1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
shtirl [24]
2 years ago
13

Using a directrix of y = −3 and a focus of (2, 1), what quadratic function is created?

Mathematics
1 answer:
Lelechka [254]2 years ago
5 0

Answer:

  y=\dfrac{1}{8}(x-2)^2-1

Step-by-step explanation:

The distance between the focus and the directrix is the vertical distance from the point to the line:

  1 -(-3) = 4

The vertex is half that distance between the point and the line, so is at x=2 and ...

  y = (-3 +1)/2 = -1

The vertical scale factor of the quadratic is 1/(4p) where p is the distance from vertex to focus. Here, that distance is 2, so the equation in vertex form is ...

  y = (1/(4·2))(x -2)² -1

  y = (1/8)(x -2)² -1

_____

<em>Check</em>

Any point on the parabola is equidistant from the focus and directrix. This is easily checked at the vertex, which is halfway between focus and directrix, and at the points having the same y-value as the focus. Those two points are (-2, 1) and (6, 1), both of which are 4 units from the focus and 4 units from the directrix.

You might be interested in
ILL GIVE BRAINLIEST HELP PLEASE
PilotLPTM [1.2K]

The solution to the system of equation is (1, 4).

In order to find this, we can first just see where the graphs intersect each other. This will give us the solution set.

As for what it represents, the x value in the increase in temperature and the y value is the increase in customers.

Therefore, we know that we want the temperature to go up by 1 (although we don't know the units) and that would result in the amount of people coming, and staying longer by 4 (again, we don't know the units of measure).

8 0
2 years ago
Four students were discussing how to find the unit rate for a proportional relationship. Which method is valid?
BigorU [14]

Answer:

A

Step-by-step explanation:

Look at the graph of the relationship. Find the y-value of the point that corresponds to x = 1. That value is the unit rate

7 0
3 years ago
Read 2 more answers
Prove that if n is a perfect square then n + 2 is not a perfect square
notka56 [123]

Answer:

This statement can be proven by contradiction for n \in \mathbb{N} (including the case where n = 0.)

\text{Let $n \in \mathbb{N}$ be a perfect square}.

\textbf{Case 1.} ~ \text{n = 0}:

\text{$n + 2 = 2$, which isn't a perfect square}.

\text{Claim verified for $n = 0$}.

\textbf{Case 2.} ~ \text{$n \in \mathbb{N}$ and $n \ne 0$. Hence $n \ge 1$}.

\text{Assume that $n$ is a perfect square}.

\text{$\iff$ $\exists$ $a \in \mathbb{N}$ s.t. $a^2 = n$}.

\text{Assume $\textit{by contradiction}$ that $(n + 2)$ is a perfect square}.

\text{$\iff$ $\exists$ $b \in \mathbb{N}$ s.t. $b^2 = n + 2$}.

\text{$n + 2 > n > 0$ $\implies$ $b = \sqrt{n + 2} > \sqrt{n} = a$}.

\text{$a,\, b \in \mathbb{N} \subset \mathbb{Z}$ $\implies b - a = b + (- a) \in \mathbb{Z}$}.

\text{$b > a \implies b - a > 0$. Therefore, $b - a \ge 1$}.

\text{$\implies b \ge a + 1$}.

\text{$\implies n+ 2 = b^2 \ge (a + 1)^2= a^2 + 2\, a + 1 = n + 2\, a + 1$}.

\text{$\iff 1 \ge 2\,a $}.

\text{$\displaystyle \iff a \le \frac{1}{2}$}.

\text{Contradiction (with the assumption that $a \ge 1$)}.

\text{Hence the original claim is verified for $n \in \mathbb{N}\backslash\{0\}$}.

\text{Hence the claim is true for all $n \in \mathbb{N}$}.

Step-by-step explanation:

Assume that the natural number n \in \mathbb{N} is a perfect square. Then, (by the definition of perfect squares) there should exist a natural number a (a \in \mathbb{N}) such that a^2 = n.

