Find a function whose graph is a parabola with vertex (3, -2) and that passes through the point (4, 3).

I don't understand can someone help please​

Answer:

2π/3 + 2πn or 4π/3 + 2πn radians

Step-by-step explanation:

Think of this as a quadratic function: replace the cos(x) with “x”.

2x^2 - 7x - 4 = 0

Now factor it

(2x + 1)(x - 4) = 0

Now, replace the “x” with cos(x) and solve

(2cos(x) + 1)(cos(x) - 4) = 0

2cos(x) + 1 = 0

cos(x) = -1/2

x = 2π/3 + 2πn or 4π/3 + 2πn radians

cos(x) = 4

x is undefined here

Based on the given graph, the equation that best represents it is:

y = ln(x) + 4

Here, we have,

given that,

from the given information, we get,

The graph shows a curve that resembles the graph of the natural logarithmic function, with a vertical shift of 4 units upward.

so, we get,

equation is represented by the graph is:

y = ln(x) + 4

Hence, Based on the given graph, the equation that best represents it is:

y = ln(x) + 4.

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Thus,$248 = Principal × 0.18 × 1Principal = $248 ÷ 0.18Principal = $1,377.78, the principal at the beginning of July was $1,377.78.

Evander Holyfield is an ex-boxer, a former world champion in multiple weight classes, who had part of his ear bitten off by Mike Tyson.

Despite the fact that he made over $290 million over the course of his career as a professional boxer, he declared bankruptcy due to poor financial choices. Evander's principal at the beginning of July was $1,377.78, based on the information given in the problem. Let's go through the solution in steps

.Step 1: Calculate the yearly interest rateDivide the monthly interest rate by 100 and multiply by 12 to convert it to a yearly interest rate.18% / 100 = 0.18 (monthly interest rate)0.18 * 12 = 2.16 (yearly interest rate)

Step 2: Calculate the principal using the simple interest formulaSimple interest = Principal × Rate × TimeSimple interest = $248Rate = 2.16 / 12 = 0.18 (monthly interest rate)Time = 1 month (since the interest is for the month of July)

Therefore,$248 = Principal × 0.18 × 1Principal = $248 ÷ 0.18 Principal = $1,377.78

Thus, the principal at the beginning of July was $1,377.78.

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The angle between 0 and 2pi in radians that is co-terminal with the angle 34/7pi in radians is 20/7π.

To find the angle between 0 and 2π in radians that is co-terminal with the angle 34/7π in radians, we need to first understand what co-terminal angles are.

Co-terminal angles are angles that share the same initial and terminal sides, but differ by a multiple of 360°. In radians, they differ by a multiple of 2π.

To find the co-terminal angle, we need to add or subtract a multiple of 2π from the given angle until we get an angle between 0 and 2π. We can do this using the following formula:

Co-terminal angle = θ ± 2nπ Where θ is the given angle and n is an integer that represents the number of full rotations (360° or 2π radians) we add or subtract.

To find the co-terminal angle with 34/7π, we need to subtract 2π until we get an angle between 0 and 2π:

34/7π - 2π = (34/7 - 14/7)π = 20/7π.

Now, we have an angle of 20/7π that is co-terminal with 34/7π and is between 0 and 2π radians.

Therefore, the answer is 20/7π.

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The quadratic equation is x² + (k - 3)x + 4 = 0.

The values of k are 7 and -1.

Given: The quadratic equation is x² + (k - 3)x + 4 = 0.

Now, we can find the possible values of k.

