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4 years ago

Write y= -4x² -16x -14 in vertex form.

Mathematics

1 answer:

adell [148]4 years ago

5 0

Put brackets around the first two terms on the right. y = (-4x^2 - 16x) - 14
Pull out the common factor in the first two terms. y = -4(x^2 + 4x) - 14
Divide the middle term by 2. Add that inside the brackets. square y = -4(x^2 +4x +(4/2)^2 - 14
y = -4(x^2 + 4x + 4) - 14 + 16 This is the step where most people stumble. The point is why is 16 added? It is because what you have done inside the brackets is multiplied 4 by - 4 (on the left). That means you have changed the equation by -16. To counter that, you must add 16 after the - 14. The result is y= -4(x^2 + 4x+4) + 2
Express the terms inside the brackets as a square. y = - 4(x + 2)^2 + 2

B and D are both wrong. B has +4 outside the brackets. It is - 4

D is wrong because there is no 4 of any kind outside the brackets.

A is incorrectly represented inside the brackets as 16. That's not right

That only leaves C. <<<< Answer

Graphs

Notice that the red parabola and the green one are the same thing.

Red: y = -4x^2 - 16x - 14

Green: y = -4(x + 2)^2 + 2

You might be interested in

A culture of bacteria has an initial population of 9300 bacteria and doubles every 3

sp2606 [1]

Answer:

The approximate population of bacteria in the culture after 10 hours is 93,738.

Step-by-step explanation:

<h3>General Concepts:</h3>

Exponential Functions.
Exponential Growth.
Doubling Time Model.
Logarithmic Form.

BPEMDAS Order of Operations:

Brackets.
Parenthesis.
Exponents.
Multiplication.
Division.
Addition.
Subtraction.

<h2>Definitions:</h2>

We are given the following Exponential Growth Function (Doubling Time Model), $\displaystyle\mathsf{P_{(t)}\:=\:P_0\cdot2^{(t/d)}}$ where:

$\displaystyle\sf{P_t\:\:\rightarrow}$ The population of bacteria after “<em>t </em>” number of hours.
$\displaystyle\sf{P_0 \:\:\rightarrow}$ The initial population of bacteria.
$\displaystyle{t \:\:\rightarrow}$ Time unit (in hours).
$\displaystyle{\textit d \:\:\rightarrow}$ Doubling time, which represents the amount of time it takes for the population of bacteria to grow exponentially to become twice its initial quantity.

<h2>Solution:</h2>

<u>Step 1: Identify the given values.</u>

$\displaystyle\sf{P_0\:=}$ 9,300.
<em>t</em> = 10 hours.
<em>d</em> = 3.

<u>Step 2: Find value.</u>

1. Substitute the values into the given exponential function.

$\displaystyle\mathsf{P_{(t)} = P_0\cdot2^{(t/d)}}$

$\displaystyle\mathsf{\longrightarrow P_{(10)} = 9300\cdot2^{(10/3)}}$

2. Evaluate using the BPEMDAS order of operations.

$\displaystyle\mathsf{P_{(10)} = 9300\cdot2^{(10/3)}\quad \Longrightarrow BPEMDAS:\:(Parenthesis\:\:and\:\:Division).}$

$\displaystyle\sf P_{(10)} = 9300\cdot2^{(3.333333)}\quad\Longrightarrow BPEMDAS:\:(Exponent).}$

$\displaystyle\sf P_{(10)} = 9300\cdot(10.079368399)\quad \Longrightarrow BPEMDAS:(Multiplication).}$

$\boxed{\displaystyle\mathsf{P_{(10)} \approx 93,738.13\:\:\:or\:\:93,738}}$

Hence, the population of bacteria in the culture after 10 hours is approximately 93,738.

<h2>Double-check:</h2>

We can solve for the amount of <u>time</u> <u>(</u><em>t</em> ) it takes for the population of bacteria to increase to 93,738.

1. Identify given:

$\displaystyle\mathsf{P_{(t)} = 93,738 }$ .
$\displaystyle\mathsf{P_0 = 9,300}$ .
<em>d </em>= 3.

