geno3141

I'm not certain that this is the easiest way, but ...

I'm going to find the inverse of y  =  3x + b

--->  x  =  3y + b

--->  x - b  =  3y

--->  y  =  (x - b)/3

Since they intersect at the point (-3, a), set the two equations equal to each other:

--->  3x + b  =  (x - b)/3

--->  9x + 3b  =  x - b

--->  8x  =  - 4b

Since x = -3 at that point:

--->  8(-3)  =  -4b

--->   -24  =  -4b

--->  b  =  6

Original equation is y  =  3x + 6

At the point (-3, a) --->  a  =  3(-3) + 6   --->   a  =  -9 + 6   --->   a = -3

For the graph shown, each box has a width and height of 1.

The graph of f(2x) would be identical to the graph of f(x) if you change the scale on the x-axis, so that the width of each box would be ½ (but keep the height the same as it was).

a)  On the original graph, there is the point (-4, 4). On the graph of f(2x), the coordinates of the point are                    (-2,4), so a = -2.

      On the original graph, there is the point (4, -4). On the graph of f(2x), the coordinates of the point are                    (2, -4), so b = 2. 

b) See the note above part a.

c)  f(2x - 8)  =  f( 2(x - 4) )  This graph makes a horizontal shift of the graph of f(2x) 4 spaces to the right.                    The graph of f(2x - 8), which is f( 2(x - 4) ), is congruent to the graph of f(2x), just move it 4                          spaces to the right.

     Moving (-2,4) four spaces to the right creates (2,4), so c = 2.

     Moving (2,-4) four spaces to the right creates (6, -4), so d = 6.

d)  Draw the graph of f(2x), then move that graph 4 spaces to the right to create f(2x - 8).

For the graph shown, each box has a width and height of 1.

The graph of f(2x) would be identical to the graph of f(x) if you change the scale on the x-axis, so that the width of each box would be ½ (but keep the height the same as it was).

a)  On the original graph, there is the point (-4, 4). On the graph of f(2x), the coordinates of the point are                    (-2,4), so a = -2.

      On the original graph, there is the point (4, -4). On the graph of f(2x), the coordinates of the point are                    (2, -4), so b = 2. 

b) See the note above part a.

c)  f(2x - 8)  =  f( 2(x - 4) )  This graph makes a horizontal shift of the graph of f(2x) 4 spaces to the right.                    The graph of f(2x - 8), which is f( 2(x - 4) ), is congruent to the graph of f(2x), just move it 4                          spaces to the right.

     Moving (-2,4) four spaces to the right creates (2,4), so c = 2.

     Moving (2,-4) four spaces to the right creates (6, -4), so d = 6.

d)  Draw the graph of f(2x), then move that graph 4 spaces to the right to create f(2x - 8).

Yes, it is.

The formula for a frustum of a cone is V = (Π·h/3)( R²+ Rr + r² )

where h is the height of the frustum and R and r are the two radii of the ends.

The formula for a frustum of a pyramid is V = (h/3)( A+ a + √(Aa) )

where h is th height of the grustum and A and a are the areas of the two ends.

(This formula was used by the ancient Egyptians, who didn't even have any algebra! or a number system as easy to use as ours!)

It appears that this is a parabola with vertex at (7.5, 4) and passing through the point (0, -3).

An equation for a parabola is  y - k = a(x - h)²   The vertex is (h, k)

So, we have:  y - 4 = a(x - 7.5)² 

To find the value for a, let's use the point (0, -3):  -3 - 4 = a(0 - 7.5)² 

--->   -7 = a(-7.5)²     --->   -7 = 56.25a   --->   a = -28/225

--->   y - 4 = (-28/225)(x - 7.5)² 

Substituting 15 for x:  y - 4 = (-28/225)(15 - 7.5)²   -->  y =  -3

Since both 1 134 218 and 1 135 620 are both even, 2 divides into both

1 134 218 ÷ 2 = 567 109

1 134 620 ÷ 2 = 567 810

After going online and finding a list of prime numbers, trying them one by one, you can get to 701:

567 109 ÷ 701 = 809

567 810 ÷ 701 = 810

Since 809 and 810 are relatively prime (no whole number divides into both of them), we can stop.

The greatest common factor is 2 x 701 = 1402.

Actually, I didn't do that. I have a calculator that, if you divide 1134218 by 1135620, you get .9987654321...

Then, by using a button that changes decimal fractions to common fractions, it gives the answer 809/810.

Dividing 1134218 by 809, the result is 1402, which is 701 x 2. Then, I went to an online list of prime numbers and finding 701, I knew that I was finished.

The fraction -4/3 can be written either as -4/3 or 4/-3.

tan is defined to be opposite divided by adjacent or, when graphed, as y / x.

If you write the fraction as -4/3, then the y-value (opposite) is -4 and the x-value (adjacent) is 3. This point is in Quadrant IV. Using the Pytagorean Theorem, the hypotenuse becomes 5. To find the sin, use the formula for sin, which is opposite over hypotenuse:  sin(x) = -4/5.

If you write the fraction as 4/-3, the y-value is 4 and the x-value is -3, placing the angle in Quadrant II. The hypotenuse is again 5 and the sin becomes 4/5.

So, there are two possible answers.

To factor -1/3 out of -12, you will need to be able to answer the question:  "-1/3 times what number will give you -12?"

To answer this you must divide -12 by -1/3:  -12 ÷ -1/3  =  -12 x 3/-1  =  36.

You can then write -12 as  (-1/3)(36):

To factor -1/3x - 12:  =  (-1/3)x + (-1/3)(36)

                                   =  (-1/3)( x + 36 ) 

Sligo has 5 letters

Galway has 6 letters

May has 4 letters

Roscommon has 9 letters

Leitrim has 7 letters

To answer part a: Which of these names have an even number of letters? List those names in set A:

     A  =  {          }

To answer part b: Which of these names have a prime number of letters? List those names in set B:

     B  =  {         }

If there are no answers that fit, leave the set empty. If there are more than one answer, list all answers, placing commas between answers.