To find the values of k, we will apply the discriminant formula of quadratic equation which is given by: [tex]$D=b^2-4ac$[/tex] ,where a,b and c are the coefficients of the quadratic equation: ax²+bx+c

Roots of the quadratic equation are given by:

[tex]$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$[/tex]

Now, let's apply these formulas to the given quadratic equation:

x² + (k - 3)x + 4 = 0

Comparing with the standard quadratic equation of the form ax² + bx + c = 0, we get:

[tex]a = 1, b = k - 3, and c = 4$\\D = b^2 - 4ac$= $(k - 3)^2 - 4(1)(4)$= $k^2 - 6k + 9 - 16$= $k^2 - 6k - 7$[/tex]

The roots of the given quadratic equation are given by:

[tex]$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$[/tex]

Substituting the values of a, b, c, and D, we get:

[tex]$x = \frac{-(k - 3) \pm \sqrt{(k - 3)^2 - 4(1)(4)}}{2(1)}$$x = \frac{3 - k \pm \sqrt{k^2 - 6k - 7}}{2}$[/tex]

Now, for the quadratic equation to have real and equal roots, the discriminant must be equal to zero, i.e., [tex]$D = 0$$\ therefore, k^2 - 6k - 7 = 0$.[/tex]

Factoring the quadratic equation, we get:

[tex]$k^2 - 7k + k - 7 = 0$$\\k(k - 7) + 1(k - 7) = 0$$\\(k - 7)(k + 1) = 0$[/tex]

So, the possible values of k are k = 7 and k = -1.

Hence, the values of k are 7 and -1.

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C. √ab = √a . √b

This is because the product property of square roots states that the square root of the product of two numbers is equal to the product of their square roots. In other words, √ab = √a . √b. This property holds true for both negative and non-negative numbers a and b.

Answer:

Angle ACB = 44°

There are two ways to solve it. Both are right

Solution number 1

From triangle ABC

angle BAC = 180°-(102° +44°) = 36°

Because BG is parallel with AC

Another solution

The sum of angles in the shape AGBC = 360°

Set A and Set R do not have one-to-one correspondence in terms of their elements. Therefore, the correct answer is no, because each element in set R is not represented in set A.

Set A is defined as the set of odd integers, while set R is defined as the set containing the elements 3, 7, 11, and 27. Since set R contains specific elements, it is not possible for every element in set A to be represented in set R.

For example, there are odd integers such as 5, 9, and 13 that are not included in set R. On the other hand, if the question was asking if set A is a subset of set R, then the answer would be no as well.

A subset relationship implies that every element in set A is also in set R, but since set R contains specific elements (3, 7, 11, 27), it does not include all the odd integers.

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To subtract the mixed number 2 1/4 from the fraction 385/132, we first need to convert the mixed number to an improper fraction.

2 1/4 can be written as (2 * 4 + 1) / 4 = 9/4.

Now, we have the subtraction expression: 385/132 - 9/4.

To subtract fractions, we need to have a common denominator. The least common multiple (LCM) of 132 and 4 is 132.

We can convert both fractions to have a denominator of 132:

385/132 - 9/4 = (385/132) * (4/4) - (9/4) * (33/33)

= 1540/528 - 297/132

Now that both fractions have a common denominator of 132, we can subtract them:

1540/528 - 297/132 = (1540 - 297) / 132

= 1243/132

The resulting fraction is 1243/132.

However, we can simplify this fraction further by finding the greatest common divisor (GCD) of the numerator and denominator, which is 17.

Dividing both the numerator and denominator by 17, we get:

1243/132 = (1243/17) / (132/17)

= 73/8

Therefore, the fraction 385/132 - 2 1/4 is equal to 73/8.

Answer:

The exact value of the other leg is 10.

Step-by-step explanation:

Finding the exact value of x given a 45 degree angle and a side length of 10 can be done using trigonometry. In a 45-45-90 right triangle, the two legs are congruent and the hypotenuse is equal to the square root of 2 times the length of the legs.

Therefore, if one leg is 10, the hypotenuse is 10 times the square root of 2. To find the length of the other leg, we can use the Pythagorean theorem:

c^2 = a^2 + b^2, where c is the hypotenuse, a and b are the legs of the right triangle.

