100 Points! Algebra question. Find the exact value of the expression. Photo attached. Please show as much work as possible. Thank you!

Domain (x-values) should represent the possible hours spent working, and the range (f(x)-values) should represent the corresponding number of assignments completed.

To determine if the given ordered pairs represent a possible domain and range for the function f(x), we need to consider the nature of the relationship between the number of assignments completed and the number of hours spent working.

Since the function models a student's daily task performance, it is reasonable to assume that the number of assignments completed will increase as the number of hours spent working increases.

The domain (x-values) should represent the possible hours spent working, and the range (f(x)-values) should represent the corresponding number of assignments completed.

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To translate f(x) up by 7 units, you need to add 7 to f(x):

   h(x) = f(x) + 7

This is adding 7 to the y-values from f(x), which is why the "+7" is outside the f(x) notation.

  h(x) = f(x) + 7

         = x²-1 +7

         = x² + 6

Sure! Let’s solve these problems one by one.

2.1 To determine the x-intercepts of f, we need to find the values of x when f(x) = 0. So we need to solve the equation -2x² - 5x + 3 = 0. Using the quadratic formula, we get that x = (-(-5) ± √((-5)² - 4(-2)(3))) / (2(-2)). This simplifies to x = (5 ± √(49)) / (-4). So the x-intercepts are (5 - √(49)) / (-4) and (5 + √(49)) / (-4).

2.2 The axis of symmetry of a parabola in the form f(x) = ax² + bx + c is given by the formula x = -b / (2a). In this case, a = -2 and b = -5, so the axis of symmetry is x = -(-5) / (2(-2)) = 5/4.

2.3 The range of a quadratic function in the form f(x) = ax² + bx + c is determined by its vertex. The x-coordinate of the vertex is given by the formula x = -b / (2a), which we already calculated to be 5/4. The y-coordinate of the vertex is given by f(5/4) = -2(5/4)² - 5(5/4) + 3 = -25/8 - 25/4 + 3 = -25/8 - 50/8 + 24/8 = -51/8.

Since a < 0, the parabola opens downwards and the vertex is the highest point on the graph. Therefore, the range of f is (-∞, -51/8].

2.4 Reflecting the graph of f in the x-axis means that we change the sign of the y-coordinates. This is equivalent to replacing f(x) with -f(x). So the equation of the reflected graph is -f(x) = 2x² + 5x - 3.

Translating this graph 4 units to the right means that we replace x with (x - 4). So the equation of the translated graph is g(x) = 2(x - 4)² + 5(x - 4) - 3. Expanding this expression, we get g(x) = 2x² - 16x + 32 + 5x - 20 - 3 = 2x² - 11x + 9.

So the equation of g in the form g(x) = a(x + p)² + q is g(x) = 2(x - (-11/4))² + (9 - (121/8)).

Answer:

28. B

29. D

30. A

Step-by-step explanation:

28.

2 5/8 yd × 5/6 yd =

= 21/8 × 5/6 yd²

= 105/48 yd²

= 35/16 yd²

= 2 3/16 yd²

Answer: B

29.

2641 becomes 6241.

In 6241, the 6 is in the thousands place.

6 × 1000 = 6000

Answer: D

30.

90 + 7 × (7 - 1) =

Use the correct order of operations.

= 90 + 7 × 6

= 90 + 42

= 132

Answer: A

The expected number of students that like maths in 200 students is 120

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

3 out of every 5 students said Math is their favorite class

This means that

P(Math) = 3/5

When simplified, we have

P(Math) = 0.6

In a selection of 200 students, the expected value is

Expected = 0.6 * 200

Evaluate

Expected = 120

Hence, the expected number of students that like maths is 120

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Question

In a recent survey, 3 out of every 5 students said Math is their favorite class.If 200 students were surveyed, how many students can be expected to choose math as their favorite class?

Answer:

4π

Step-by-step explanation:

Circumference = π X D (D = diameter = 2 X radius).

Diameter = 32.

Circumference = π (32) = 32π.

Divided into 8 slices = 1/8 of the circumference = 32π/8 = 4π.

Using trigonometric identities, it has been proven that:

cos²θ cot²θ  = cot²θ  - cos²θ

To prove that a given equation is an identity, what we need to do is to apply known identities to prove that one side of the equation can be transformed into the other.

To prove that an equation is not an identity, we need to find one input at which the two sides of the equation have different values.

