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3 years ago

What is the equation of the quadratic graph with a focus of (3, -1) and a directrix of y = 1?

Mathematics

1 answer:

nydimaria [60]3 years ago

7 0

Answer:

The equation of the quadratic graph is y = $-\frac{1}{4}$ (x - 3)²

Step-by-step explanation:

The standard form of the equation of the quadratic graph is

(x - h)² = 4p(y - k), where

The vertex of the parabola is (h, k)

The focus is (h, k + p)

The directrix is at y = k - p

∵ The focus is (3, -1)

∵ The focus is (h, k + p)

→ Compare them

∴ h = 3

∴ k + p = -1 ⇒ (1)

∵ The directrix is at y = 1

∵ The directrix is at y = k - p

→ Compare them

∴ k - p = 1 ⇒ (2)

→ Add equations (1) and (2) to eliminate p

∵ (k + k) + (p - p) = (-1 + 1)

∴ 2k + 0 = 0

∴ 2k = 0

→ Divide both sides by 2

∴ K = 0

→ Substitute the value of k in equation (1) to find p

∵ 0 + p = -1

∴ p = -1

→ Substitute the values of h, k, and p in the form of the equation above

∵ (x - 3)² = 4(-1)(y - 0)

∴ (x - 3)² = -4(y)

∴ (x - 3)² = -4y

→ Divide both sides by -4

∴ $-\frac{1}{4}$ (x - 3)² = y

→ Switch the two sides

∴ y = $-\frac{1}{4}$ (x - 3)²

∴ The equation of the quadratic graph is y = $-\frac{1}{4}$ (x - 3)²

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The table below summarizes baseline characteristics on patients participating in a clinical trial

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The best and most correct answer among the choices provided by the question is the fifth choice which is both A and B.

Age and total cholesterol are baseline characteristics are significantly different (at the 0.05 level of significance) between treatment groups.
I hope my answer has come to your help. God bless and have a nice day ahead!

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4 years ago

The port of South Louisiana, located along 54 miles of the Mississippi River between New Orleans and Baton Rouge, is the largest

Ksenya-84 [330]

Answer:

a) 0.7287

b) 0.9663

c) 0.237

d) 3.65 tons of cargo per week or more that will require the port to extend its operating hours.

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 4.5 million tons of cargo per week

Standard Deviation, σ = 0 .82 million tons

We are given that the distribution of number of tons of cargo handled per week is a bell shaped distribution that is a normal distribution.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

a) P( port handles less than 5 million tons of cargo per week)

P(x < 5)

$P( x < 5) = P( z < \displaystyle\frac{5 - 4.5}{0.82}) = P(z < 0.609)$

Calculation the value from standard normal z table, we have,

$P(x < 5) =0.7287= 72.87\%$

b) P( port handles 3 or more million tons of cargo per week)

$P(x \geq 3) = P(z \geq \displaystyle\frac{3-4.5}{0.82}) = P(z \geq −1.82926)\\\\P( z \geq −1.82926) = 1 - P(z < -1.829)$

Calculating the value from the standard normal table we have,

$1 - 0.0337 = 0.9663 = 96.63\%\\P( x \geq 3) = 96.63\%$

c)P( port handles between 3 million and 4 million tons of cargo per week)

$P(3 \leq x \leq 4) = P(\displaystyle\frac{3 - 4.5}{0.82} \leq z \leq \displaystyle\frac{4-4.5}{0.82}) = P(-1.829 \leq z \leq -0.609)\\\\= P(z \leq -0.609) - P(z < -1.829)\\= 0.271-0.034 = 0.237= 23.7\%$

$P(3 \leq x \leq 4) = 23.7\%$

d) P(X=x) = 0.85

We have to find the value of x such that the probability is 0.85.

P(X > x)

$P( X > x) = P( z > \displaystyle\frac{x - 4.5}{0.82})=0.85$

$= 1 -P( z \leq \displaystyle\frac{x - 4.5}{0.82})=0.85$

$=P( z \leq \displaystyle\frac{x - 4.5}{0.82})=0.15$

Calculation the value from standard normal z table, we have,

$P( z \leq -1.036) = 0.15$

$\displaystyle\frac{x - 4.5}{0.82} = -1.036\\x = 3.65$

Thus, 3.65 tons of cargo per week or more that will require the port to extend its operating hours.

8 0

3 years ago

Suppose astronomers built a 50-meter telescope. how much greater would its light-collecting area be than that of the 10-meter ke

drek231 [11]

Answer: Its light-collecting area would be 25 times greater than that of the 10-meter keck telescope.

Step-by-step explanation:

1. To solve this problem you must apply the formula for calculate the area of a circle, which is shown below:

$A=r^{2}\pi$

Where r is the radius of the circle.

2. The diameter of a circle is:

$D=\frac{r}{2}$

Where r is the radius of the circle.

3. Therefore, keeping this on mind, you have that the light-collecting area of a 50-meter keck telescope is:

$A_1=(\frac{50m}{2})^{2}\pi=1963,49m^{2}$

4. And the light-collecting area of a 10-meter keck telescope is:

$A_2=(\frac{10m}{2})^{2}\pi=78.53m^{2}$

5. Divide $A_1$ by $A_2$ , then:

$\frac{1963.49m^{2}}{78.53m^{2}}=25$

6. Therefore, its light-collecting area would be 25 times greater than that of the 10-meter keck telescope.

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4 years ago

Daniel is playing on the swings at the playground. At the top of his swing, he will have _____.

ioda

Answer: all potential energy and no kinetic energy

Step-by-step explanation:

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3 years ago

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The quotient of two numbers is negative. It must be true that? A. Neither number is negative. B. One of the numbers is negative

denpristay [2]

n mathematics, a negative number is a real number that is less than zero. Negative numbers represent opposites. If positive represents movement to the right, negative represents movement to the left. If positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal. They are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, a decrease in some quantity may be thought of as a negative increase. If a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common sense idea of an opposite is reflected in arithmetic. For example, − − 3 = 3 because the opposite of an opposite is the original thing.

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3 years ago