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3 years ago

For the function f(x) = −2(x + 3)2 − 1, identify the vertex, domain, and range.

Mathematics

1 answer:

Jet001 [13]3 years ago

4 0

Answer:

Vertex: (-3, -1)

Domain: All real numbers or (-∞, ∞)

Range: (-∞, -1]

Step-by-step explanation:

Quadratic Function: f(x) = a(bx - h)² + k

(h, k) is vertex

Step 1: Find vertex

f(x) = -2(x + 3)² - 1

h = -3

k = 1

Vertex = (-3, 1)

Step 2: Graph

When we graph, we should see our domain and range.

Domain is all x-values that can be inputted into the function.

Range is all y-values that can be outputted from the function.

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How to rename the number 24 ten thousands?

kifflom [539]

Its 24,000 if the number goes to ten thousand's place..

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3 years ago

Find the absolute extrema for f(x,y)=4-x^2-y^4+1/2y^2 over the closed disk D:x^2+y^2 is less than or equal to 1

algol [13]

Find the critical points of $f(x,y)$ :

$\dfrac{\partial f}{\partial x}=-2x=0\implies x=0$

$\dfrac{\partial f}{\partial y}=y-4y^3=y(1-4y^2)=0\implies y=0\text{ or }y=\pm\dfrac12$

All three points lie within $D$ , and $f$ takes on values of

$\begin{cases}f(0,0)=4\\f\left(0,-\frac12\right)=\frac{65}{16}\\f\left(0,\frac12\right)=\frac{65}{16}\end{cases}$

Now check for extrema on the boundary of $D$ . Convert to polar coordinates:

$f(x,y)=f(\cos t,\sin t)=g(t)=4-\cos^2-\sin^4t+\dfrac12\sin^2t=3+\dfrac32\sin^2t-\sin^4t$

Find the critical points of $g(t)$ :

$\dfrac{\mathrm dg}{\mathrm dt}=3\sin t\cos t-4\sin^3t\cos t=\sin t\cos t(3-4\sin^2t)=0$

$\implies\sin t=0\text{ or }\cos t=0\text{ or }\sin t=\pm\dfrac{\sqrt3}2$

$\implies t=n\pi\text{ or }t=\dfrac{(2n+1)\pi}2\text{ or }\pm\dfrac\pi3+2n\pi$

where $n$ is any integer. There are some redundant critical points, so we'll just consider $0\le t< 2\pi$ , which gives

$t=0\text{ or }t=\dfrac\pi3\text{ or }t=\dfrac\pi2\text{ or }t=\pi\text{ or }t=\dfrac{3\pi}2\text{ or }t=\dfrac{5\pi}3$

which gives values of

$\begin{cases}g(0)=3\\g\left(\frac\pi3\right)=\frac{57}{16}\\g\left(\frac\pi2\right)=\frac72\\g(\pi)=3\\g\left(\frac{3\pi}2\right)=\frac72\\g\left(\frac{5\pi}3\right)=\frac{57}{16}\end{cases}$

So altogether, $f(x,y)$ has an absolute maximum of 65/16 at the points (0, -1/2) and (0, 1/2), and an absolute minimum of 3 at (-1, 0).

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3 years ago

Which line has a slope of 1/2 and a y-intercept of 2?

AnnZ [28]

Answer:

it would be the bottom left graph

Step-by-step explanation:

Rise over Run

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3 years ago

A survey is made to determine the number of households having electric appliances in a certain city. It is found that 75% have r

Mashcka [7]

Answer:

The probability that a household has at least one of these appliances is 0.95

Step-by-step explanation:

Percentage of households having radios P(R) = 75% = 0.75

Percentage of households having electric irons P(I) = 65% = 0.65

Percentage of households having electric toasters P(T) = 55% = 0.55

Percentage of household having iron and radio P(I∩R) = 50% = 0.5

Percentage of household having radios and toasters P(R∩T) = 40% = 0.40

Percentage of household having iron and toasters P(I∩T) = 30% = 0.30

Percentage of household having all three P(I∩R∩T) = 20% = 0.20

Probability of households having at least one of the appliance can be calculated using the rule:

P(at least one of the three) = P(R) +P(I) + P(T) - P(I∩R) - P(R∩T) - P(I∩T) + P(I∩R∩T)

P(at least one of the three)=0.75 + 0.65 + 0.55 - 0.50 - 0.40 - 0.30 + 0.20 P(at least one of the three) = 0.95

The probability that a household has at least one of these appliances is 0.95

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3 years ago

Solve the equation

12345 [234]

Answer:

Step-by-step explanation:

x+(x+1)+(x+2)+(x+3)=-210

x+1x+1+1x+2+1x+3=-210

4x+6=-210

4x=-210-6

4x/4x=-216/4

x=54

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3 years ago