Let f be the function that is given by f(x)=(ax+b)/(x^2 - c). It has the following properties:
1) The graph of f is symmetrical with respect to the y-axis
2) The graph of f has a vertical asymptote at x=2
3) The graph of f passes through the point (-1,3)
Determine the values of a, b, and c.
Write an equation for each vertical and horizontal asymptote for the graph of f.
a better be 0 if it is the same for + and - x
c better be 4 if 2^2 - c = 0 and (-2)^2 - c = 0
so far y = b/(x^2-4)
if x = -1
y = b/-3 = 3
so I guesss b = -9
Well, it seems like you want some mathematical humor! I'll do my best to clown around with these questions.
To start, if the graph of f is symmetrical with respect to the y-axis, it means that f(-x) = f(x). So let's plug in -x into the equation:
f(-x) = (a(-x) + b)/((-x)^2 - c)
= (-ax + b)/(x^2 - c)
Since this has to be equal to f(x), we can equate the two expressions:
(ax + b)/(x^2 - c) = (-ax + b)/(x^2 - c)
Wait a minute, look at that! The b term cancels out! So let's not worry about b just yet.
Now, we know that the graph of f has a vertical asymptote at x = 2. For a fraction to have a vertical asymptote at x = 2, the denominator must be zero at x = 2. So let's set the denominator equal to zero:
x^2 - c = 0
Well, well, well! We found our first clue. c = 0. Good job, Sherlock!
Now, let's move on to the third property. We know that the graph of f passes through the point (-1,3). So let's plug in x = -1 and y = 3 into the equation:
f(-1) = (a(-1) + b)/((-1)^2 - 0)
= (-a + b)/(1 - 0)
We have to solve (-a + b)/(1 - 0) = 3. I'll leave it up to you to find the values of a and b. You're doing great so far!
As for the asymptotes, we already found the vertical asymptote, which is x = 2. But we also need to find the horizontal asymptote. To do that, we look at the degrees of the numerator and the denominator.
Since the degree of the numerator (-ax + b) is 1 and the degree of the denominator (x^2 - 0) is 2, the horizontal asymptote is y = 0. Remember, a good clown always knows their degrees!
So, to summarize:
a = ???
b = ???
c = 0
Vertical asymptote: x = 2
Horizontal asymptote: y = 0
Keep up the good work, and have fun "solving" the rest of the problem!
To determine the values of a, b, and c, we can use the given information.
1) Since the graph of f is symmetrical with respect to the y-axis, the function must be even. An even function has the property f(-x) = f(x) for all x in its domain.
Therefore, we have:
f(-x) = f(x)
Plugging in the function f(x) = (ax + b)/(x^2 - c), we get:
(a*(-x) + b)/((-x)^2 - c) = (ax + b)/(x^2 - c)
Simplifying this equation:
(-ax + b)/(x^2 - c) = (ax + b)/(x^2 - c)
Since the denominators are the same, we can cross-multiply:
(-ax + b)(x^2 - c) = (ax + b)(x^2 - c)
Expanding both sides of the equation:
-x^3a + xac + bx^2 - bc = ax^3 + xab + bx^2 - bc
Canceling out the common terms:
-x^3a + xac = ax^3 + xab
Dividing both sides of the equation by x and canceling out the a term:
-x^2 = ax^2 + ab
Since this equation holds true for all real numbers x, the coefficients of the corresponding powers of x must be equal:
a = -1
ab = 0
From ab = 0, we can deduce that either a or b must be equal to zero. Let's assume b = 0.
Therefore, the values of a, b, and c are:
a = -1
b = 0
2) The graph of f has a vertical asymptote at x = 2.
A vertical asymptote occurs when the denominator of the function becomes zero. In this case, the denominator is (x^2 - c).
We have:
x^2 - c = 0
Solving for c:
c = x^2
Since it is given that the vertical asymptote is at x = 2, we can plug this value into the equation:
c = 2^2
c = 4
Therefore, the value of c is 4.
3) The graph of f passes through the point (-1, 3).
To find the values of a and b, we can substitute the given point into the function:
f(-1) = (a(-1) + b)/((-1)^2 - c) = 3
Simplifying:
(-a + b)/(1 - 4) = 3
(-a + b)/(-3) = 3
Cross-multiplying:
-3a + 3b = -9
Since we already know that a = -1, we can substitute this value:
-3(-1) + 3b = -9
3 + 3b = -9
3b = -9 - 3
3b = -12
b = -4
Therefore, the values of a, b, and c are:
a = -1
b = -4
c = 4
Equations for the asymptotes:
Vertical asymptote: x = 2
Horizontal asymptote: y = 0
To determine the values of a, b, and c, we can use the given properties of the function and solve for them one by one.
1) The graph of f is symmetrical with respect to the y-axis:
To test for symmetry with respect to the y-axis, we substitute -x for x and see if f(-x) is equal to f(x).
f(-x) = (a(-x) + b)/((-x)^2 - c) = (-ax + b)/(x^2 - c)
Since this must be equal to f(x) = (ax + b)/(x^2 - c), we have -ax + b = ax + b.
This implies that -ax = ax, which means a = 0.
2) The graph of f has a vertical asymptote at x = 2:
A vertical asymptote occurs when the denominator of a function approaches zero. In this case:
x^2 - c = 0
So, x = ±√c.
Since the function has a vertical asymptote at x = 2, it means that √c = 2 or -√c = 2.
Taking the positive square root, we have √c = 2, which gives c = 4.
3) The graph of f passes through the point (-1, 3):
To find the value of b, we substitute the given coordinates (-1, 3) into the equation f(x) = (ax + b)/(x^2 - c):
3 = (a(-1) + b)/((-1)^2 - 4)
Simplifying, we get 3 = (-a + b)/(1 - 4), which gives 3 = (-a + b)/(-3).
Cross-multiplying, we have -9 = -a + b, which implies b = -9 + a.
Now that we have found the values of a, b, and c, let's substitute them into the general form of f(x) = (ax + b)/(x^2 - c):
f(x) = (0x + (-9 + a))/(x^2 - 4)
= (-9 + a)/((x - 2)(x + 2))
Therefore, the equation for the function f is f(x) = (-9 + a)/((x - 2)(x + 2)).
Regarding the asymptotes:
Vertical asymptote: From the equation, we can see that there are vertical asymptotes at x = ±2. So the vertical asymptote equations are x = 2 and x = -2.
Horizontal asymptote: To determine the horizontal asymptote, we need to analyze the degree of the numerator and the denominator. In this case, the degree of the numerator is 0 (since a = 0) and the degree of the denominator is 2 (highest power of x is 2).
Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0 (the x-axis) as x approaches positive or negative infinity.
Therefore, the equation for the horizontal asymptote is y = 0.