State an equation of a rational function that satisfies the given conditions: vertical asymptote at x=5, horizontal asymptote at y=-3, and x-intercept is 5/2.
Need help solving.
vertical asymptote at x=5
y = 1/(x-5)
horizontal asymptote at y=-3
y = -3x/(x-5)
x-intercept is 5/2
Change the x to (x - 5/2)
y = -3(x - 5/2)/(x-5)
See the graph at
www.wolframalpha.com/input/?i=-3(x+-+5%2F2)%2F(x-5)
Sure thing! Let's call the rational function "f(x)".
Since the vertical asymptote is at x = 5, we know that (x - 5) is a factor in the denominator of f(x).
Since the horizontal asymptote is at y = -3, we know that the degree of the numerator is equal to the degree of the denominator. Therefore, we can assume that both the numerator and denominator are linear.
Given that the x-intercept is 5/2, we know that (2x - 5) is a factor in the numerator of f(x).
Putting all these pieces together, we can write the equation as:
f(x) = (2x - 5) / (x - 5)
Hope that helps! Now, let's crack some jokes! Why don't scientists trust atoms? Because they make up everything!
To find an equation of a rational function that satisfies these conditions, consider the following steps.
Step 1: Vertical Asymptote:
Given that the vertical asymptote is at x = 5, we can include (x - 5) in the denominator.
Step 2: Horizontal Asymptote:
Given that the horizontal asymptote is at y = -3, the highest power of x in the numerator should match the highest power of x in the denominator. In this case, since there are no terms in the numerator, the degree of the denominator should be one.
Step 3: X-intercept:
Given that the x-intercept is at x = 5/2, we can include (x - 5/2) in the numerator.
Putting all these steps together, the equation of the rational function can be written as:
f(x) = (x - 5/2) / (x - 5)
This equation satisfies the given conditions of a vertical asymptote at x = 5, a horizontal asymptote at y = -3, and an x-intercept at x = 5/2.
I hope this helps! Let me know if you have any further questions.
To find an equation of a rational function that satisfies the given conditions, we can start by considering the characteristics of the rational function.
A rational function can be defined by two polynomial expressions, one for the numerator (top) and the other for the denominator (bottom). Let's call the numerator polynomial N(x) and the denominator polynomial D(x).
Based on the given conditions:
1. Vertical asymptote at x=5: This means that the denominator polynomial D(x) must have a factor of (x - 5) in order for the function to be undefined at x=5.
2. Horizontal asymptote at y=-3: As x approaches positive or negative infinity, the function value should approach -3. This means that the degree of the denominator polynomial must be equal to or greater than the degree of the numerator polynomial.
3. x-intercept is 5/2: This means that the function value is equal to zero when x=5/2. Therefore, the numerator polynomial N(x) must have a factor of (x - 5/2) for the function to intersect the x-axis at x=5/2.
With these conditions in mind, we can start constructing the equation.
Let's consider the form:
f(x) = N(x)/D(x)
Since we know that the vertical asymptote is at x=5, the denominator polynomial must have a factor of (x - 5):
D(x) = (x - 5)
We also know that the function intersects the x-axis at x=5/2, so the numerator polynomial must have a factor of (x - 5/2):
N(x) = (x - 5/2)
To satisfy the horizontal asymptote at y=-3, we can set the degrees of N(x) and D(x) to be the same. We can achieve this by introducing another factor of (x - 5) to the numerator:
N(x) = (x - 5/2)(x - 5)
Putting it all together, we have the equation:
f(x) = (x - 5/2)(x - 5)/(x - 5)
Simplifying:
f(x) = (x^2 - (5/2)x - 5x + (25/2))/(x - 5)
Further simplification:
f(x) = (x^2 - (15/2)x + (25/2))/(x - 5)
So, an equation of a rational function that satisfies the given conditions is:
f(x) = (x^2 - (15/2)x + (25/2))/(x - 5)