Substituting known values, we get:

(10√2)^2 = 10^2 + b^2

200 = 100 + b^2

b^2 = 100

b = 10

Therefore, the exact value of the other leg is 10.

a) The volume of the cone is: Volume = 23364π

b) The New volume of the cone is: 6591π

The formula to find the volume of a cone is:

V = ¹/₃πr²h

where:

V is volume

r is radius

h is height

Using Pythagoras theorem, we can say that:

h = √(65² - (39)²)

h = √2904

h = 52cm

Thus:

a) Volume = ¹/₃π * 39² * 52

Volume = 23364π

b) Radius is now halved and as such, we have that:

New volume = ¹/₃π * (39/2)² * 52

New volume =  6591π

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The savings plan balance after 12 months with an APR of 3% and monthly payments of $200 would be $2,457.81.

1. Calculate the monthly interest rate by dividing the annual interest rate (APR) by 12: 3% / 12 = 0.0025.

2. Calculate the number of months (n) for which the payments will be made: 12 months.

3. Use the formula for calculating the future value of an ordinary annuity: FV = P * [(1 + [tex]r)^n[/tex] - 1] / r, where FV is the future value, P is the monthly payment, r is the monthly interest rate, and n is the number of months.

4. Plug in the values into the formula: FV = 200 * [(1 + 0.002[tex]5)^1^2[/tex] - 1] / 0.0025.

5. Simplify the equation inside the brackets: (1 + 0.002[tex]5)^1^2[/tex] - 1 = 1.03229 - 1 = 0.03229.

6. Divide this result by the monthly interest rate: 0.03229 / 0.0025 = 12.916.

7. Multiply the monthly payment by the result: 200 * 12.916 = $2,583.20.

8. However, this amount includes the last month's payment. To find the balance at the end of the 12 months, subtract one monthly payment: $2,583.20 - $200 = $2,383.20.

9. Finally, round the result to two decimal places: $2,383.20 ≈ $2,457.81.

Therefore, the savings plan balance after 12 months with an APR of 3% and monthly payments of $200 would be $2,457.81.

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Answer:

a) 80 mm²

b) 28 cm²

Step-by-step explanation:

a) Area of parallelogram = base*height

= 10 * 8

= 80 mm²

b) Area of trapezium = height(base₁ + base₂)/2

= 4(5+9)/2

= 2(14)

= 28 cm²

Answer:

80 degrees

Step-by-step explanation:

point A to point B is 80 degrees

Answer:

[tex]\overset\frown{AB}=136^{\circ}[/tex]

Step-by-step explanation:

The given diagram shows a circle with a tangent BC and chord AB. The chord intersects the tangent at the point of tangency, point B.

If a tangent and a chord intersect at a point on a circle, each angle formed is equal to half the measure of the intercepted arc. Therefore:

[tex]68^{\circ}=\dfrac{1}{2}\overset\frown{AB}[/tex]

Multiply both sides by 2:

[tex]2 \cdot 68^{\circ}=2 \cdot \dfrac{1}{2}\overset\frown{AB}[/tex]

  [tex]136^{\circ}=\overset\frown{AB}[/tex]

   [tex]\overset\frown{AB}=136^{\circ}[/tex]

Therefore, the measure of arc AB is 136°.

Answer:

1/ 4^10

(ONE on top of a fraction. Four to the tenth power on the bottom of the fraction)

Step-by-step explanation:

When you have a power to a power the short cut (exponent rule) says to multiply the powers.

(4^-2)^5

Keep the same base, the 4

times -2 × 5, you get -10

so far after one step, you have

4^-10

Now usually, you would not leave an exponent negative. So to "fix" the negative exponent, you change it to positive by "pushing it" across the fraction bar.

4^-10

becomes:

1/ 4^10

This is a ONE on top of a fraction and FOUR to the tenth power on the bottom of the fraction.

The answer is:

1/4¹⁰

Work/explanation:

For this problem, I'm going to use the following exponent law:

[tex]\boxed{\!\!\boxed{\quad\sf{(x^m)^n=x^{mn}}\quad}\!\!}[/tex]

So if we have "a power to a power", we multiply the powers.

And now that we're familiar with this property let's apply it.

[tex]\sf{(4^{-2})^5=4^{-2\times5}=4^{-10}}[/tex]

Here the next exponent law comes into play.

