The x-intercepts are marked with points located at (5,0) and (-6,0), while the y-intercept is marked with point located at (0,3/2). The asymptotes are y=1, x=-5, and x=4. Give your formula as a reduced rational function.
The horizontal asymptote at y=1 indicates that the numerator and denominator are of the same degree.
The vertical asymptotes indicate that
y = p(x) / (x+5)(x-4)
for some quadratic polynomial p(x)
The x-intercepts then dictate that
y = a(x+6)(x-5) / (x+5)(x-4)
The y-intercept at (0,3/2) means that
a(6)(-5) / (5)(-4) = 3/2
a = 3/2 * 20/30 = 1
So, y = (x+6)(x-5) / (x+5)(x-4) = (x^2+x-30)/(x^2+x-20)
See the graph at
https://www.wolframalpha.com/input/?i=(x%5E2%2Bx-30)%2F(x%5E2%2Bx-20)+for+-7+%3C%3D+x+%3C%3D+6
Sure! I'd be happy to help. First, let's find the equation of the rational function.
Since the x-intercepts are (5,0) and (-6,0), we know that the equation includes the factors (x - 5) and (x + 6).
And since the y-intercept is (0,3/2), we know that the equation includes the factor (y - 3/2).
Lastly, since there are vertical asymptotes at x = -5 and x = 4, we know that the equation includes the factors (x + 5) and (x - 4) in the denominator.
Putting it all together, the equation of the rational function is:
(y - 3/2) / ((x - 5)(x + 6)(x + 5)(x - 4))
And there you have it! That's the reduced rational function. But don't worry, it won't reduce your sense of humor!
To find the equation of the rational function, we can start by determining the factors of the equation using the given intercepts and asymptotes.
The x-intercepts are at (5,0) and (-6,0). This means that the factors of the equation will be (x - 5) and (x + 6).
The y-intercept is at (0,3/2), which means that the constant term of the equation will be 3/2.
The equation has asymptotes at y = 1, x = -5, and x = 4. This means that the equation will have factors of (y - 1), (x + 5), and (x - 4) in the denominator.
So, the equation of the rational function, in factored form, will be:
f(x) = (x - 5)(x + 6) / (y - 1)(x + 5)(x - 4)
To express it as a rational function in terms of x only, we can substitute y = f(x):
f(x) = (x - 5)(x + 6) / [(f(x) - 1)(x + 5)(x - 4)]
Now, let's solve for f(x):
f(x)[(f(x) - 1)(x + 5)(x - 4)] = (x - 5)(x + 6)
Expanding the equation:
f(x)[(f(x) - 1)(x^2 + x - 20)] = (x^2 + x - 30)
Distributing f(x):
[f(x)^2 - f(x)](x^2 + x - 20) = (x^2 + x - 30)
Expanding further:
f(x)^2(x^2 + x - 20) - f(x)(x^2 + x - 20) = (x^2 + x - 30)
Now, we can simplify and solve for f(x):
f(x)^2(x^2 + x - 20) - f(x)(x^2 + x - 20) - (x^2 + x - 30) = 0
Let a = f(x):
a^2(x^2 + x - 20) - a(x^2 + x - 20) - (x^2 + x - 30) = 0
(a^2 - a)(x^2 + x - 20) - (x^2 + x - 30) = 0
(x^2 + x - 20)(a^2 - a) - (x^2 + x - 30) = 0
Now, we can factor out the common quadratic term (x^2 + x - 20):
(x^2 + x - 20)[(a^2 - a) - 1] = 0
Since (x^2 + x - 20) cannot be equal to 0 (it's not an asymptote), we can set the other factor equal to 0:
(a^2 - a) - 1 = 0
a^2 - a - 1 = 0
This is a quadratic equation in terms of a. We can solve it using the quadratic formula:
a = [ -(-1) ± √((-1)^2 - 4(1)(-1)] / (2(1))
Simplifying:
a = [ 1 ± √(1 + 4)] / 2
a = [ 1 ± √5] / 2
So, the reduced rational function, in terms of x, is:
f(x) = [ 1 ± √5] / 2
To find the equation of a rational function, we need to consider the given information about the x-intercepts, y-intercept, and asymptotes.
The x-intercepts are the points where the graph of the function intersects the x-axis. In this case, the x-intercepts are (5,0) and (-6,0). This means that the function, when equals zero, has x-values of 5 and -6. Therefore, we can write two factors of the numerator as (x - 5) and (x + 6).
The y-intercept is the point where the graph of the function intersects the y-axis. In this case, the y-intercept is given as (0, 3/2). So, when x is zero, the function equals 3/2. This means we have a constant in the numerator, specifically (3/2).
Next, let's consider the asymptotes. The vertical asymptotes occur when the denominator of the rational function is equal to zero. Here, we have two vertical asymptotes: x = -5 and x = 4. So, we can have two factors in the denominator as (x + 5) and (x - 4).
Lastly, we have the horizontal asymptote, which is y = 1. This means that as x approaches positive or negative infinity, the function approaches the value of 1. Therefore, we don't have any x terms in the numerator, as the function approaches a constant value of 1. We can write this as (1) in the numerator.
Putting all the factors together, we can write the equation of the rational function as:
f(x) = (x - 5)(x + 6) / ((x + 5)(x - 4))
This is the reduced form of the given rational function equation.