Find an equation of a rational function that satisfies the given conditions:
Vertical asymptotes: 𝑥 = −5, 𝑥 = 2
Horizontal asymptotes: 𝑦 = −2
𝑥-intercepts: 𝑥 = −6, 𝑥 = 4
𝑓(−2) = −4
𝑥-intercepts: 𝑥 = −6, 𝑥 = 4
----> we will need f(x) = a(x+6)(x-4) as a start
Vertical asymptotes: 𝑥 = −5, 𝑥 = 2
----> adjust to f(x) = a(x+6)(x-4)/(b(x+5)(x-2))
Horizontal asymptotes: 𝑦 = −2
----> a/b = -2 , or a = -2b
so far we have:
f(x) = -2b(x+6)(x-4)/(b(x+5)(x-2) = -2(x+6)(x-4)/((x+5)(x-2))
we also need
f(-2) = -4
f(-2) = -2(4)(-6) / (3)(-4) = 4 , well that was lucky
f(x) = -2(x+6)(x-4)/((x+5)(x-2))
verify by entering my equation into
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extra credit: how would you satisfy the additional condition that the y-intercept is at (0,10) ?
To find an equation of a rational function that satisfies the given conditions, we need to consider the properties of rational functions.
First, let's consider the vertical asymptotes. The vertical asymptotes occur at values of x where the denominator of the rational function becomes zero. Given that the vertical asymptotes are x = -5 and x = 2, we can start with the denominator as (x + 5)(x - 2).
Next, let's consider the horizontal asymptote. The horizontal asymptote describes the behavior of the rational function as x approaches positive or negative infinity. Since the horizontal asymptote is y = -2, we know that the degree of the numerator and denominator of the rational function should be the same. In this case, we want the leading terms to be of equal degree to ensure the horizontal asymptote is y = -2. So, let's assume the numerator and denominator have the same degree and consider it as (ax + b)/(x + 5)(x - 2), where a and b are constants to be determined.
Now let's consider the x-intercepts. The x-intercepts occur where the numerator of the rational function becomes zero. Given that the x-intercepts are x = -6 and x = 4, we can set up the numerator as (x + 6)(x - 4).
Putting it all together, we have the equation of the rational function as:
f(x) = (ax + b)/(x + 5)(x - 2)
To determine the values of a and b, we can use the given condition that f(-2) = -4. Substituting x = -2 and y = -4 into the equation, we can solve for a and b.
-4 = (a(-2) + b)/((-2) + 5)((-2) - 2)
-12 = -2a - 3b
To make this system of equations solvable, let's solve for a or b in one of the equations.
-2a = -12 - 3b
2a = 12 + 3b
a = 6 + (3/2)b
Now, we can substitute the expression for a into the second equation:
6 + (3/2)b = 6 + (3/2)b
0 = 3b
This implies that b = 0. Now, substitute b = 0 back into a = 6 + (3/2)b:
a = 6 + (3/2)(0)
a = 6
So, the equation of the rational function that satisfies the given conditions is:
f(x) = (6x)/(x + 5)(x - 2)