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3 years ago

Find an equation in standard form of the parabola passing through these points: (0,-6),(-1,-12),(3,0)

Mathematics

1 answer:

Elan Coil [88]3 years ago

3 0

One form of the equation of a parabola is
y = ax² + bx + c

The curve passes through (0,-6), (-1,-12) and (3,0). Therefore
c = - 6 (1)
a - b + c = -12 (2)
9a + 3b + c = 0 (3)

Substitute (1) into (2) and into (3).
a - b -6 = -12
a - b = -6 (4)
9a + 3b - 6 = 0
9a + 3b = 6 (5)

Substitute a = b - 6 from (4) into (5).
9(b - 6) + 3b = 6
12b - 54 = 6
12b = 60
b = 5
a = b - 6 = -1

The equation is
y = -x² + 5x - 6

Let us use completing the square to write the equation in standard form for a parabola.
y = -[x² - 5x] - 6
= -[ (x - 2.5)² - 2.5²] - 6
= -(x - 2.5)² + 6.25 - 6

y = -(x - 2.5)² + 0.25
This is the standdard form of the equation for the parabola.
The vertex us at (2.5, 0.25).
The axis of symmetry is x = 2.5
Because the leading coefficient is -1 (negative), the curve opens downward.
The graph is shown below.

Answer: y = -(x - 2.5)² + 0.25

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OC and OR are similar. Find the length of QP.

Elena L [17]

Given:

Circle C and circle R are similar.

The length of arc AB is $s = \frac{22 \pi}{9}$

The radius of circle C (AC) = 4 unit

The radius of circle R (QR) =6 unit

To find the length of arc QP.

Formula

The relation between s, r and $\theta$ is

$arclength = 2\pi r \frac{\theta}{360}$

where,

s be the length of the arc

r be the radius

$\theta$ be the angle.

Now,

For circle C

Taking r = 4

According to the problem,

$2 \pi r \frac{\theta}{360} = \frac{22 \pi}{9}$

or, $2r \frac{\theta}{360} = \frac{22}{9}$ [ eliminating $\theta$ from both side]

or, $\theta = \frac{(22)(360)}{(9)(2)(4)}$

or, $\theta = 110^\circ$

Again,

For circle R

Taking, r = 6 and $\theta = 110^\circ$ we get,

The length of arc QP is

$arc length = 2\pi (6)(\frac{110}{360} )$

or, $arclength = \frac{11 \pi}{3}$

Hence,

The length of QP is $\frac{11 \pi}{3}$ . Option C.

5 0

3 years ago

Kellie is playing with a toy building block that has the shape of a rectangular prism.

Lena [83]

I think the volume is 200

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3 years ago

An electronics company produces transistors, resistors, and computer chips. Each transistor requires 3 units of copper, 1 unit

padilas [110]

Answer:

An electronics company can be produce 350 transistors and 340 computer chips, they can´t produce resistors.

Step-by-step explanation:

1. We will name the variables for transistors, resistors and the computer chips.

a = Transistors

b= Resistors

c = Computer chips

2. We propose three linear equations, one for the copper, one for the zinc and one for the glass.

$\left \{ {{3a+3b+2c=1730} \atop {a+2b+c=690}}\atop {2a+b+2c=1380}} \right.$

3. We write the matrix form as Ax=d

$A=\left(\begin{array}{ccc}3&3&2\\1&2&1\\2&1&2\end{array}\right)$

$x=\left(\begin{array}{ccc}a\\b\\c\end{array}\right)$

$A=\left(\begin{array}{ccc}1730\\690\\1380\end{array}\right)$

With this formula the solution of x is $x=\frac{d}{A}$ or $x=A^{-1}d$

4. We will find the inverse matrix $A^{-1}$ using the formula:

$A^{-1} = \frac{1}{detA} (C_{A})^{T}$

a. det A

$det A=\left[\begin{array}{ccc}3&3&2\\1&2&1\\2&1&2\end{array}\right] =3*(4-1)-3*(2-2)+2*(1-4)=9-0-6=3$

b. $(C_{A})^{T}$

$C_{A}=\left(\begin{array}{ccc}4-1&.(2-2)&1-4\\-(6-2)&6-4&-(3-6)\\3-4&-(3-2)&6-3\end{array}\right)$

$C_{A}=\left(\begin{array}{ccc}3&.0&-3\\-4&2&3\\-1&-1&3\end{array}\right)$

$(C_{A}) ^T=\left(\begin{array}{ccc}3&0&-3\\-4&2&3\\-1&-1&3\end{array}\right)^T$

$(C_{A}) ^T=\left(\begin{array}{ccc}3&-4&-1\\0&2&-1\\-3&3&3\end{array}\right)$

c. $A^{-1}$

$A^{-1}=\frac{1}{3} \left(\begin{array}{ccc}3&-4&-1\\0&2&-1\\-3&3&3\end{array}\right)$

5. As $x=\frac{d}{A}$ or $x=A^{-1}d$ , the solution of x is:

$x=\frac{1}{3}\left(\begin{array}{ccc}3&-4&-1\\0&2&-1\\-3&3&3\end{array}\right)\left(\begin{array}{ccc}1730\\690\\1380\end{array}\right)$

$x=\frac{1}{3}\left(\begin{array}{ccc}(3*1730)+(-4*690)+(-1*1380)\\(0*1730)+(2*690)+(-1*1380)\\(-3*1730)+(3*690)+(3*1380)\end{array}\right)$

$x=\frac{1}{3}\left(\begin{array}{ccc}1050\\0\\1020)\end{array}\right)$

$X=\left[\begin{array}{ccc}350\\0\\340\end{array}\right]$

Therefore:

a= 350 Transistors

b=0 Resistors

c=340 Computer chips

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2 years ago

Which statement about 6x2 + 7x – 10 is true?

amm1812

Answer:

(x+2)

Step-by-step explanation:

factor out the equation and get

(x+2)(6x-5)

7 0

3 years ago

Read 2 more answers

When looking for a job, Micah is told that the monthly salary has a range of $876. He also knows that the starting salary is $1,

blondinia [14]

This statement describes a range of Micah's salary.

The range is a number that describes the distance from the beginning of something to the end of something.

In this scenario the beginning would be the beginning salary, in the end salary would be where you stop at end at after adding $876.

$1793(starting) + $876 = $2669

The maximum amount of money that Micah could earn would be $2669.

8 0

3 years ago