The trigonometric equation is given as:

cos²θ cot²θ

We know that:

cos²θ = 1 - sin²θ

cot²θ = cos²θ/sin²θ

Thus, our original expression is now:

(1 - sin²θ) * cos²θ/sin²θ

Expanding the bracket gives us:

(cos²θ/sin²θ) - cos²θ

= cot²θ  - cos²θ

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Answer:

$41.25

Step-by-step explanation:

8.25x5=41.25

The addition of the fractions, 41 2/9 + 18 2/5, is 59 28/45

From the question, we are to add the given fractions by following the given steps

The given fractions are:

41 2/9 + 18 2/5

Writing each mixed number as the sum of a whole and a fraction

41 2/9 = 41 + 2/9

and

18 2/5 = 18 + 2/5

Adding the whole numbers

41 + 18 = 59

Finding the LCD of the fractions 2/9 and 2/5

LCD = 45

Rewriting the sum of the fractions with a common denominator

2/9 + 2/5 = 10/45 + 18/45 = (10 + 18)/45 = 28/45

Thus,

41 2/9 + 18 2/5 = 59 + 28/45

41 2/9 + 18 2/5 = 59 28/45

Hence,

The addition of the fractions results into 59 28/45

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The amount each member needs to pay is $27.50

If 1 in 20 people has their scooters stolen or broken, that means 2 people will need to file a claim for their scooters (1/20 * 40).

We multiply the number of claims by the price of each scooter to determine the total amount of insurance required: 2 * $550 = $1100.

The cost of the insurance will be covered by equal contributions from each of the 40 participants.

Therefore, the amount each member needs to pay is $1100 / 40 = $27.50.

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The two prime numbers that add up to 100005, based on the properties of prime numbers would be 2 and 100003.

To find the two prime numbers, the relevant relationship would be  the Goldbach's Conjecture. This posits that every even number greater than 2 can be expressed as the sum of two prime numbers.

So, to write an odd number as the sum of two prime numbers, one of the prime numbers has to be 2 because we know that 2 is the only even prime number.

The other number would therefore be:

= 100005 - 2

= 100003

100003 is a prime number so the two prime numbers are 2 and 100003.

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The least number of votes necessary from each group to pass the policy is 596 votes from technical managers and 119 votes from administrative managers.

We know that 30% of technical managers vote in favor, which means 0.3x technical managers will vote in favor.

Similarly, 60% of administrative managers vote in favor, which means 0.6y administrative managers will vote in favor.

The total number of managers who vote in favor is given by the sum of technical and administrative managers who vote in favor:

0.3x + 0.6y

We want this total number of votes to be at least 250, so we can write the inequality:

0.3x + 0.6y ≥ 250

Since the total number of managers in the company is 715, we also have the constraint:

x + y ≤ 715

To find the least number of votes from each group necessary to pass the policy, we can solve this system of inequalities.

0.3x + 0.6y ≥ 250 ---(1)

x + y ≤ 715 ---(2)

To simplify equation (1), we can multiply both sides by 10 to remove the decimal:

3x + 6y ≥ 2500

Now, we have the following system:

3x + 6y ≥ 2500 ---(3)

x + y ≤ 715 ---(4)

To solve this system, we can use a method like substitution or elimination.

Let's solve using the elimination method:

Multiply equation (4) by 3:

3x + 3y ≤ 2145 ---(5)

y ≥ 118.33

Since y represents the number of administrative managers, it must be a whole number, so we take the next higher integer, y = 119.

Substitute this value of y into equation (4):

x + 119 ≤ 715

x ≤ 596

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The exponential expression which is equivalent to [tex]8 \sqrt{y}[/tex]  is  [tex]y^\frac{1}{8}[/tex].

Powers can simply be expressed in concise form using exponential expressions. The exponent shows how many times the base has been multiplied. Thus, for the number 32, it may be expressed as 2 2 2 2 25 where 2 is the "base" and 5 is the "exponent". "Two to the fifth power" is how we translate this statement.

The power of a power rule  stress that in a case whereby power is being raised to another power then the exponents can undergo multiplication where the base is been left alone.

Given that [tex]8\sqrt{y}[/tex]

= [tex]y^\frac{1}{8}[/tex]

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Answer: To determine which equation gives the line shown on the graph, we need to examine the slope (m) and y-intercept (b) of the line.

Looking at the given options, we can determine the equation by comparing the slope and y-intercept values with the line on the graph.

The line on the graph passes through the point (0, 1) and (6, 13). We can calculate the slope using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line.

