788   so we can write the equation:

=> 0.12x - 0.05y = $ 1788

From (1)

=> 0.12x - 0.05(18,300 - x) = $ 1788

=> 0.12x - 915 + 0.05x = 1788

=> 0.17x - 915 = 1788

=> 0.17x = 873

=> x = 15900

So, we invested $ 15900 in the stock that paid 12% annual interest  

The remaining amount that suffered a 5% loss is y = 18,300 - 15900

= $ 2,400

Therefore,

We invested $ 15900 in the stock that paid 12% annual interest, and the remaining amount that suffered a 5% loss is $ 2,400.

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Answer: Rectangle

Step-by-step explanation: The shape created by the cross-section is a rectangle because the cross-section shows a 2-dimensional slice of a 3-dimensional rectangular prism. Since a rectangular prism has rectangular faces, any cross-section taken perpendicular to the length of the prism will be a rectangle.

The solution to the system of equations using matrices is x = 45, y = 15, and z = 30.

A scalar value that can be calculated from a matrix's elements is the determinant. When a square matrix is used to transform vectors, the determinant is a measurement of how much the matrix "stretches" or "shrinks" space. In linear algebra, the determinant is employed in a variety of operations, including as the computation of a matrix's inverse, the description of a matrix's eigenvalues and eigenvectors, and the resolution of linear equation systems. In specifically, the existence of a unique solution, the absence of a solution, or an unlimited number of solutions to a system of linear equations can be determined using the determinant of the coefficient matrix.

The given equation are:

2x -y + 3z = 180

-4x + 2y + 3z = 225

3x - 4y = 270

Writing the equations in matrix form we have:

[tex]\begin{bmatrix} 2 & -1 & 3 \\ -4 & 2 & 3 \\ 3 & -4 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 180 \\ 225 \\ 270 \end{bmatrix}[/tex]

Multiplying the inverse of the coefficient matrix we have:

[tex]\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 2 & -1 & 3 \\ -4 & 2 & 3 \\ 3 & -4 & 0 \end{bmatrix}^{-1} \begin{bmatrix} 180 \\ 225 \\ 270 \end{bmatrix}[/tex]

Now,

[tex]\begin{bmatrix} 2/23 & 5/46 & -3/23 \\ 2/23 & 1/23 & 5/23 \\ -3/23 & -5/46 & 2/23 \end{bmatrix}[/tex]

Multiplying this by the vector on the right-hand side gives:

[tex]\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 2/23 & 5/46 & -3/23 \\ 2/23 & 1/23 & 5/23 \\ -3/23 & -5/46 & 2/23 \end{bmatrix} \begin{bmatrix} 180 \\ 225 \\ 270 \end{bmatrix} = \begin{bmatrix} 45 \\ 15 \\ 30 \end{bmatrix}[/tex]

Hence, the solution to the system of equations is x = 45, y = 15, and z = 30.

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Thus, the solution of the given expression is found as  [tex]2xy^{2} \sqrt{13xy }[/tex] using the prime factorization.

A natural number other than 1 whose only factors are 1 and itself is said to have a prime factor. In actuality, the initial set of prime numbers are 2, 3, 5, 7, 11, and so forth.

Nevertheless, we may also apply the so-called prime factorization, which actually involves using factor trees, for numbers. The technique of expressing all numbers as the product of primes is known as prime factorization.

Given that:

= [tex]\sqrt{52x^{3 }y^{5} }[/tex]

Find the prime factors of 52:

52 = 2 x 2 x 13

x³ = x².x

y⁵ = y².y².y

Put obtained values in expression:

= [tex]\sqrt{(2)(2)(13).x^{2 }.x.y^{2} .y^{2}. y }[/tex]

Now, for the value to take out of the square root, make a pair with the power two and bring it as 1.

= [tex]2xy^{2} \sqrt{13xy }[/tex]

Thus, the solution of the given expression is found as  [tex]2xy^{2} \sqrt{13xy }[/tex] using the prime factorization.

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the system of equations is: [tex]300x + 450y = 6000[/tex]

[tex]y = x + 5[/tex]

Let's define the variables:

Let "x" be the number of cans of soup purchased by Samuel.

Let "y" be the number of frozen dinners purchased by Samuel.

