Write the equation of the rational function that passes through the points (0,0) and (4,8/7), has the x-axis as a horizontal asymptote, and has 2 vertical asymptotes at x=3 and x=-3.

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To find the equation of the rational function, we need to consider the given information.

First, we know that the x-axis serves as a horizontal asymptote. This means that as x approaches positive or negative infinity, the function approaches zero. Therefore, the numerator of our rational function should have a degree less than the denominator.

Next, we are given that the function has vertical asymptotes at x = 3 and x = -3. This implies that the denominator should contain factors of (x - 3) and (x + 3).

Now, let's consider the points (0,0) and (4, 8/7). We can use these points to determine the value of the numerator.

Since the point (0,0) lies on the graph, it means that when x = 0, the function evaluates to y = 0. Therefore, the numerator of our rational function must have a factor of x in order for y to be zero at x = 0.

Now let's consider the other point (4, 8/7). When x = 4, the function evaluates to y = 8/7. This gives us another piece of information about the numerator.

Putting all this information together, we can write the equation of the rational function in the following form:

f(x) = (Ax)(x - 3)(x + 3) / (x - 3)(x + 3)

where A is a constant that needs to be determined.

To find the value of A, we can substitute the x and y values from the point (4, 8/7) into the equation:

8/7 = (A(4)(4 - 3)(4 + 3)) / ((4 - 3)(4 + 3))
8/7 = (A(4)(7)) / (1)(7)
8/7 = 4A

Solving for A, we have:
A = (8/7) / 4
A = 2/7

Therefore, the equation of the rational function that satisfies the given conditions is:

f(x) = (2/7)(x)(x - 3)(x + 3) / (x - 3)(x + 3)

To find the equation of the rational function that satisfies these conditions, we need to consider the properties of rational functions and use the given information.

First, let's consider the horizontal and vertical asymptotes. If the x-axis is a horizontal asymptote, it means that as x approaches positive or negative infinity, the function approaches the x-axis. This implies that the degree of the numerator is less than or equal to the degree of the denominator.

Since the vertical asymptotes are at x = 3 and x = -3, it means that the denominator must have factors of (x - 3) and (x + 3). Therefore, the denominator must be some form of (x - 3)(x + 3).

Next, let's find the equation of the rational function that passes through the given points (0,0) and (4,8/7). To do this, we can use the fact that the equation of a rational function can be written in the form:

f(x) = (a(x - h)(x - k))/(b(x - c)(x - d))

where (h, k) is a point on the graph and c, d are the vertical asymptotes.

Let's substitute the point (0,0) into the equation to find a and b:

0 = (a(0 - h)(0 - k))/(b(0 - c)(0 - d))

Since (0,0) implies a zero numerator, we have:

0 = (0 - h)(0 - k)

This implies that one of the factors (x - h) or (x - k) is 0. Therefore, either h or k must be zero.

If h = 0, then the equation becomes:

0 = (0 - 0)(0 - k) = 0

This still holds true for any value of k.

Similarly, if k = 0, then the equation becomes:

0 = (0 - h)(0 - 0) = 0

Again, this holds true for any value of h.

Therefore, we can conclude that h = 0 and k = 0.

Now, we have the equation:

f(x) = (a(x - 0)(x - 0))/(b(x - c)(x - d))

Simplifying this equation, we get:

f(x) = (ax^2)/(b(x - c)(x - d))

We have two remaining unknowns, a and b. To find them, we can substitute the other given point (4,8/7) into the equation:

8/7 = (a(4^2))/(b(4 - c)(4 - d))

8/7 = (16a)/(b(4 - c)(4 - d))

We can rearrange this equation to solve for a:

a = (8/7)(b(4 - c)(4 - d))/(16)

Now, let's substitute this value of a into our original equation:

f(x) = ((8/7)(b(4 - c)(4 - d))x^2)/(16b(x - c)(x - d))

Simplifying, we find:

f(x) = (8(b(4 - c)(4 - d))x^2)/(112(x - c)(x - d))

Finally, to express this equation without fractions in the numerator, we can multiply both numerator and denominator by 112:

f(x) = (8(b(4 - c)(4 - d))x^2)/(112(x - c)(x - d))

This gives us the equation of the rational function that passes through the points (0,0) and (4,8/7), has the x-axis as a horizontal asymptote, and has vertical asymptotes at x = 3 and x = -3.