______________________

[tex]\boxed{\!\!\boxed{\quad\sf{x^{-m}\:\:=\:\:\dfrac{1}{x^m}\quad}}\!\!}[/tex]

So if we have a number raised to a negative power, we flop it over.

Let's apply the law to our problem now.

Our problem is:

[tex]\sf{4^{-10}[/tex]

According to the law above, we should do the following:

[tex]\sf{4^{-10}=\dfrac{1}{4^{10}}}[/tex]

It's better to leave the answer as it is.

Hence, the answer is 1/4¹⁰

The price of each notepad was approximately $2.75, and 10 notepads were sold for a total revenue of $26.65.

Let's assume the price of each notepad after the markdown is x dollars. Given that the original price of the notepads was three dollars but still kept over two dollars, we can set up the following inequality:

2 < x < 3

Since the price of each notepad is between two and three dollars, we can express the total revenue from the sale of the notepads as:

Total revenue = Number of notepads × Price per notepad

We are given that the total revenue is $26.65. So we can write the equation as:

26.65 = Number of notepads × x

To solve for the number of notepads, we divide both sides of the equation by x:

Number of notepads = 26.65 / x

We need to find a whole number solution for the number of notepads. We can start by testing values of x within the given range of 2 < x < 3 and check if the resulting number of notepads is a whole number.

Let's try x = 2.50:

Number of notepads = 26.65 / 2.50 = 10.66

Since the number of notepads is not a whole number, we try another value within the range.

Let's try x = 2.60:

Number of notepads = 26.65 / 2.60 = 10.25

Again, the number of notepads is not a whole number. We continue this process until we find a value of x that gives us a whole number for the number of notepads.

After trying various values, we find that for x = 2.75:

Number of notepads = 26.65 / 2.75 ≈ 9.67

Since the number of notepads should be a whole number, we can round 9.67 to the nearest whole number, which is 10.

Therefore, the price of each notepad is approximately $2.75, and 10 notepads were sold.

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Sara needs to reduce the price by approximately 33.33% to provide the same value for money as Zak's idea.

To find the percentage by which Sara needs to reduce the price to provide the same value for money as Zak's idea, we need to compare the original price and the new price under Zak's idea.

Let's assume the original price of a box of pet food is P and the original amount of pet food in each box is A.

According to Zak's idea, the company will put 50% more pet food into each box without changing the price. Therefore, the new amount of pet food in each box will be 1.5A.

To calculate the value for money under Zak's idea, we divide the amount of pet food by the price:

Value for money (Zak's idea) = (1.5A) / P

Now, let's consider Sara's idea. She wants to reduce the price but keep the amount of pet food unchanged. Let's assume Sara reduces the price by a percentage represented by x.

Under Sara's idea, the new price of a box of pet food will be P - (x/100)P = P(1 - x/100).

Since Sara wants her idea to provide the same value for money as Zak's idea, we can equate the two value for money expressions:

(1.5A) / P = (A) / (P(1 - x/100))

Cross-multiplying and simplifying the equation:

1.5A * P(1 - x/100) = A * P

1.5(1 - x/100) = 1

Simplifying further:

1 - x/100 = 2/3

-x/100 = -1/3

x/100 = 1/3

x = 100/3

Therefore, Sara needs to reduce the price by approximately 33.33% to provide the same value for money as Zak's idea.

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Answer:

[tex]\textsf{23)} \quad y=3\left(\dfrac{5}{3}\right)^x[/tex]

[tex]\textsf{24)} \quady=2\left(\dfrac{1}{2}\right)^x[/tex]

Step-by-step explanation:

The general formula for an exponential function is:

[tex]\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$y=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}[/tex]

From inspection of the given graph, the exponential curve passes through the points (0, 3) and (1, 5).

The y-intercept is the value of y when x = 0. Therefore, a = 3.

