Determine the equation of the parabola that opens to the right, has focus (13,-6),
and a focal diameter of 28.

Using the function f(x) = 1/2 x and the inverse of the function f⁻¹(x) = 2x, the solutions of the given are :

f(2) = 1, f⁻¹(1) = 2, f⁻¹(f(2)) = 2

f⁻¹(-2) = -4, f(-4) = -2, f(f⁻¹(-2)) = -2

Given a function,

f(x) = 1/2 x

And an inverse function,

f⁻¹(x) = 2x

We have to find the values of the following.

f(2) = 1/2 × 2 = 1

f⁻¹(1) = 2 × 1 = 2

f⁻¹(f(2)) = f⁻¹(1) = 2 × 1 = 2

f⁻¹(-2) = 2 × -2 = -4

f(-4) = 1/2 × 4 = -2

f(f⁻¹(-2)) = f (-4) = 1/2 × 4 = -2

So, in general, f⁻¹(f(x)) = f(f⁻¹(x))

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Answer:

a. 2

Step-by-step explanation:

We Know

Side AB = 7

Side XY = 14

7 → 14, we time 2. So, the scale factor is 2.

The reasoning for the third step of the proof is the SAS Triangle Congruence Theorem, which establishes the congruence of the two triangles involved.

In the given proof, we are trying to prove that AC is equal to ED in a rectangle ABCD. The reasoning for the third step of the proof can be explained as follows:

Statement 3: ZBAD = ZCDA

Reason: SAS Triangle Congruence Theorem

Explanation: The SAS (Side-Angle-Side) Triangle Congruence Theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

In this case, we are considering triangles ZBA and ZDA. We know that AB is congruent to AD (statement 2, "AB & CD") and that the angles ZBA and ZDA are right angles since ABCD is a rectangle (statement 1, "Quadrilateral ABCD is a rectangle").

Therefore, by the SAS Triangle Congruence Theorem, we can conclude that triangle ZBA is congruent to triangle ZDA. And since corresponding parts of congruent triangles are congruent (CPCTC), we can say that angle ZBAD is congruent to angle ZCDA.

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a) The terminal point is 0.896 radius length.

b) The value returned is -0.896.

We have,

The circle's radius is 3 units long and the terminal point is located at (−2.69,−1.33)

a. Since the circle's radius is 3 units long, we divide the x-coordinate by 3:

x-coordinate of terminal point: -2.69

Number of radius lengths to the right: -2.69 / 3 ≈ -0.896

However, since the angle is measured from the 3-o'clock direction, we consider it to be in the clockwise direction.

Thus, the number of radius lengths to the right is

Number of radius lengths to the right: -(-0.896) = 0.896

Therefore, the terminal point is 0.896 radius length.

b. Using Trigonometry

cos(h) = x-coordinate of terminal point / radius length

cos(h) = -2.69 / 3 ≈ -0.896

c. As, θ = h. From the given information, we have:

θ ≈ -0.896 radians

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Step-by-step explanation:

y = 1/3 x +4  

       is the equation of a line in the form  

                 y = mx+b      where m =slope   and    b = the y-axis intercept

                           so  m = slope = 1/3   for this line

Answer:

[tex]\huge\boxed{\sf Slope = 1/3}[/tex]

Step-by-step explanation:

[tex]\displaystyle y=\frac{1}{3} x + 4[/tex]

y = mx + c

Where m is slope and c is y-intercept.

So,

By comparing both the equations, we get:

m = 1/3

[tex]\rule[225]{225}{2}[/tex]

The result when (-2/3m + 4) is added to (5/6m - 1/2) is 4 - 1/3m.

Hence, the correct answer is option A: -1/6m + 7/2.

To find the result when (-2/3m + 4) is added to (5/6m - 1/2), we need to combine like terms.

First, let's simplify the expression -2/3m + 4:

This can be rewritten as (4 - 2/3m).

Next, let's simplify the expression 5/6m - 1/2:

This can be rewritten as (5/6m - 3/6m).

Now, let's add the two expressions together:

(4 - 2/3m) + (5/6m - 3/6m)

Combining the constants and the terms with "m," we get:

4 + 5/6m - 2/3m - 3/6m

To simplify further, we need to find a common denominator for the fractions:

4 + 5/6m - (2/3m + 3/6m)

The common denominator for the fractions is 6:

4 + 5/6m - (4/6m + 3/6m)

Now, let's combine the like terms:

4 + 5/6m - 7/6m

Finally, we can simplify the expression:

4 - 2/6m

Simplifying further, we have:

4 - 1/3m.

