Write an equation to represent a rational function with the following conditions:
- reciprocal of a quadratic function
- asymptotes with the equations x = -5, x= 4/3, and y = 0
- y < 0 whenever x < -5
- y-intercept of 1.5
To represent a rational function with these conditions, we'll start by representing the reciprocal of a quadratic function, which has the form 1/(ax^2 + bx + c).
Next, let's consider the asymptotes. We have three asymptotes: x = -5, x = 4/3, and y = 0. For the vertical asymptotes, we need factors that will make the denominator equal to zero at x = -5 and x = 4/3. Thus, the denominator will be of the form (x - (-5))(x - (4/3)) = (x + 5)(x - 4/3).
For the horizontal asymptote, since y = 0, we know that the degree of the numerator is less than or equal to the degree of the denominator. This means that the quadratic in the denominator must have a higher degree than the numerator.
Considering the y-intercept of 1.5, we can set x = 0 in our equation to find the value of the numerator. Since the denominator has factors of (x + 5) and (x - 4/3), the y-intercept occurs when x = 0, so we can write the numerator as k(x - 0) = kx. Since the y-intercept is 1.5, we can set up the equation as follows: 1.5 = k(0) = 0, which means k = 1.5.
Putting everything together, our rational function equation is:
f(x) = (1.5x)/((x + 5)(x - 4/3))
This equation satisfies the conditions of being the reciprocal of a quadratic function, having the given asymptotes, y < 0 whenever x < -5, and having a y-intercept of 1.5.
To create an equation that represents a rational function with these conditions, we can start by constructing the equation in factored form.
First, let's start with the asymptotes. Since the asymptotes are defined by x = -5 and x = 4/3, we can incorporate these into our factored form as (x + 5) and (x - 4/3).
Next, we need to consider the reciprocal of a quadratic function. This means that the numerator will be a constant, and the denominator will be a quadratic function.
So, our equation in factored form would be:
ƒ(x) = k / [(x + 5)(x - 4/3)]
To determine the constant, k, we can use the y-intercept. It is given that the y-intercept is 1.5. The y-intercept occurs when x = 0, so let's substitute those values into the equation:
1.5 = k / [(0 + 5)(0 - 4/3)]
Simplifying this equation, we get:
1.5 = k / (-20/3)
To solve for k, we can multiply both sides by (-20/3):
k = 1.5 * (-20/3)
k = -30/3
k = -10
Now that we have the value of k, we can substitute it back into the equation:
ƒ(x) = -10 / [(x + 5)(x - 4/3)]
Thus, the equation to represent the given conditions is:
ƒ(x) = -10 / [(x + 5)(x - 4/3)]
the roots are -5 and 4/3, so start with
y = a/((x+5)(3x-4))
This also has y=0 as the horizontal asymptote
When x < -5, both factors are negative, so a must be negative
Now, using the point (0,3/2), we get
3/2 = a/(-5 * -19)
a = 142.5
But that is positive. So we need to make that denominator positive, without changing the asymptotes. One way is to square a factor.
y = a/((x+5)^2 (3x-4))
Now at x=0,
3/2 = a/(25 * -4)
a = -150
and so
y = -150/((x+5)^2 (3x-4)) = -150/(3x^3+26x^2+35x-100)