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What is the equation of a parabola with (−2, 4) as its focus and y = 6 as its directrix? Enter the equation in the box.

antiseptic1488 [7]

Check the picture below. So the parabola looks more or less like so.

let's recall that the vertex is half-way between the focus point and the directrix, at "p" units away from both.

Let's notice that the focus point is below the directrix, that means the parabola is vertical, namely the squared variable is the "x", and it also means that it's opening downwards as you see in the picture, namely that "p" is negative, in this case "p" is 1 unit, and thus is -1.

$\bf \textit{parabola vertex form with focus point distance} \\\\ \begin{array}{llll} 4p(x- h)=(y- k)^2 \\\\ \stackrel{\textit{we'll use this one}}{4p(y- k)=(x- h)^2} \end{array} \qquad \begin{array}{llll} vertex\ ( h, k)\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix} \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \begin{cases} h=-2\\ k=5\\ p=-1 \end{cases}\implies 4(-1)(y-5)=[x-(-2)]^2\implies -4(y-5)=(x+2)^2 \\\\\\ y-5=-\cfrac{1}{4}(x+2)^2\implies y=-\cfrac{1}{4}(x+2)^2+5$

6 0

3 years ago

The question is in the screenshot attached. Thank you!

Rzqust [24]

Answer:

Domain: -4 < x < 4

Zeros: (-2, 0), (0, 0) and (2, 0)

The function is positive if: 0 < x < 2

The function is negative if: -4 < x < 0 and 2 < x < 4

Step-by-step explanation:

Domain of the function are those x values where the function is defined, For this case, -4 < x < 4

Zeros of a function are those x values where y = 0, that is, the graph intersect x-axis. For this case, the points are: (-2, 0), (0, 0) and (2, 0)

The function is positive if the graph o the function is above x-axis. For this case, the function is positive at the interval (0, 2)

The function is negative if the graph o the function is below x-axis. For this case, the function is negative at the intervals (-4, 0) and (2, 4)

7 0

3 years ago

9 in. 6 in. the volume of the party closest. In cubic inches

prohojiy [21]

I think it’s 54

Good luckkk

4 0

3 years ago

I need help w/ this!! thank you

Fittoniya [83]

<h3>Answer:</h3>

$\boxed{\pink{\sf \leadsto Yes \ there \ is \ a \ solution \ of \ the \ given \ inequality .}}$

<h3>Step-by-step explanation:</h3>

A inequality is given to us and we need to convert it into standard form and see whether if it has a solution . So let's solve the inequality.

The inequality given to us is :-

$\bf\implies |2y + 3 | - 1 \leq 0 \\\\\bf\implies |2y+3|\leq 1 \\\\\bf\implies (|2y+3|)^2 \leq 1^2 \\\\\bf\implies (2y+3)^2 \leq 1 \\\\\bf\implies (2y)^2+3^2+2(2y)(3) \leq 1 \\\\\bf\implies 4y^2+9+12y - 1 \leq 0 \\\\\bf\implies 4y^2+12y+8 \leq 0 \\\\\bf\implies 4( y^2 + 3y + 2 ) \leq 0 \\\\\bf\implies y^2+3y +2 \leq 0 \:\:\bigg\lgroup \purple{\bf Standard \ form \ of \ inequality }\bigg\rgroup \\\\\bf\implies y^2y+2y+y+2 \leq 0 \\\\\bf\implies y(y+2)+1(y+2)\leq 0 \\\\\bf\implies ( y+2)(y+1)\leq 0 \\\\\bf\implies \boxed{\red{\bf y \leq (-2) , (-1) }}$

Let's plot a graph to see its interval . Graph attached in attachment .

Now we can see that the Interval notation of would be ,

$\boxed{\boxed{\orange \tt \purple{\leadsto}y \in [-2,-1] }}$

<h3>Hence the standard form of inequality is y²+3y +2 ≤ 0 and the Solution set of the inequality is [ -2 , -1 ] .</h3>

4 0

2 years ago

Find the 45th term of the arithmetic sequence: −9, −2, 5, 12, ...

Nat2105 [25]

You’ll get your answer by adding +7 to each number.

3 0

3 years ago

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