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1. The number of members who play soccer only is 59.

2. The number of members who play tennis or soccer, but not rugby, is 37.

3. The number of members who play exactly two sports is 20.

4. The number of members who play at least one sport is 195.

5. If 60 members play soccer, then 7 members play tennis.

1. To find the number of members who play soccer only, we need to subtract the members who play soccer and other sports from the total number of soccer players.

Soccer only = Total soccer players - (Soccer and Rugby) - (Soccer and Tennis) + (All three sports)

Soccer only = 68 - 6 - 7 + 4

Soccer only = 59

Therefore, 59 members play soccer only.

2. To find the number of members who play tennis or soccer but not rugby, we need to subtract the members who play all three sports from the total number of tennis and soccer players.

Tennis or Soccer (not rugby) = (Tennis + Soccer) - (All three sports)

Tennis or Soccer (not rugby) = 43 + 68 - 4

Tennis or Soccer (not rugby) = 107

Therefore, 107 members play tennis or soccer but not rugby.

3. To find the number of members who play exactly two sports, we need to add the members who play tennis and soccer (TS), tennis and rugby (TR), and soccer and rugby (RS).

Exactly two sports = TS + TR + RS

Exactly two sports = 7 + 11 + 6

Exactly two sports = 24

Therefore, 24 members play exactly two sports.

4. To find the number of members who play at least one sport, we can add the number of members who play each sport individually (Tennis, Rugby, Soccer) and then subtract the number of members who play none of the sports.

At least one sport = Tennis + Rugby + Soccer - None

At least one sport = 43 + 84 + 68 - 0

At least one sport = 195

Therefore, 195 members play at least one sport.

5. If 60 members play soccer, we can calculate the number of members who play tennis by subtracting the members who play both sports (TS) and the members who play soccer only from the total number of soccer players.

Tennis = Total soccer players - (Soccer only) - (TS)

Tennis = 60 - 59 - 7

Tennis = -6

It is not possible to have a negative number of members playing tennis. Please double-check the provided information or calculations.

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The coordinates of the vertices after a dilation with a scale factor of 1/3 considered at the origin include:

Q' (-3, -3).

R' (1, -3).

S' (3, 3).

T' (-1, 3).

In Geometry, a dilation is a type of transformation which typically changes the side lengths of a geometric object, but not its shape.

In this scenario and exercise, we would dilate the coordinates of the vertices by applying a scale factor of 1/3 that is centered at the origin as follows:

Coordinate Q (-9, -9) → Q' (-9 × 1/3, -9 × 1/3) = Coordinate Q' (-3, -3).

Coordinate R (3, -9) → R' (3 × 1/3, -9 × 1/3) = Coordinate R' (1, -3).

Coordinate S (9, 9) → S' (9 × 1/3, 9 × 1/3) = Coordinate S' (3, 3).

Coordinate T (-3, 9) → T' (-3 × 1/3, 9 × 1/3) = Coordinate T' (-1, 3).

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The function f(x) = (x+2)²(x-1)(x-5)² has-2 is a zero with multiplicity 2

5 is a zero with multiplicity 1 and the graph is tangent to x-axis at x = 1

For the function f(x) = (x+2)²(x-1)(x-5)²

-2 is a zero with multiplicity 2

This is represented by the factor (x+2)², indicating that -2 is a zero that appears twice in the function.

5 is a zero with multiplicity 1
This is represented by the factor (x-5), indicating that 5 is a zero that appears once in the function.

The graph is tangent to the x-axis at x = 1:

This is represented by the factor (x-1), indicating that the graph touches the x-axis at x = 1.

Therefore, the function f(x) = (x+2)²(x-1)(x-5)² has all the specified attributes.

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To simplify the given expression: 4x + 8 + (6x - 7) - (8x + 9), we can follow the order of operations (parentheses, exponents, multiplication and division, and addition and subtraction) to simplify it step by step.

First, let's simplify the expression inside the parentheses:

6x - 7 becomes 6x - 7

8x + 9 becomes 8x + 9

Now, let's simplify the entire expression:

4x + 8 + 6x - 7 - 8x - 9

Combining like terms:

(4x + 6x - 8x) + (8 - 7 - 9)

2x - 8

Therefore, the simplified form of the expression 4x + 8 + (6x - 7) - (8x + 9) is 2x - 8.

Answer:

Step-by-step explanation:

Ukraine wants to join the european union.Due to this Russia realize that if ukraine join the european union then if war started between Ukraine and any other country like Russia it means Ukraine and european union both attack on this country.

