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Here's a "formula"  that can be used to find a new point :

New Point '   =  [ Old Point  - Dilation Center ] *Scale Factor  + Dilation Center

Note that   (a , b)  - (c, d)  =  (a - c, b - d)

(a, b) + (c , d)  =  ( a + c, b + d)     and

(a, b) * S  =  (Sa, Sb)

So.....to transform (0, - 4)   as follows

New Point'  =    [  ( 0 - 4) - ( 2, - 7) ] * 3  +  ( 2, -7)  =

[ ( -2, 3) ] * 3  + (2, -7)  =

(- 6, 9)  + ( 2, - 7)   =

(-4, 2)

You can compute the other points yourself

a)

If we suppose that c varies inversely with d, then this is what that statement is telling you:

• If d increases, then c decreases.
• If d decreases, then c increases.

The most basic example of this occurence is \(y=\frac{1}{x}\). As you think of larger values of x, the input, then the output, y, will decrease. 

There is only one difference from the previous function to this particular problem: We do not know the rate in which the inverse relation occurs. For example, if the numerator of the previous function was a 2, then the rate of the inverse function would double. Let's name the rate of the inverse relation "k." We, therefore, make the following equation.

\(c=\frac{k}{d}\)

b)

We can use the previous equation to figure out the rest. It involves basic substitution.

Now, let's find d when c=68:

Hello? Any1 online or looking at this post?

Note that we can write

    1

 ______    -   3            as

 2x - 10

    1

_______     -  3     =

2 ( x - 5)

 1             1

___  *   ________    -  3

 2          (x - 5)

This  graph  shifts  the parent graph to  the right by 5 units, down by 3 units,  and vertically "compresses"  the parent by a factor of (1/2)

Well, we may not need that nice diagram to answer that question!

The fourth vertex of the quadrilateral is located at \((-2,-7)\), which is the same y-coordinate as the center of dilation, \((2,-7)\). Let's assume that the center of dilation is actually the origin of the coordinate plane. 

This means that, from the original origin, we have effectively shifted the entire coordinate plane 2 units to the right and 7 units downward. Since the coordinates of the fourth vertex lie on the same y-coordinate, it is much easier to notice that the distance from the vertex to the "new origin" is 4 units. This means that the center of dilation is at \((0,0)\) and that the fourth vertex is located at \((-4,0)\)

If the question asks for a dilation by a scale factor of 3, then the distance of the vertex from our "new origin" also triples. The distance, therefore, would be 12, so the coordinate of the dilated point would be \((-12,0)\).

Of course, we must account for the fact that we shifted the Cartesian plane for convenience sake. In order to reverse these effects, we must move the plane 2 units to the left and 7 units upward, so the coordinates of the fourth vertex would become \((-10,-7)\)

\(\frac{z^2-4}{z-3}/\frac{z+2}{z^2+z-12}\)

Switch the division sign to multiplication and switch the top and bottom equations on the second fraction.

\(\frac{z^2-4}{z-3}*\frac{z^2+z-12}{z+2}\)

Simplify the polynomial

\(\frac{(z+2)(z-2)}{z-3}*\frac{(z+4)(z-3)}{z+2}\)

You can cross out from top to bottom, top left to bottom right, and bottom left to top right.

\((z-2)(z+4)\)

Multiply your answer.

\(z^2-2z-8\)

---------

To find out the restrictions, you need to take the bottom number of each fraction and make it equal to zero.

\(z-3=0\)

\(z=3\)

...

\(z^2+z-12=0\)

\((z+4)(z-3)=0\)

\(z+4=0, z-3=0\)

\(z=-4, z=3\)

You can't use -4 or 3 as an answer to this problem.

Thank you so much for all of your help, I really appreciate it!

That's quite a nice diagram, CoopTheDupe!

Unfortunately, my nitpicky mindset noticed that \((-2,-7)\), a coordinate of the original quadrilateral is graphed incorrectly (as \((2,7)\).

Thanks so much for answering but you actually marked the 4th corner of the original shape wrong its expose to be (-2,-7) not (2,-7) however it is expose to still be centered at (2,-7). this should effect the answer do you think you could help me fix it so it is correct?

Very nice, Coop  !!!

thank u

In this diagram, \(m\angle BDA = 45^{\circ}\). \(\angle BDA\) represents one and only one angle. It appears as if you do not understand the notation of angles. 