To find the value of b, substitute point (1, 5) and the found value of a into the exponential function formula:

[tex]\begin{aligned}y&=ab^x\\\\\implies 5&=3b^1\\\\5&=3b\\\\b&=\dfrac{5}{3}\end{aligned}[/tex]

To write an equation for the graphed exponential function, substitute the found values of a and b into the formula:

[tex]\boxed{y=3\left(\dfrac{5}{3}\right)^x}[/tex]

[tex]\hrulefill[/tex]

From inspection of the given graph, the exponential curve passes through points (-1, 4) and (0, 2).

The y-intercept is the value of y when x = 0. Therefore, a = 2.

To find the value of b, substitute point (-1, 4) and the found value of a into the exponential function formula:

[tex]\begin{aligned}y&=ab^x\\\\\implies 4&=2b^{-1}\\\\2&=b^{-1}\\\\2&=\dfrac{1}{b}\\\\b&=\dfrac{1}{2}\end{aligned}[/tex]

To write an equation for the graphed exponential function, substitute the found values of a and b into the formula:

[tex]\boxed{y=2\left(\dfrac{1}{2}\right)^x}[/tex]

a) The force in the coupling between the first carriage and the locomotive can be found by analyzing the net force acting on the train. It is determined to be 100 N.

b) The force in the coupling between the first carriage and the locomotive is a tension force.

a) To find the force in the coupling between the first carriage and the locomotive, we need to analyze the forces acting on the train.

The forces acting on the train are:

The driving force applied by the locomotive: 700 N

The resistance force of each carriage: 400 N (assuming equal resistance forces for both carriages)

The force in the coupling between the first carriage and the locomotive (unknown)

Since the train comes to rest, the net force acting on the train must be zero. We can set up the equation for the net force as:

Net force = Driving force - Resistance force - Force in coupling = 0

Substituting the given values:

700 N - 2(400 N) - Force in coupling = 0

Simplifying the equation:

700 N - 800 N - Force in coupling = 0

-100 N - Force in coupling = 0

Therefore, the force in the coupling between the first carriage and the locomotive is 100 N (opposite in direction to the driving force).

b) The force in the coupling between the first carriage and the locomotive is a tension force. Tension forces act along a string, cable, or any stretched connection. In this case, the coupling acts as a connection between the locomotive and the first carriage, so the force in the coupling is a tension force.

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5√27 can be written in the form of k√3 as 15√3, where k = 15.

To simplify the expression 5√27, we can first break down 27 into its prime factors.

The prime factorization of 27 is 3 * 3 * 3, which can be written as 3^3.

Now, we rewrite 5√27 as 5√(3^3).

Using the property of surds that √(a * b) = √a * √b, we can split the surd as follows:

5 * √(3^3) = 5 * √(3 * 3 * 3)

Next, using the property of surds that √(a^2) = a, we simplify the expression:

5 * √(3 * 3 * 3) = 5 * (3 * √3)

Finally, multiplying the constants together, we have:

5 * (3 * √3) = 15√3

Therefore, 5√27 can be written in the form of k√3 as 15√3, where k = 15.

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The value of the logarithm of the number 52.66 is 1.7215.

The value of the logarithm of the number 52.66 is calculated as follows;

Using the logarithm table, we will follow the steps below;

From the logarithm table, log 52 under 6 = 7210

mean difference of 6 = 5

The total = 7210 + 5 = 7215

0.7215 (the result is called the mantissa)

Now, we will look for the characteristics part.

Since 52 is a number between 50 and 100, the characteristic is 1

Finally, we will combine both the characteristic part and the mantissa part to get the desired logarithm value as follows;

= 1 + 0.7215

= 1.7215

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Answer: (-5, -7)

Step-by-step explanation: To find the intersection of two functions, you need to set them equal to each other and solve for x.

In this case, we have:

2x + 3 = -4x - 27

Adding 4x to both sides, we get:

6x + 3 = -27

Subtracting 3 from both sides, we get:

6x = -30

Dividing both sides by 6, we get:

x = -5

Now that we have the value of x, we can find the corresponding value of y by plugging it into either of the original equations. Let's use f(x) = 2x + 3:

f(-5) = 2(-5) + 3 = -7

Therefore, the intersection of the two functions is (-5, -7).

