Consider the function y=(kx)/(x-k)
a. Show that y is symmetric with respect to the line y=x.
b. Write the equations of the horizontal and vertical asymptotes of y.
c. Find the point(s) of intersection of y with the line x-y=-2k, and find the slope of the tangent line at the point(s) of intersection.
Please show step-by-step work in your answers. Thanks in advance!!! :)
a) take the inverse of your function and show that the original equation comes back at you.
b) for VA, the denominator is zero, so x-k = 0 is the vertical asymptote
for a HA, I visualize what happens to kx/(x-k) as x -- ∞
I see y = k
c) I solved y = kx/(x-k) with y = x+2k and got
x = ± k√2
put each of those values back into your straight line equation to get the corresponding y's.
take the derivative of the first function using the quotient rule, sub in the points you found to get the slope.
Now that you have the slope and the point, find the straight line equation of the tangent.
(I am assuming you know how to follow those steps)
a)
the reflection matrix about the line y=x is
0 1
1 0
In other words, new x = old y
and new y = old x
So what happens if we do this operation on and x old = a and a y old = b? (a,b)
the original would be:
b = (ka)/(a - k)
find new x and new y if the symmetry exists (b,a)
a = (kb)/(b-k)
If that is true, we have the symmetry.
so solve for b
b a - k a = k b
b (a-k) = k a
b = (ka)/(a-k)
a. To show that the function is symmetric with respect to the line y = x, we need to show that if we substitute x with y and y with x in the equation, we get an equivalent equation.
Substituting x with y and y with x in the equation y = (kx)/(x - k), we get:
x = (ky)/(y - k)
Now, let's multiply both sides of the equation by (y - k):
x(y - k) = ky
Expanding the left side and rearranging the equation, we have:
xy - kx = ky
Moving all terms to one side, we get:
xy - ky = kx
Factoring out x and y:
y(x - k) = kx
Dividing both sides by (x - k), we get:
y = (kx)/(x - k)
This is the same equation as the original one, so the function is symmetric with respect to the line y = x.
b. To find the horizontal asymptote, we need to determine the behavior of the function as x approaches positive infinity and negative infinity.
As x approaches positive infinity, the term kx in the numerator dominates, and the term x - k in the denominator becomes insignificant. This leads to the numerator getting infinitely larger than the denominator, resulting in y approaching positive infinity. Therefore, y = +∞ is the horizontal asymptote as x approaches positive infinity.
As x approaches negative infinity, the term kx in the numerator dominates again, and the term x - k in the denominator becomes insignificant. This leads to the numerator getting infinitely larger than the denominator, resulting in y approaching negative infinity. Therefore, y = -∞ is the horizontal asymptote as x approaches negative infinity.
To find the vertical asymptote, we need to set the denominator equal to zero and solve for x:
x - k = 0
x = k
So, x = k is the vertical asymptote.
c. To find the point(s) of intersection of y with the line x - y = -2k, we can substitute y = (kx)/(x - k) into the equation and solve for x.
Substituting y in the equation:
x - (kx)/(x - k) = -2k
Multiply both sides by (x - k) to eliminate the denominator:
(x - k)(x - (kx)/(x - k)) = -2k(x - k)
Expanding and simplifying the equation:
(x - k)(x(x - k) - kx) = -2k(x - k)
(x - k)(x^2 - kx - kx) = -2kx + 2k^2
(x - k)(x^2 - 2kx + 2k^2) = -2kx + 2k^2
Expanding further:
x^3 - 2kx^2 + 2k^2x - kx^2 + 2k^2x - 2k^3 = -2kx + 2k^2
Combining like terms:
x^3 - 3kx^2 + 4k^2x - 2k^3 = -2kx + 2k^2
Moving all terms to one side:
x^3 - 3kx^2 + 4k^2x + 2kx - 2k^2 - 2k^3 = 0
Rearranging and factoring the equation:
x^3 - (2k + 3k)x^2 + (4k^2 + 2k)x - (2k^2 + 2k^3) = 0
x(x^2 - (2k + 3k)x + (4k^2 + 2k)) - (2k^2 + 2k^3) = 0
Now, we can try to solve for x using various methods, such as factoring, using the quadratic formula, or numerical methods like approximations.
To find the slope of the tangent line at the point(s) of intersection, we will need to find the derivative of the function y = (kx)/(x - k) and evaluate it at the x-values from the previous step. However, since we have not determined the x-values yet, we cannot find the slope of the tangent line at this point.
Sure! Let's go through each part of the question step by step:
a. To show that the function y = (kx)/(x-k) is symmetric with respect to the line y = x, we need to show that f(x) = f(y), where f(x) represents the function y = (kx)/(x-k).