Assume by contradiction that n + 2 is indeed a perfect square. Then there should exist another natural number b \in \mathbb{N} such that b^2 = (n + 2).

Note, that since (n + 2) > n \ge 0, \sqrt{n + 2} > \sqrt{n}. Since b = \sqrt{n + 2} while a = \sqrt{n}, one can conclude that b > a.

Keep in mind that both a and b are natural numbers. The minimum separation between two natural numbers is 1. In other words, if b > a, then it must be true that b \ge a + 1.

Take the square of both sides, and the inequality should still be true. (To do so, start by multiplying both sides by (a + 1) and use the fact that b \ge a + 1 to make the left-hand side b^2.)

b^2 \ge (a + 1)^2.

Expand the right-hand side using the binomial theorem:

(a + 1)^2 = a^2 + 2\,a + 1.

b^2 \ge a^2 + 2\,a + 1.

However, recall that it was assumed that a^2 = n and b^2 = n + 2. Therefore,

\underbrace{b^2}_{=n + 2)} \ge \underbrace{a^2}_{=n} + 2\,a + 1.

n + 2 \ge n + 2\, a + 1.

Subtract n + 1 from both sides of the inequality:

1 \ge 2\, a.

\displaystyle a \le \frac{1}{2} = 0.5.

Recall that a was assumed to be a natural number. In other words, a \ge 0 and a must be an integer. Hence, the only possible value of a would be 0.

Since a could be equal 0, there's not yet a valid contradiction. To produce the contradiction and complete the proof, it would be necessary to show that a = 0 just won't work as in the assumption.

If indeed a = 0, then n = a^2 = 0. n + 2 = 2, which isn't a perfect square. That contradicts the assumption that if n = 0 is a perfect square, n + 2 = 2 would be a perfect square. Hence, by contradiction, one can conclude that

\text{if $n$ is a perfect square, then $n + 2$ is not a perfect square.}.

Note that to produce a more well-rounded proof, it would likely be helpful to go back to the beginning of the proof, and show that n \ne 0. Then one can assume without loss of generality that n \ne 0. In that case, the fact that \displaystyle a \le \frac{1}{2} is good enough to count as a contradiction.

7 0
3 years ago
a store has 300 televisions on order, and 80% are high definition. how many televisions on order are HD?
Korolek [52]
240 Televisions are on HD.

8 0
2 years ago
Read 2 more answers
Explain please
TiliK225 [7]

Answer:

n=150

Step-by-step explanation:

(-17+n/5)-13=0

-17= -17/1 = -17(5)/5

-17(5)+n/5 = n-85/5

(n-85)/5-13=0

13=13/1 =13(5)/5

(n-85)-(13(5))/5 =n-150/5

(n-150)/5=0

n-150=0

n=150

7 0
2 years ago
Other questions:
  • City Lake Waterpark has many attractions, one of which is the Super Slide. Are the people who ride the Super Slide the populatio
    14·1 answer
  • Sean b= 5,79; c= 10,4,el angulo A= 54,46°, el angulo C mide ?
    12·1 answer
  • X2-2xy+y2=30 solve the equation
    5·1 answer
  • Find the radian measure of an angle of 110º.
    8·2 answers
  • The ratio of girls to boys in the geography bee was 5 to 4. There were 36 total students. How many were boys?
    14·1 answer
  • In the figure below, segment CD is parallel to segment EF and point H bisects segment DE :
    12·1 answer
  • Factored form of x^2-3x-4
    13·2 answers
  • Please help me with this question
    12·1 answer
  • Help? please- Can someone help me solve for x. Like you don't need to tell me the answer just how to. It's just a little hard fo
    6·2 answers
  • A shopkeeper had 3sacks of sugar each weighing a mass of 90kg.he repacked the sugar in 0.00025tonnes packets.how many packets di
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!