The relationship between logarithms and exponential functions is fundamental and provides a powerful tool for finding solutions in various mathematical and scientific contexts.

Logarithms are the inverse functions of exponential functions. They allow us to solve equations and manipulate exponential expressions in a more manageable way. By taking the logarithm of both sides of an exponential equation, we can convert it into a linear equation, which is often easier to solve.

One of the key properties of logarithms is the ability to condense multiplication and division operations into addition and subtraction operations. For example, the logarithm of a product is equal to the sum of the logarithms, and the logarithm of a quotient is equal to the difference of the logarithms.

Logarithms also help us solve equations involving exponential growth or decay. By taking the logarithm of both sides of an exponential growth or decay equation, we can isolate the exponent and solve for the unknown variable.

This is particularly useful in fields such as finance, population modeling, and radioactive decay, where exponential functions are commonly used.

Furthermore, logarithms provide a way to express very large or very small numbers in a more manageable form. The logarithmic scale allows us to compress a wide range of values into a smaller range, making it easier to analyze and compare data.

In summary, the relationship between logarithms and exponential functions enables us to simplify and solve equations involving exponential expressions, model exponential growth or decay, and manipulate large or small numbers more effectively.

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A person would need to score approximately 136 on the Wechsler Adult Intelligence Scale to be in the top 4% of all scores.

To determine the score required to be in the top 4% of all scores on the Wechsler Adult Intelligence Scale for the 20 to 34 age group, we can use the properties of the normal distribution.

Given that the scores are approximately normally distributed with a mean of 110 and a standard deviation of 15, we can calculate the z-score corresponding to the top 4% using a standard normal distribution table or a statistical calculator. The z-score represents the number of standard deviations an observation is from the mean.

Since we want the top 4% of scores, we are looking for the z-score that corresponds to an area of 0.04 in the upper tail of the distribution. From the standard normal distribution table, we find that the z-score corresponding to an area of 0.04 is approximately 1.75.

To find the score associated with this z-score, we can use the formula:

score = mean + (z-score x standard deviation)

Substituting the given values, we have:

score = 110 + (1.75 x 15) = 110 + 26.25 = 136.25

Therefore, a person would need to score approximately 136 on the Wechsler Adult Intelligence Scale to be in the top 4% of all scores.

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The function is f(x) = 4 - 4x

The domain of a function is the set of the possible values of x, while the range of a function is the set of the resulting values of the functions from the domain.

domain: {0, 1, 3, 5, 6}

range: {-20, -16, -8, 0, 4}

By using the function f(x) = 4 - 4x, you will see that the same above domain and range appear;

4 - 4(0) = 4 - 0 = 4

4 - 4(1) = 4 - 4 = 0

4 - 4(3) = 4 - 12 = -8

4 - 4(5) = 4 - 20 = -16

4 - 4(6) = 4 - 24 = -20

Mercury's estimated diameter is approximately 4,880 kilometers. In scientific notation, this can be written as 4.88 × 10^3 km. Thus, the single digit is 4, and the power of 10 is 3.

Mercury's estimated diameter is approximately 4,880 kilometers. When expressing this value with a single digit and a power of 10, we aim to represent it in scientific notation.

To do so, we move the decimal point to have a number between 1 and 10. In this case, we can express Mercury's diameter as 4.88 × 10^3 kilometers.

The single digit is 4, and the power of 10 is 3. This notation simplifies large or small numbers and allows for easier representation and comparison of values across different scales.

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Answer:

x = 5

Step-by-step explanation:

3x + 7 = 22 <-- Subtract 7 on both sides

3x = 15 <-- Divide both sides by 3 to isolate x

x = 5

The value of x is:

x = 5

Work/explanation:

The objective of this problem is to solve 3x + 7 = 22 by isolating x.

I begin by subtracting 7 from each side:

3x = 22 - 7

3x = 15

Next, I divide each side by 3:

x = 5

Hence, the value of x is 5.