Find a rational function that satisfies the given conditions.
Vertical asymptotes x=-2,x=7
Horizontal asymptote y=7/2
x-intercept (−5, 0)
vertical asymptotes: 1/((x+2)(x-7))
root at x = -5:
(x+5)/((x+2)(x-7))
horizontal asymptote. we need the degree to be equal, so try
7x(x+5)
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2(x+2)(x-7)
The problem here is that now we have another x-intercept at (0,0)
So, let's work with
7(x+5)^2
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2(x+2)(x-7)
That gives us an intercept, but it does not cross the x-axis.
We could go with something like
7(x^2+1)(x+5)
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2(x+2)(x-7)^2
That gives us the same degree top and bottom, a single crossing at -5, and still the two vertical asymptotes.
To find a rational function that satisfies the given conditions, we need to consider the properties of asymptotes and the given x-intercept.
First, let's determine the vertical asymptotes. We are given that x = -2 and x = 7 are the vertical asymptotes. This means that the rational function will have factors in the denominator that correspond to these values, resulting in undefined values at those points.
Next, let's determine the horizontal asymptote. We are given that the horizontal asymptote is y = 7/2. This means that as x approaches positive or negative infinity, the function's values approach 7/2.
Now, let's consider the x-intercept. We are given that the x-intercept is (-5, 0). This means that when x = -5, the function has a value of 0.
Based on this information, we can construct a rational function as follows:
1. Since there are vertical asymptotes at x = -2 and x = 7, we need factors of (x + 2) and (x - 7) in the denominator.
2. Since the horizontal asymptote is y = 7/2, the degree of the numerator should be less than or equal to the degree of the denominator. To achieve this, we can use a numerator with one factor of (x) and a constant.
Putting all these elements together, the rational function that satisfies the given conditions is:
f(x) = (ax + b) / [(x + 2)(x - 7)]
where a and b are constants that can be determined by using the x-intercept.
Using the x-intercept (-5, 0), we can substitute these coordinates into the function:
0 = (a(-5) + b) / [((-5) + 2)((-5) - 7)]
0 = (-5a + b) / (-21)
0 = -5a + b
To find the values of a and b, we need another equation. We can use the fact that the function should approach the horizontal asymptote y = 7/2 as x approaches infinity or negative infinity.
As x approaches infinity, the function should approach a single number, which in this case is 7/2. Therefore, we can set the limit of the function as x approaches infinity equal to 7/2:
lim(x->∞) [f(x)] = 7/2
By simplifying and plugging in our expression for f(x), we get:
lim(x->∞) [(ax + b) / [(x + 2)(x - 7)]] = 7/2
[lim(x->∞) (ax + b)] / [lim(x->∞) [(x + 2)(x - 7)]] = 7/2
(a*x + b) / (∞) = 7/2
As x approaches infinity, the x terms in the numerator and denominator become dominant, so we can ignore the constant term 'b':
(a*x) / (∞) = 7/2
This implies that a = 7/2.
Now that we have the value of a, we can substitute it back into the equation we obtained earlier from the x-intercept:
0 = -5a + b
0 = -5(7/2) + b
0 = -35/2 + b
b = 35/2
Therefore, the rational function that satisfies the given conditions is:
f(x) = (7/2 * x + 35/2) / [(x + 2)(x - 7)]
To find a rational function that satisfies these conditions, we need to consider the characteristics of a rational function.
A rational function is in the form of f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. The degree of the numerator (P(x)) must be less than or equal to the degree of the denominator (Q(x)).
Given the conditions:
1) Vertical asymptotes at x = -2 and x = 7: This means that the denominator, Q(x), must have factors of (x + 2) and (x - 7) for these vertical asymptotes.
2) Horizontal asymptote at y = 7/2: This means that the degree of the numerator, P(x), must be less than or equal to the degree of the denominator, Q(x). In this case, the degrees should match and be equal to 1.
3) x-intercept at (-5, 0): This means that the function passes through the point (-5, 0), which gives us one of the factors of the numerator.
Let's proceed with finding the rational function:
Step 1: Start with the denominator Q(x). We know that it must have the factors (x + 2) and (x - 7) to have vertical asymptotes at x = -2 and x = 7.
Q(x) = (x + 2)(x - 7)
Step 2: Determine the degree of the numerator P(x) based on the horizontal asymptote at y = 7/2. Since the degrees of P(x) and Q(x) should match, we need the numerator to be linear (degree 1).
P(x) = ax + b (where a and b are constants)
Step 3: Use the x-intercept (-5, 0) to find one of the factors of the numerator equation.
Since the function passes through (-5, 0), we can substitute these values into the equation P(x) = ax + b:
0 = a(-5) + b
0 = -5a + b
Step 4: Combine the numerator P(x) and denominator Q(x) to form the rational function.
f(x) = P(x)/Q(x)
f(x) = (ax + b) / [(x + 2)(x - 7)]
We have the equation for the function, but we still need to determine the values of a and b.
Step 5: Use the equation we derived from the x-intercept to find the values of a and b.
0 = -5a + b
Since the function does not cross the x-axis at any other point, we can substitute x = -5 back into the rational function:
0 = (-5a + b) / [(-5 + 2)(-5 - 7)]
0 = (-5a + b) / (-35)
Simplifying the equation further:
-5a + b = 0
Since we have two equations -5a + b = 0 and 0 = -5a + b, it implies that b = 5a.
Plugging in the value of b into one of the equations, we have:
-5a + 5a = 0
This equation simplifies to 0 = 0, which means that a can take any value.
Therefore, the rational function that satisfies the given conditions is:
f(x) = (ax + 5a) / [(x + 2)(x - 7)]