Let f(x)=a(x-p)^2+q be a quadratic function, where a,p,q are constants and a≠0. f(x) is translated up by k units, translated right by c units, and then take a reciprocal of the translated function. Define g(x) be the function after performing all the transformations.
a. Find the conditions on the constants a,q and k so that g(x) does not have a vertical asymptote.
b. Find the x value(s) so that there is/are invariant point(s) linking f(x) and 1/(f(x)).
translations give you
a((x-c)-p)^2+q+k
so,
g(x) = 1/(a(x-c-p)^2+q+k)
= 1/(a(x-(c+p))^2 + (q+k))
= 1/a(x^2-2(c+p)x+(c+p)^2 + (q+k))
= 1/(ax^2 - 2a(c+p)x + a(c+p)^2+q+k)
to have no vertical asymptote, the denominator cannot be zero, which means that the discriminant is negative.
4a^2(c+p)^2 - 4a(a(c+p)^2+q+k) < 0
4a^2(c+p)^2 - 4a^2(c+p)^2 - 4a(q+k) < 0
4a(q+k) > 0
By invariant, do you mean where f(x)=g(x)?
If so, then just set up the equation and solve for x. It'll be a quartic, so good luck there.
d
d
d
d
d
d
d
d
d
d
d
a. To find the conditions on the constants a, q, and k so that g(x) does not have a vertical asymptote, we need to consider the reciprocal function.
First, let's translate f(x) up by k units. This can be done by adding k to the constant term q in f(x), giving us f(x) = a(x-p)^2 + q + k.
Next, let's translate f(x) right by c units. This can be done by replacing x with (x - c) in f(x). So, our new function becomes f(x) = a((x - c) - p)^2 + q + k.
Finally, let's take the reciprocal of the translated function. The reciprocal of any non-zero number x is given by 1/x. So, the reciprocal of f(x) is 1 / (a((x - c) - p)^2 + q + k).
For g(x) to have a vertical asymptote, the denominator of the reciprocal function cannot be equal to zero. So, we need to find the conditions under which the denominator can be zero.
Setting the denominator equal to zero, we get:
a((x - c) - p)^2 + q + k = 0.
Now, solve for x:
a((x - c) - p)^2 = -q - k,
((x - c) - p)^2 = (-q - k) / a,
(x - c) - p = ±sqrt((-q - k) / a),
x - c = p ± sqrt((-q - k) / a),
x = c + p ± sqrt((-q - k) / a).
As we can see, the denominator of the reciprocal function will be zero when x is equal to c + p ± sqrt((-q - k) / a). To avoid having vertical asymptotes, this means that the expression inside the square root must be negative, since the square root of a negative number is imaginary.
Therefore, the condition for g(x) to not have a vertical asymptote is:
(-q - k) / a < 0.
b. To find the x value(s) that give invariant points linking f(x) and 1/(f(x)), we need to find where f(x) is equal to its reciprocal, 1/(f(x)).
Taking the reciprocal of f(x), we get:
1 / (f(x)) = 1 / (a(x - p)^2 + q).
We want to solve for x when f(x) is equal to 1/(f(x)), so we can set up an equation:
a(x - p)^2 + q = 1 / (a(x - p)^2 + q).
Now, multiply both sides of the equation by (a(x - p)^2 + q) to eliminate the denominator:
(a(x - p)^2 + q)^2 = 1.
Expanding the equation:
a^2(x - p)^4 + 2aq(x - p)^2 + q^2 = 1.
This is a quartic equation in the variable (x - p)^2. Let's substitute a new variable, let's say u = (x - p)^2, and rewrite the equation:
a^2u^2 + 2aqu + q^2 = 1.
This is now a quadratic equation in u. We can solve it for u. Once we have the value(s) of u, we can substitute back x = p ± sqrt(u) to find the x value(s) that give invariant points linking f(x) and 1/(f(x)).
a. To find the conditions on the constants a, q, and k so that g(x) does not have a vertical asymptote, let's go step by step through the transformations:
1. Translation up by k units: This translates the function vertically, shifting it upwards. The function becomes f(x) + k.
2. Translation right by c units: This translates the function horizontally, shifting it to the right. The function becomes f(x - c) + k.
3. Taking a reciprocal: This flips the function vertically, turning any vertical asymptotes into horizontal asymptotes. The function becomes 1 / (f(x - c) + k).
Now, for g(x) = 1 / (f(x - c) + k) to not have a vertical asymptote, the denominator of the expression must not equal zero. In other words, we need to find the values of x for which f(x - c) + k ≠ 0.
Since f(x - c) is a quadratic function, we can write it as f(x - c) = a(x - c - p)^2 + q. Therefore, we need to solve the equation:
a(x - c - p)^2 + q + k ≠ 0
b. To find the x value(s) where there is/are invariant point(s) linking f(x) and 1/(f(x)), we need to find the values of x for which f(x) equals its reciprocal.
Recall that the reciprocal of f(x) is given by 1 / f(x) = 1 / (a(x - p)^2 + q). Therefore, we need to solve the equation:
f(x) = 1 / f(x)
Substituting f(x) with a(x - p)^2 + q, we get:
a(x - p)^2 + q = 1 / (a(x - p)^2 + q)
Now, we can solve this equation for the x value(s) that satisfy it.