Whitney needs 5/4 cup of sugar in total to bake all the brownies.

To calculate the total amount of sugar needed to bake all the brownies, we need to add the quantities of sugar required for each type of brownie.

According to the information given, one type of brownie requires 1/2 cup of sugar and the other type requires 3/4 cup of sugar.

To find the total amount of sugar needed, we add these two quantities:

1/2 cup + 3/4 cup

To add fractions, we need to have a common denominator.

The common denominator is 4:

(1/2) x (2/2) cup + (3/4) x (4/4) cup

This simplifies to:

2/4 cup + 12/16 cup

Now, we can add the fractions:

(2/4) cup + (12/16) cup = (4/8) cup + (12/16) cup

Next, we find a common denominator, which is 16:

(4/8) x (2/2) cup + (12/16) x (1/1) cup

This simplifies to:

8/16 cup + 12/16 cup

Finally, we add the fractions:

8/16 cup + 12/16 cup = 20/16 cup

Since 20/16 is an improper fraction, we can simplify it by dividing both the numerator and denominator by their greatest common divisor, which is 4:

(20/4)/(16/4) cup = 5/4 cup

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Answer:

Step-by-step explanation:

y  =  -5x^2

if x  =  0 , then

y  =  -5(0)

y  =  0

Answer: No

To check if (-5,6) is a solution to the equation y=-7, we need to substitute -5 for x and 6 for y in the equation.

If the equation is true, then the ordered pair is a solution to the equation.

If we put (-5,6) into y=-7, we get 6=-7, which is not true.

So, (-5,6) is not a solution to the equation y=-7.

An equation can be written to determine the number of students that went on this trip is: C. [tex]\frac{1259}{s} + 12=\frac{1259}{s+9}[/tex]

In order to write an equation to determine the number of students that went on this trip, we would have to assign a variable and an expression to the number of students who went on the trip and the original number of students who had planned to go on the trip as follows;

Since a total of $1259 was collected from all of the students with 9 students dropping out and their portion of the money collected given back to them while the students that are all still going must contribute an additional $12, an equation to determine the number of students that went on this trip is given by;

[tex]\frac{1259}{s} + 12=\frac{1259}{s+9}[/tex]

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Complete Question:

A high school class is planning a field trip to an art museum in their city. The class collected a total of $1259 from all of the students, which will go towards renting a bus, museum admission, and lunch.

However, a week before the trip 9 students had to drop out and their portion of the collected money had to be given back to them. Due to this, the students that are all still going must contribute an additional $12 each to cover all of the cost.

Now let s be the number of students who went on the trip and s + 9 be the original number of students who had planned to go on the trip

What equation can be written to determine the number of students that went on this trip?

The opening balance is 100.00 A. Its placement on the credit side implies funds deposited before the statement period. The specific impact on Gloria's finances and the value of B are not provided.

Based on the provided information, let's address each question:

3.2.1 The opening balance is 100.00 A (currency is not specified). This amount represents the initial balance in the bank account before any transactions occur.

3.2.2 When the opening balance is on the credit side of the bank statement, it indicates that there were funds deposited or credited to the account before the statement period. In this case, the opening balance of 100.00 A suggests that there were funds deposited or credited into the account before the statement period began.

3.2.3 If Gloria decides to withdraw money at Capital Bank ATM, her account balance will decrease by the amount of the withdrawal. The specific amount of the withdrawal is not mentioned in the provided information, so we cannot determine the exact impact on her finances. However, it's important to note that ATM withdrawals may also incur transaction fees, which could further reduce her account balance.

3.2.4 Based on the information given, the value of B is not explicitly provided. It seems to be missing or not specified in the provided data. Without knowing the value of B, it is not possible to calculate it.

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The solution to the inequality indicates that the attendance in 2009 must be 826,000 or more.

Let's assume the attendance in 2009 is represented by the variable A. According to the given information, the attendance in 2008 was at least 16,700 less than the attendance in 2009.

Mathematically, we can express this as:

A - 16,700 ≥ 809,300

To find the attendance in 2009, we need to solve this inequality.

First, we add 16,700 to both sides of the inequality:

A - 16,700 + 16,700 ≥ 809,300 + 16,700

Simplifying, we have:

A ≥ 826,000

Therefore, the attendance in 2009 is greater than or equal to 826,000.

In this case, the inequality A ≥ 826,000 indicates that the attendance in 2009 must be equal to or greater than 826,000 for the statement to hold true.

