Question

Write an equation for the translation of y=6/x that has the asymptotes x=4 and y=5

Answer 1

What is the 50th term of the sequence that begins -2,4,10,16?

Answer 2

What is the 20th term of the sequence that begins -4,8,-16,32

Answer 3

To translate the function y=6/x with asymptotes x=4 and y=5, we can use the following equation:

y - 5 = k * (6 / (x - 4))

where k is a constant that determines the rate of the translation. If we let k = 1/2, for example, the equation becomes:

y - 5 = 1/2 * (6 / (x - 4))

Expanding this, we can simplify it to:

2y - 10 = 6 / (x - 4)

Multiplying both sides by (x - 4), we get:

(2y - 10) * (x - 4) = 6

Expanding this, we can rearrange it to the standard form of a hyperbola:

2xy - 8y - 10x + 40 = 6

And finally, we can simplify it to:

2xy - 10x - 8y + 34 = 0

This is the equation of the translated function, which has asymptotes x=4 and y=5.

Answer 4

The given sequence starts with -2, 4, 10, 16, which has a pattern of adding 6 to the previous term. To find the 50th term, we can use the formula:

nth term = a + (n-1)d

where a is the first term, d is the common difference, and n is the term number we want to find.

In this case, a = -2, d = 6, and n = 50. Plugging these values into the formula, we get:

50th term = -2 + (50-1)6
= -2 + 49*6
= -2 + 294
= 292

Therefore, the 50th term of the given sequence is 292.

Answer 5

The given sequence starts with -4, 8, -16, 32, which has a pattern of multiplying the previous term by -2. To find the 20th term, we can use the formula:

nth term = a x r^(n-1)

where a is the first term, r is the common ratio, and n is the term number we want to find.

In this case, a = -4, r = -2, and n = 20. Plugging these values into the formula, we get:

20th term = -4 x (-2)^(20-1)
= -4 x (-2)^19
= -4 x (-524288)
= 2097152

Therefore, the 20th term of the given sequence is 2097152.