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2 years ago

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Translations Use the translation (x,y)(x+5,y-9) for questions 1-7. 1. What is the image of A(-1,3)? 2. What is the image of B(2,

5)? 3. What is the image of C(4,-2)? 4. What is the image of A'? 5. What is the preimage of D'(12,7)? 6. What is the image of A"?

1 answer:

lara [203]2 years ago

7 0

Answer:

The answer is c hopes this helps

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Consider the equation below. (If you need to use -[infinity] or [infinity], enter -INFINITY or INFINITY.)f(x) = 2x3 + 3x2 − 180x

soldier1979 [14.2K]

Answer:

(a) The function is increasing $\left(-\infty, -6\right) \cup \left(5, \infty\right)$ and decreasing $\left(-6, 5\right)$

(b) The local minimum is x = 5 and the maximum is x = -6

(c) The inflection point is $x = -\frac{1}{2}$

(d) The function is concave upward on $\left(- \frac{1}{2}, \infty\right)$ and concave downward on $\left(-\infty, - \frac{1}{2}\right)$

Step-by-step explanation:

(a) To find the intervals where $f(x) = 2x^3 + 3x^2 -180x$ is increasing or decreasing you must:

1. Differentiate the function

$\frac{d}{dx}f(x) =\frac{d}{dx}(2x^3 + 3x^2 -180x) \\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\f'(x)=\frac{d}{dx}\left(2x^3\right)+\frac{d}{dx}\left(3x^2\right)-\frac{d}{dx}\left(180x\right)\\\\f'(x) =6x^2+6x-180$

2. Now we want to find the intervals where f'(x) is positive or negative. This is done using critical points, which are the points where f'(x) is either 0 or undefined.

$f'(x) =6x^2+6x-180 =0\\\\6x^2+6x-180 = 6\left(x-5\right)\left(x+6\right)=0\\\\x=5,\:x=-6$

These points divide the number line into three intervals:

$(-\infty,-6)$ , $(-6,5)$ , and $(5, \infty)$

Evaluate f'(x) at each interval to see if it's positive or negative on that interval.

$\left\begin{array}{cccc}Interval&x-value&f'(x)&Verdict\\(-\infty,-6)&-7&72&Increasing\\(-6,5)&0&-180&Decreasing\\(5, \infty)&6&72&Increasing\end{array}\right$

Therefore f(x) is increasing $\left(-\infty, -6\right) \cup \left(5, \infty\right)$ and decreasing $\left(-6, 5\right)$

(b) Now that we know the intervals where f(x) increases or decreases, we can find its extremum points. An extremum point would be a point where f(x) is defined and f'(x) changes signs.

We know that:

f(x) increases before x = -6, decreases after it, and is defined at x = -6. So f(x) has a relative maximum point at x = -6.

f(x) decreases before x = 5, increases after it, and is defined at x = 5. So f(x) has a relative minimum point at x = 5.

(c)-(d) An Inflection Point is where a curve changes from Concave upward to Concave downward (or vice versa).

Concave upward is when the slope increases and concave downward is when the slope decreases.

To find the inflection points of f(x), we need to use the f''(x)

$f''(x)=\frac{d}{dx}\left(6x^2+6x-180\right)\\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\f''(x)=\frac{d}{dx}\left(6x^2\right)+\frac{d}{dx}\left(6x\right)-\frac{d}{dx}\left(180\right)\\\\f''(x) =12x+6$

We set f''(x) = 0

$f''(x) =12x+6 =0\\\\x=-\frac{1}{2}$

Analyzing concavity, we get

$\left\begin{array}{cccc}Interval&x-value&f''(x)\\(-\infty,-1/2)&-2&-18\\(-1/2,\infty)&0&6\\\end{array}\right$

The function is concave upward on $(-1/2,\infty)$ because the f''(x) > 0 and concave downward on $(-\infty,-1/2)$ because the f''(x) < 0.

f(x) is concave down before $x = -\frac{1}{2}$ , concave up after it. So f(x) has an inflection point at $x = -\frac{1}{2}$ .

7 0

3 years ago

Which construction is partially represented by the diagram on the baseball field? Explain the next set of instructions to correc

spayn [35]

Answer:

Construction of an angle bisector is partially represented by the diagram on the baseball field.

Step-by-step explanation:

Angle bisector of an angle bisects it into two equal angles.

Construction of an angle bisector is partially represented by the diagram on the baseball field.

Consider the figure:

From point B, draw an arc by opening the compass up to the same extend as we did while drawing arc from point A.

Take the point of intersection of both the arcs as C.

Join OC.

OC is the angle bisector of the angle.

5 0

3 years ago

Find the area of the figure below.

Fudgin [204]

Answer:

Area = 42 in^2

Step-by-step explanation:

here's your solution

=> divide above picture in two parts

=> rectangle and traingle

=> area of rectangle = length* width

=> area of rectangle = (5*6)in.

=> area of rectangle = 30 in^2

=> now, area of traingle = 1/2*base *height

=> area of traingle = 1/2*4*6

=> area of traingle = 12 in^2

=> area of figure = 30 in^2 + 12 in^2

=> Area = 42 in^2

hope it helps

3 0

3 years ago

Read 2 more answers

For f(x)= x+5 and g(x)=4x+2 <br> find (fog)(x) And (gof)(x)

Svetllana [295]

Composing functions means that the input of the outer functions is the output of the inner function.

In fact, you can rewrite the circle notation as

$(f\circ g)(x)=f(g(x)),\quad (g\circ f)(x)=g(f(x))$

So, we can substitute g(x) with its expression:

$(f\circ g)(x)=f(g(x))=f(4x+2)$

And since f(x)=x+5, we simply have to add 5 to its input:

$f(4x+2)=(4x+2)+5=4x+7$

Similarly, we have, substituting f with its expression,

$(g\circ f)(x)=g(f(x))=g(x+5)$

And since g(x)=4x+2, we have to multiply the input by 4 and add 2:

$g(x+5)=4(x+5)+2=4x+20+2=4x+22$

7 0

3 years ago

Two consecutive numbers have a sum of 346. What are the numbers?

seraphim [82]

Answer:

81,82,83

Step-by-step explanation:

What three consecutive integers have a sum of 246? Which means that the first number is 81, the second number is 81 + 1 and the third number is 81 + 2. Therefore, three consecutive integers that add up to 246 are 81, 82, and 83.

7 0

3 years ago