Using the points (0, 1) and (6, 13):

m = (13 - 1) / (6 - 0)

m = 12 / 6

m = 2

Now, let's compare the slope and y-intercept values with the given equation options:

   y = 2x - 4

   y = x + 4

   y = 2x + 4

   y = -x - 4

From the calculation above, we determined that the slope of the line on the graph is 2. Comparing this with the slope values in the equation options, we find that options 1 and 3 have a matching slope of 2.

Next, we can compare the y-intercept of the line on the graph, which is (0, 1), with the y-intercept values in the equation options:

   y = 2x - 4 -> y-intercept: -4

   y = x + 4 -> y-intercept: 4

   y = 2x + 4 -> y-intercept: 4

   y = -x - 4 -> y-intercept: -4

From the comparison, we see that option 3, y = 2x + 4, matches both the slope and y-intercept of the line on the graph.

Therefore, the equation that gives the line shown on the graph is y = 2x + 4.

Step-by-step explanation:

The first term of the sequence is u₁ = 2.

In a geometric sequence, each term is found by multiplying the previous term by a constant ratio. Let's denote the first term as u₁ and the common ratio as r.

Given information:

u₂ = 6

u₆ = 486

We can use the formula for the nth term of a geometric sequence:

uₙ = u₁ x r^(n-1)

Plugging in the values for the second term (n = 2) and the sixth term (n = 6), we get:

u₂ = u₁ x r²⁻¹ = u₁ x r

u₆ = u₁ x r⁶⁻¹ = u₁ x r⁵

We can set up a system of equations with these two equations:

u₂ = 6 --> u₁r = 6 ...(1)

u₆ = 486 --> u₁ r⁵ = 486 ...(2)

Dividing equation (2) by equation (1), we get:

(u₁r⁵) / (u₁r) = (486) / 6

Simplifying the equation:

r⁴ = 81

Taking the fourth root of both sides:

r = √(81)

Therefore,

u₁ = 6/3

u₁ = 2.

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The slope of the line represented by the equation √3y - x - 2√x = 0 is

(8√x + 4)/3.

We have,

To find the slope of the line represented by the equation √3y - x - 2√x = 0, we need to rearrange the equation into the slope-intercept form,

y = mx + b, where m represents the slope.

Let's begin by isolating the term containing y:

√3y = x + 2√x

To solve for y, we square both sides of the equation to eliminate the square root:

(√3y)² = (x + 2√x)²

3y = (x + 2√x)(x + 2√x)

3y = (x² + 4x√x + 4x√x + 4x)

3y = (x² + 8x√x + 4x)

Now, we divide both sides of the equation by 3 to isolate y:

y = (x² + 8x√x + 4x)/3

Now, the equation is in the form y = mx + b, where the slope is represented by the coefficient of x, which is (8√x + 4)/3.

Therefore,

The slope of the line represented by the equation √3y - x - 2√x = 0 is

(8√x + 4)/3.

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The value of the missing terms are;

1. N = 24

2. 144

3. 6

4. 1000

5. 12.75

To determine the value, we have that the proportion is determined by;

1. 4/25 = N/150

cross multiply the values, we have;

25N = 150(4)

multiply the values

25N = 600

N = 24

2. 48/75 = N/225

cross multiply the values

75N = 10800

Divide the values

N = 144

3. N/16 = 30/80

cross multiply the values

80N = 480

N = 6

4. 125/N = 2/16

cross multiply the values

2N = 2000

N = 1000

5. 2.75/11 = N/51

cross multiply the values

11N = 140.25

N = 12.75

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The solution is: the price for the 1st week after Valentine's Day will be $60.

Here, we have,

given that,

For Valentine's Day in February, Gabriel's Appliances stocks hundreds of red heart-shaped vacuum cleaners. The next week Gabriel still has hundreds of them in stock and tells the manager,

"Discount them 25% each week until they are gone."

For the week of Valentine's Day, the vacuums were priced at $80.

so, we get,

The price of red heart-shaped vacuum cleaners for the week of Valentine's Day is $80.

now, as per he told, we get,

the price for the 1st week after Valentine's Day will be with 25% discount.

so, the price for the 1st week after Valentine's Day will be

= $80 - $80*25%

=$80 - 20

=$60

Hence, The solution is: the price for the 1st week after Valentine's Day will be $60.

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The new equation would be:

g(x) = 9x - 7

Option D, g(x) = 9x - 7, represents the equation of the new graph after shifting 7 units down.

If the graph of f(x) = 9x is shifted 7 units down, the equation of the new graph can be obtained by subtracting 7 from the original function. Therefore, the correct equation would be:

g(x) = 9x - 7

Option D, g(x) = 9x - 7, represents the equation of the newly transformed graph after shifting 7 units down.