To do this, we substitute y for x and x for y in the function and then simplify it. So we have:
f(y) = (ky)/(y-k)
To show f(x) = f(y), we need to rearrange the equation f(y) = (ky)/(y-k) so that it is f(x) = (kx)/(x-k).
To do this, we substitute y for x and x for y in the equation f(y) = (ky)/(y-k).
So we have:
f(y) = (ky)/(y-k)
f(x) = (kx)/(x-k)
Since f(x) = f(y), the function y = (kx)/(x-k) is symmetric with respect to the line y = x.
b. To find the horizontal asymptote, we need to determine the limit of the function as x approaches positive or negative infinity.
As x approaches positive or negative infinity, the value of (kx)/(x-k) becomes very large.
Therefore, the horizontal asymptote is y = 1.
To find the vertical asymptote, we need to determine the values of x that make the denominator zero.
Setting x - k = 0, we get x = k.
Therefore, the vertical asymptote is x = k.
c. To find the point(s) of intersection of the function y = (kx)/(x-k) with the line x - y = -2k, we need to solve the two equations simultaneously.
Substitute y = (kx)/(x-k) into the equation x - y = -2k:
x - ((kx)/(x-k)) = -2k
To simplify, we can multiply both sides by (x - k) to eliminate the denominator:
(x - k)(x - ((kx)/(x-k))) = -2k(x - k)
Expand and simplify:
x(x - k) - kx = -2kx + 2k^2
x^2 - kx - kx = -2kx + 2k^2
x^2 - 4kx + 2k^2 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-(-4k) ± sqrt((-4k)^2 - 4(1)(2k^2))) / (2(1))
Simplifying and factoring:
x = (4k ± sqrt(16k^2 - 8k^2)) / 2
x = (4k ± sqrt(8k^2)) / 2
x = (4k ± 2k * sqrt(2)) / 2
x = 2k ± k * sqrt(2)
So we have two possible solutions for x:
x1 = 2k + k * sqrt(2)
x2 = 2k - k * sqrt(2)
To find the corresponding y-values, we substitute each x-value into the function y = (kx)/(x-k):
y1 = (k(2k + k * sqrt(2))) / ((2k + k * sqrt(2)) - k)
= (2k^2 + k^2 * sqrt(2)) / (k + k * sqrt(2))
= (3k^2 * sqrt(2) + 2k^2) / (k + k * sqrt(2))
y2 = (k(2k - k * sqrt(2))) / ((2k - k * sqrt(2)) - k)
= (2k^2 - k^2 * sqrt(2)) / (k - k * sqrt(2))
= (k^2 - k^2 * sqrt(2)) / (-k + k * sqrt(2))
So, the two points of intersection are:
(x1, y1) = (2k + k * sqrt(2), (3k^2 * sqrt(2) + 2k^2) / (k + k * sqrt(2)))
(x2, y2) = (2k - k * sqrt(2), (k^2 - k^2 * sqrt(2)) / (-k + k * sqrt(2)))
To find the slope of the tangent line at the point(s) of intersection, we can take the derivative of the function with respect to x and then substitute the x-values of the points of intersection:
The derivative of y = (kx)/(x-k) is given by:
dy/dx = (k * (x-k) - kx) / (x-k)^2
Substituting x1 = 2k + k * sqrt(2):
dy/dx = (k * ((2k + k * sqrt(2))-k) - k*(2k + k * sqrt(2))) / ((2k + k * sqrt(2))-k)^2
Simplifying:
dy/dx = (k * (k * sqrt(2))) / ((k + k * sqrt(2)))^2
dy/dx = (k^2 * sqrt(2)) / (k^2 + 2k^2 * sqrt(2) + k^2 * 2)
dy/dx = (k^2 * sqrt(2)) / (k^2 * 2 + 2k^2 * sqrt(2) + k^2 * 2)
dy/dx = (k^2 * sqrt(2)) / (4k^2 + 2k^2 * sqrt(2))
Simplifying:
dy/dx = sqrt(2) / (4 + 2 * sqrt(2))
This is the slope of the tangent line at the point (2k + k * sqrt(2), (3k^2 * sqrt(2) + 2k^2) / (k + k * sqrt(2))).
Similarly, substituting x2 = 2k - k * sqrt(2) into the derivative equation, we can find the slope of the tangent line at the point (2k - k * sqrt(2), (k^2 - k^2 * sqrt(2)) / (-k + k * sqrt(2))).
I hope this helps! Let me know if you have any further questions.