In order to identify \(\angle BDA\) in your individual diagram, so the following:

1) Draw a segment from point B to point D. 

2) Draw a segment from point D to point A.

3) Point D should create an angle that is bounded between two segments: \(\overline{BD}\text{ and }\overline{DA}\). This is where the angle is located. 

Remember that the central letter of the notation indicates the location of the angle. The other letters indicate what segment bound the angle between. Does this make sense?

.45 D   =  135        divide both sides by  .45

135 / .45   =   D   =    300 miles

So you have a quad that looks like this. (Orange is original points, black is making up the shape)
 

Draw up 3 lines connecting the center to the 3 points. (I'm gonna use Red, Blue, and Green).

Since it's a dilation of 3, you're going to take each line and extend it 2 more times straight. Then mark the new points. (I'm going to use purple)

Connect the new lines.

The new points should be (2, -7), (-4, 3), (5, 5), and (8, -1).

1) Given the proportion a/b = 11/20 , what ratio completes the equivalent proportion a/11 = ? (The answer needs to be a fraction)

Notice something, RP

If

5/10   =  1/2        then it is also true that  the ratio of the numerators to the ratio of the denominators is the same....that is :

5/1  =  10/2

This implies   that

a /b   =  11/20      means that

a / 11  =  b / 20

Do you see this??

2) A model is made of a car. The car is 3 meters long and the model is 4 centimeters long. What is the ratio of the length of the car to the length of the model?

Equate units in terms of centimeters

3 m  =   300  cm

Ratio  :   actual length  / model length   =    300 cm / 4 cm   =   75 / 1   =  75 : 1

6  is correct for the second one  !!!

OK....for the last....you have it set up correctly

5(3y - 8)  =  12y    simplify

15y - 40   =  12y      subtract  15y   from both sides

-40  =   -  3y        divie both sides by  - 3

40/3   =  y

2. 2/5=m/15

Cross multiply: 30=5m

Divide by 5

6=m

4. 3y-8/12=y/5

Cross multiply: (3y-8)(5)=12(y)

ugh idk T_T

HAHAHA!!!!....don't know about that...but...I do know a few answers, here and there...

We can solve everyone of these by cross-multiplication...so

6/a  =  18 /24      cross-multiply

24 * 6  =  18 * a

144  =  18 *a      divide both sides by   18

8  =  a

I'll  do  the third one....you try the other two, RP....let me know if you get stuck

n - 6 / 3n  =  n - 5 / 3n + 1        cross-multiply

(n - 6) (3n + 1)  =  3n ( n - 5)      simplify

3n^2 - 18n + 1n - 6  =  3n^2 -  15n      subtract 3n^2  from both sides

-17n - 6  =  - 15n       add  17n  to both sides

-6   =  2n      divide both sides by 2

-3  =  n

Thank you Cphill

You are a Math saver!!

No....I just did that for convenience, Manuel....your solutions are correct  !!!

6. A salsa recipe uses green pepper, onion, and tomato in the extended ratio 2 : 5 : 8. How many cups of onion are needed to make 105 cups of salsa?

There are   2 + 5 + 8   =   15  parts.....and  the onion  is  (5/15)  = 1/3

So.....  105 (1/3)   =  35  cups of onion

7. The measures of the angles of a triangle are in the extended ratio 2 : 3 : 5. What is the measure of the smallest angle?

There are  2 + 3 + 5    =  10  parts.....

And the smallest angle is  2/10  of this

So.....(2/10) 180   =  2 * 180 / 10   =  2 * 18   =   36°

do I have to include the brackets in my answer?

\(\frac{\frac{x^2+x-12}{x-2}}{\frac{3x^2+11x-4}{x^2-4}}\)

Simplify the polynomials.

\(\frac{\frac{(x+4)(x-3)}{x-2}}{\frac{(3x-1)(x+4)}{(x+2)(x-2)}}\)

Multiply the top and bottom fractions by (x+2)(x-2).

\(\frac{(x+4)(x-3)(x+2)}{(3x-1)(x+4)}\)

Cross out like terms.

\(\frac{(x-3)(x+2)}{(3x-1)}\)

Multiply the top polynomial out.

\(\frac{x^2-x+6}{3x-1}\)

Just a different way to look at it. :D