2. Substitute the values into the given exponential function.

$\displaystyle\mathsf{P_{(t)} = P_0\cdot2^{(t/d)}}$

$\displaystyle\mathsf{\longrightarrow 93,378 = 9,300\cdot2^{(t/3)}}$

3. Divide both sides by 9,300:

$\displaystyle\mathsf{\longrightarrow \frac{93,378}{9,300} = \frac{9,300\cdot2^{(t/3)}}{9,300}}$

$\displaystyle\mathsf{\longrightarrow 10.07936840 = 2^{(t/3)}}$

4. Transform the right-hand side of the equation into logarithmic form.

$\boxed{\displaystyle\mathsf{\underbrace{ x = a^y}_{Exponential\:Form} \longrightarrow \underbrace{y = log_a x}_{Logarithmic\:Form}}}$

$\displaystyle\mathsf{\longrightarrow 10.07936840 = \bigg[\:\frac{t}{3}\:\bigg]log(2)}$

5. Take the <em>log</em> of both sides of the equation (without rounding off any digits).

$\displaystyle\mathsf{log(10.07936840) = \bigg[\:\frac{t}{3}\:\bigg]log(2)}$

$\displaystyle\mathsf{\longrightarrow 1.003433319 = \bigg[\:\frac{t}{3}\:\bigg]\cdot(0.301029996)}$

6. Divide both sides by (0.301029996).

$\displaystyle\mathsf{\frac{1.003433319}{0.301029996} = \frac{\bigg[\:\frac{t}{3}\:\bigg]\cdot(0.301029996) }{0.301029996}}$

$\displaystyle\mathsf{\longrightarrow 3.3333333 = \frac{t}{3}}$

7. Multiply both sides of the equation by 3 to isolate "<em>t</em>."

$\displaystyle\mathsf{(3)\cdot(3.3333333) = \bigg[\:\frac{t}{3}\:\bigg]\cdot(3)}$

$\boxed{\displaystyle\mathsf{t\approx10}}$

Hence, it will take about 10 hours for the population of bacteria to increase to 93,378.

__________________________________

Learn more about Exponential Functions on:

brainly.com/question/18522519

7 0

2 years ago

(8+5i) + (10+6i)-(3+6i)

Natali [406]

Answer:

15 + 5i

Step-by-step explanation:

$(8+5i) + (10+6i)-(3+6i) \\ \\ = 8 + 5i + 10 + 6i - 3 - 6i \\ \\ =( 8 + 10 - 3 )+ (5i + 6i - 6i )\\ \\ = 15 + 5i$

8 0

3 years ago

Find the length of the curve given by ~r(t) = 1 2 cos(t 2 )~i + 1 2 sin(t 2 ) ~j + 2 5 t 5/2 ~k between t = 0 and t = 1. Simplif

xxMikexx [17]

Answer:

The length of the curve is

L ≈ 0.59501

Step-by-step explanation:

The length of a curve on an interval a ≤ t ≤ b is given as

L = Integral from a to b of √[(x')² + (y' )² + (z')²]

Where x' = dx/dt

y' = dy/dt

z' = dz/dt

Given the function r(t) = (1/2)cos(t²)i + (1/2)sin(t²)j + (2/5)t^(5/2)

We can write

x = (1/2)cos(t²)

y = (1/2)sin(t²)

z = (2/5)t^(5/2)

x' = -tsin(t²)

y' = tcos(t²)

z' = t^(3/2)

(x')² + (y')² + (z')² = [-tsin(t²)]² + [tcos(t²)]² + [t^(3/2)]²

= t²(-sin²(t²) + cos²(t²) + 1 )

................................................

But cos²(t²) + sin²(t²) = 1

=> cos²(t²) = 1 - sin²(t²)

................................................

So, we have

(x')² + (y')² + (z')² = t²[2cos²(t²)]

√[(x')² + (y')² + (z')²] = √[2t²cos²(t²)]

= (√2)tcos(t²)

Now,

L = integral of (√2)tcos(t²) from 0 to 1

= (1/√2)sin(t²) from 0 to 1

= (1/√2)[sin(1) - sin(0)]

= (1/√2)sin(1)

≈ 0.59501

8 0

3 years ago

Erin has quarters and nickels for a total of 16 coins. The value of his coins is $2.60. Let represent the number of quarters and

Aneli [31]

Note: Consider we need to find each type of coins he has.

Given:

Erin has quarters and nickels for a total of 16 coins.

The value of his coins is $2.60.

To find:

The number of each type of coins.

Solution:

Let x represent the number of quarters and y represent the number of nickels.

According to the question,

Total coins : $x+y=16$ ...(i)

Total value : $0.25x+0.05y=2.60$ ...(ii)

Multiply equation (i) by 0.25 and subtract the result from (ii).

$0.25x+0.05y-0.25(x+y)=2.60-0.25(16)$

$0.25x+0.05y-0.25x-0.25y=2.60-4$

$-0.20y=-1.40$

Divide both sides by -0.20.

$y=7$

Put y=7 in (i).

$x+7=16$

$x=16-7$

$x=9$

Therefore, the number of quarters is 9 and number of nickels is 7.

6 0

3 years ago

Write the ratios for sin M, cos M, and tan M.

Dominik [7]

Answer:

Sin M= opposite/hypotenuse, cos M= adjacent/Hypotenuse, tan M= opposite/adjacent

Step-by-step explanation:

7 0

2 years ago

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