If the attendance in 2009 is less than 826,000, it would not satisfy the given condition that the attendance in 2008 was at least 16,700 less than the attendance in 2009.

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(3,1)

Ordered pairs are always ordered as (x,y).  Point B has an x-value of 3 and a y-value of 1.

Answer:

2y² + 3

Step-by-step explanation:

width of one side is 4y² - y + 6.

two equal sides of rectangle = 2(4y² - y +6) = 8y² - 2y + 12.

perimeter = 2 lengths + 2 widths

12y² - 2y + 18 = 2 lengths + 8y² - 2y + 12

2 lengths = (12y² - 2y + 18) - (8y² - 2y + 12)

= 4y² +6

length = 1/2 (4y² +6)

= 2y² + 3

The coordinates of the point of intersection of the tangent and the secant are;

T(a, b) = (9/5, -14/5)

C = (0, -1)

A tangent is a line or segment that touches a circle only once.

The equation of the line AN is; y - 2 = ((-2 - 2)/(2 - 3)) × (x - 3)

y = 4·x - 12 + 2 = 4·x - 10

y = 4·x - 10

The coordinates of the center of the circle is; ((-5 + 3)/2, (4 + 2)/2) = (-1, 3)

The radius of the circle is; √((3 - (-1))² + (2 - 3)²) = √(17)

The equation of the circle is; (x + 1)² + (y - 3)² = 17

The point C indicates;

(0 + 1)² + (y - 3)² = 17

(y - 3)² = 17 - 1 = 16

y - 3 = √(16) = ±4

y = 4 + 3 = 7, and y = -4 + 3 = -1

The coordinates of the point C is; (0, -1)

The slope of AC = (2 - (-1))/(3 - 0) = 1

AC is perpendicular to BC

The slope of BC = -1

The equation of BC is therefore; y - 4 = -(x - (-5)) = -x - 5

y = -x - 1

The equation of the segment AN is; y - 2 = ((2 - (-2))/(3 - 2))×(x - 3)

y - 2 = 4·(x - 3)

y = 4·x - 12 + 2 = 4·x - 10

-x - 1 = 4·x - 10

x-coordinate of T = 9/5

The y-coordinate of T = -9/5 - 1  = -14/5

Therefore, at the point T, we get; (9/5, -14/5)

a = 9/5, b = -14/5, C = (0, -1)

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Answer:

B and C

Step-by-step explanation:

Answer:  B and C

Step-by-step explanation:

Pythagorean Theorem:

c²=a²+b²              >where a and be are the shorter legs and c is the

                                longer/hypotenuse

Let's plug each into the pythagorean theorem

A)   Not a right triangle

33² = 5² + (√8)²

1089 = 25 +8

1089 = 33             >Not equal,  The statement is not true, so this is not a

                                 right triangle.

B)  Yes, it is a right triangle

17²=8²+15²

289=64+225

289=289       They are equal, true statement, so yes it is a right triangle

C) No, it is not a right triangle

2² = (√2)²+(√2)²

4 = 2+2

4=4              They are equal, true statement, so yes it is a right triangle

To find the probability that two distinct numbers chosen randomly from the set {-2, -6/5, -1/3, 0, 1/2, 5/6, 3} will be the slopes of two perpendicular lines, we need to determine the total number of possible pairs of distinct numbers and the number of pairs that result in perpendicular slopes.

Total number of possible pairs:

Since we are choosing two distinct numbers from a set of seven, the total number of possible pairs is given by the combination formula:

C(7, 2) = 7! / (2!(7 - 2)!) = 7! / (2! * 5!) = 21

Pairs resulting in perpendicular slopes:

For two lines to be perpendicular, the product of their slopes must be -1. So, we need to count the number of pairs whose product is -1.

The set contains two negative numbers {-2, -6/5} and one positive number {3}. None of these numbers have a reciprocal within the set that gives a product of -1. Therefore, no pairs can be formed from these numbers that result in perpendicular slopes.

Hence, the number of pairs resulting in perpendicular slopes is 0.

Therefore, the probability of randomly choosing two distinct numbers from the set {-2, -6/5, -1/3, 0, 1/2, 5/6, 3} that will be the slopes of two perpendicular lines is 0/21, which simplifies to 0.