This equation means that for any x-value, the corresponding y-value on the new graph will be 7 units below the y-value on the original graph.

Options A and B are incorrect because they add 7 to the original function, which would shift the graph 7 units up rather than down.

Option C is incorrect because it suggests a shift to the right or left by 7 units, which is not the required transformation on the graph in this case. The correct transformation is a vertical shift downwards.

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The coordinates of (1/2, y), and the trigonometric ratios of tangents, indicates;

y = [√3]/[7]

The trigonometric ratios expresses the relationships between the angles of a right triangle and two of the sides of the right triangle.

The x-value of the point in the diagram = (1/2)

The length of the radius of the unit circle, r = 1

Therefore; x/r = cos(θ)

Therefore; cos(θ) = (1/2)/1 = (1/2)

Therefore, θ = arccos(1/2) = 60°

The trigonometric ratio SOH CAH TOA, of tangents indicates;

tan(θ) = Opposite/Adjacent

Therefore;

tan(60°) = y/(1/2) = √3

y = (1/2) × √3

y = (√3)/2

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φ is an irrational mathematical constant approximately 1.618 which is known as Phil.

In mathematics, the expression "φ(u, p1, .......pn)" typically represents a function φ that takes a set of parameters p1, p2, ..., pn, and an input value u.

The function φ then performs some operations or computations using these parameters and returns a result based on the input value u. The exact interpretation and definition of the function φ and its parameters would depend on the specific context or mathematical framework in which it is used.

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The exact value of sin(π/8) is:

[tex]sin(\frac{\pi}{8}) = \frac{\sqrt{2-\sqrt{2}} }{2}[/tex]

Trigonometry deals with the relationship between the ratios of the sides of a right-angled triangle with its angles.

Recall that the half angle formula of sine in trigonometry is:

[tex]sin(\frac{A}{2}) = \±\sqrt{\frac{1-cosA}{2} }[/tex]

Thus, we can use the half-angle formula to find the exact value of sin(π/8):

In this case, A = π/8. Thus,

[tex]sin(\frac{\pi}{8}) = \±\sqrt{\frac{1-cos(\frac{\pi}{4} )}{2} }[/tex]

The value of cos(π/4) is √2/2. Thus,

[tex]sin(\frac{\pi}{8}) = \±\sqrt{\frac{1-\frac{\sqrt{2} }{2}}{2} }[/tex]

[tex]sin(\frac{\pi}{8}) = \±\sqrt{\frac{\frac{2-\sqrt{2} }{2}}{2} }[/tex]

[tex]sin(\frac{\pi}{8}) = \±\sqrt{\frac{2-\sqrt{2}}{4}} }[/tex]

[tex]sin(\frac{\pi}{8}) =\± \frac{\sqrt{2-\sqrt{2}} }{2}[/tex]

Since π/8 is in the first quadrant, the sine function will be positive. Therefore, the exact value is:

[tex]sin(\frac{\pi}{8}) = \frac{\sqrt{2-\sqrt{2}} }{2}[/tex]

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The values of x that make the inequality true are (b) 1, (c) 0 and (e) 1 1/4

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

-1/4 < x

This expression can be rewritten as

x > -1/4

This means that the true values of x are values above -1/4

Using the above as a guide, we have the following:

The values of the inequality are (b) 1, (c) 0 and (e) 1 1/4

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Answer: C

Step-by-step explanation:

The characteristics that describe the function correctly would be B. The function is decreasing and has a y - intercept at ( 0, - 2 ).

When looking at a linear function as the one shown in this graph, you can tell when it is increasing if the line is moving such that the value of y is increasing. This means that the slope would be diagonal and trending up.

In this case, the diagonal is trending down so this function is decreasing. This rules out C and D.

The y - intercept is the point where the line crosses the y - intercept and as we can see on the graph, this is at point ( 0, - 2 ).

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Answer:

the watch was 1 year old when David's father lost it.

Step-by-step explanation:

Let's break down the information given:David found the vintage watch 38 years after his father lost it.The watch's date showed that it had worked for over two years after his father lost it.When David found the watch, it was 3 times as old as when his father lost it.

Let's assume the age of the watch when David's father lost it as X years.According to the information, the watch worked for over two years after his father lost it. Therefore, we can represent the age of the watch when David found it as (X + 2) years.We are also given that when David found the watch, it was 3 times as old as when his father lost it. So we can write the equation:(X + 2) = 3XSimplifying the equation:

2 = 3X - X

2 = 2X

X = 1

Answer:

Step-by-step explanation:

multiply by .1  which means move decimal over once

4.8