According to the given information, each can of soup has 300 mg of sodium and each frozen dinner has 450 mg of sodium. Samuel purchased 5 more frozen dinners than cans of soup, so we can write the following equations:

The total sodium from cans of soup: 300x mg

The total sodium from frozen dinners: 450y mg

The total sodium from both cans of soup and frozen dinners: [tex]6000 mg[/tex]

So the first equation is:

[tex]300x + 450y = 6000[/tex]  (since the total sodium from cans of soup and frozen dinners is 6000 mg)

Since Samuel purchased 5 more frozen dinners than cans of soup, we can write the second equation:

[tex]y = x + 5[/tex] (since y is 5 more than x)

Therefore,  the system of equations is:

[tex]300x + 450y = 6000[/tex]

[tex]y = x + 5[/tex]

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Answer: 32

Step-by-step explanation:

     We will use the Order of Operations. First, we will do parentheses (), then brackets [], then lastly the braces {}, then lastly add 12.

Given:

     {[2+(11+9)]-2)+12

Add 11 to 9:

     {[2+20]-2)+12

Add 2 to 20:

     {22-2)+12

Subtract 2 from 20:

     20 + 12

Add 20 and 12:

     32

Therefore , the solution of the given problem of unitary method comes out to be any other restrictions or criteria would determine how many possible methods there are to shade 3/8 of the square.

Use the tried-and-true fundamental method, the actual variables, and any relevant information gleaned from general and specific questions to complete expression the assignment. Customers may be given another chance to taste the products in response. If these adjustments don't happen, we'll lose out on significant advancements in our understanding of programmes.

Here,

The phrase "shade 3/8 of the square" might be interpreted in a variety of ways, which might result in various solutions. Here are a few potential explanations and their accompanying responses:

1) The most straightforward interpretation is to shade 3/8 of the square as a single, continuous area.

There are countless ways to shade precisely 3/8 of the square in this interpretation.

2) Discrete Interpretation: Using a grid or discrete cells to shade 3/8 of the square.

The number of alternative ways relies on the size of the grid or the number of discrete cells if the square is divided into a grid or cells and the requirement is to shade exactly 3/8 of the cells. F

3) Interpretation based on constraints: Shading 3/8 of the square with specific restrictions.

In conclusion, the exact interpretation and any other restrictions or criteria would determine how many possible methods there are to shade 3/8 of the square. It is challenging to estimate the precise number of potential solutions in the absence of more details.

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the probability that the child randomly arranges the letters in the order TIGER is 1/120.

What is Probability?

Probability is the study of the chances of occurrence of a result, which are obtained by the ratio between favorable cases and possible cases.

There are 5 blocks with 5 different letters. Since we want the child to arrange the letters in the order TIGER, there is only one way to arrange them correctly out of a total of 5! = 120 possible arrangements.

Therefore, the probability that the child randomly arranges the letters in the order TIGER is 1/120.

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Answer:

y = 1/28(x²)

Step-by-step explanation:

Not totally sure about this, but here's what I think is the answer:

focus (0,7)

directrix y = -7

Always a good idea to plot the point (0,7) and the line y = -7 so you can visualize what the parabola looks like.

From the graph, you can see that p = 7  (halfway between the focus and the directrix).

The vertex of this parabola is (0,0), which are your (h, k) values.  You can see that from the graph.  The parabola opens UP.

4p(y-k) = (x-h)²  →  4p(y-0) = (x-0)²

4py = x²

y = x²/4p = x²/4(7) = x²/28

y = 1/28(x²)

sorry if this isn't the right answer, but I tried  (I'm not a paid expert, just a fellow student trying to learn math)!

a. TRUE, b. TRUE, c. TRUE, d. TRUE, e. TRUE, f. FALSE, g. TRUE

a. TRUE: The probability of an event can never be negative, and can at most be equal to 1, which represents certainty.

b. TRUE: The sample space is the set of all possible outcomes of an experiment, and the probability of the sample space is always equal to 1, since one of the outcomes must occur.

c. TRUE: If the events in a sequence are disjoint, then they have no outcomes in common, so the probability of the union of the events is the sum of the probabilities of the individual events.

d. TRUE: If one event is a subset of another event, then the probability of the subset is less than or equal to the probability of the superset. This follows from the fact that the subset contains fewer outcomes than the superset.

e. TRUE: The probability of the union of two events is the probability of the first event plus the probability of the second event, minus the probability of the intersection of the events, which is the probability of both events occurring together. This is known as the inclusion-exclusion principle.

f. FALSE: The formula (P(A ∩ B) = P(A)P(B)) only applies to independent events, but not all independent events are mutually exclusive. For example, if A is the event of rolling a 4 on a die, and B is the event of rolling an even number, then A and B are independent, but not mutually exclusive.

g. TRUE: If two events are mutually exclusive, then they have no outcomes in common, so the occurrence of one event tells us that the other event cannot occur. This dependence means that the events are not independent.

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