The area of the base of the cylinder is A = 16π units²

Given data ,

Let the area of the base of the cylinder be A

Now , the radius of the cylinder is r = 4 units

And , the height of the cylinder is h = 6 units

where A = πr²

The base of the cylinder is a circular disk of radius r units

On simplifying , we get

A = π ( 4 )²

A = 16π units²

So , the value of A is 16π units²

Hence , the area of the base is A = 16π units²

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Answer:

Step-by-step explanation:

We know that,

sum of three side of triangle is 180°

So,

x + 72° + 90° = 180°

x + 162° = 180°

x = 180° - 162°

x = 18°

[tex]\underline{\rule{190pt}{4pt}}[/tex]

The value of x in the triangle is 18 degrees.

The given triangle is a right angled triangle.

We have to find the value of x.

As the triangle is a right angle one angle is 90 degrees.

And the other angle is 72 degrees.

The missing angle is x which we have to find.

By angle sum property we know that the sum of three angles is 180 degrees in a triangle.

72+90+x=180

Combine the like terms:

162+x=180

Subtract 162 from both sides:

x=180-162

When one hundred sixty two is subtracted from one hundred eighty, we get 18.

x=18 degrees

Hence, 18 degrees is the value of x in the triangle.

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The Simplified form of the expression (x+2)(x-2) - (x+3)² is -13 - 6x.

The expression (x+2)(x-2) - (x+3)², we'll follow the distributive property and simplify each term.

First, let's expand (x+2)(x-2):

(x+2)(x-2) = x(x) + x(-2) + 2(x) + 2(-2)

          = x² - 2x + 2x - 4

          = x² - 4

Next, let's expand (x+3)²:

(x+3)² = (x+3)(x+3)

      = x(x) + x(3) + 3(x) + 3(3)

      = x² + 3x + 3x + 9

      = x² + 6x + 9

Now, we can substitute these expansions into the original expression:

(x+2)(x-2) - (x+3)² = (x² - 4) - (x² + 6x + 9)

To simplify further, let's distribute the negative sign to every term inside the parentheses:

(x² - 4) - (x² + 6x + 9) = x² - 4 - x² - 6x - 9

Now, we can combine like terms:

(x² - x²) + (-4 - 9) - 6x = -13 - 6x

Therefore, the simplified form of the expression (x+2)(x-2) - (x+3)² is -13 - 6x.

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Answer: To calculate the expected value, we need to multiply each possible outcome by its probability and then add up the results.

The probability of selecting a gold marble is 3/41, the probability of selecting a silver marble is 8/41, and the probability of selecting a black marble is 30/41.

The winnings/losses associated with each outcome are $3 for gold, $2 for silver, and -$1 for black.

Step-by-step explanation:

Therefore, the expected value can be calculated as follows:

(3/41) x $3 + (8/41) x $2 + (30/41) x (-$1)

= $0.073

The expected value is $0.073, which means that on average, you can expect to win $0.073 per game if you play this game many times.

Therefore, if you play this game, you can expect to win a small amount of money on average, but it is not a guaranteed win.

References:

Grinstead, C.M. and Snell, J.L. (2006). Introduction to Probability. American Mathematical Society.

"Expected Value," Investopedia, accessed May 14, 2023, https://www.investopedia.com/terms/e/expectedvalue.asp.

Answer:

y = 4.71 and y = -8.71

Step-by-step explanation:

Step 1:  Subtract 5 from both sides:

((y + 2)^2 + 5 = 50) - 5

(y + 2)^2 = 45

Step 2:  Take the square root of both sides, subtract 2 from both sides, and round to the nearest hundredth to solve for y.  

Positive answer:

√(y + 2)^2 = √45

(y + 2 = √45) - 2

y = √45 - 2

y = 4.708203932

y = 4.71

Thus, one answer for y (rounded to the nearest hundredth) is approximately 4.71.

Negative answer:

√(y + 2)^2 = -√45

(y + 2 = -√45) - 2

y = -√45 - 2

y = -8.708203932

y ≈ - 8.71

Thus, the other answer for y (rounded to the nearest hundredth) approximately. -8.71

Optional Step 3:  We can check our work by plugging in 4.71 and -8.71 for y and seeing whether we get 50.  Because they're rounded, you won't get exactly 50, but the answer should be close enough for us to trust our approximations:

Plugging in 4.71 for y:

(4.71 + 2)^2 + 5 = 50

(6.71)^2 + 5 = 50

45.0241 + 5 = 50

50.0241 > 50

Our answer is close enough to 50 that we can trust our approximation.

Plugging in -8.71 for y:

(-8.71 + 2)^2 + 5 = 50

(-6.71)^2 + 5 = 50

45.0241 + 5 = 50

50.0241 > 50

Again, our answer is close enough to 50 that we can trust our approximation.