Write an equation for the translation of y=6/x that has the asymptotes x=4 and y=5
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.
What is the 50th term of the sequence that begins -2,4,10,16?
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.
What is the 20th term of the sequence that begins -4,8,-16,32
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.
To translate the function y = 6/x with asymptotes at x = 4 and y = 5, you can first adjust the x-axis by shifting it 4 units to the right, which corresponds to substituting (x - 4) for x. Then, for the y-axis, shift it 5 units up by adding 5 to the equation. Combining these adjustments, the translation equation becomes:
y = 6 / (x - 4) + 5
To find the equation for the translation of the function y = 6/x with the asymptotes x = 4 and y = 5, we need to shift the original function horizontally and vertically.
First, let's focus on the horizontal shift. The equation y = 6/x has the vertical asymptote x = 0, which means that the graph approaches infinity as x approaches 0. To shift this asymptote to x = 4, we need to replace x with (x - 4) in the equation. This horizontal shift is in the negative direction because x = 4 is to the right of x = 0. Therefore, our equation becomes:
y = 6/(x - 4)
Next, let's consider the vertical shift. The equation y = 6/x has no vertical intercept and a horizontal asymptote at y = 0. To shift the horizontal asymptote from y = 0 to y = 5, we need to add 5 to the equation. This vertical shift is in the positive direction because y = 5 is above y = 0. Thus, our final equation for the translation becomes:
y = 6/(x - 4) + 5