Write the equation of the function that passes through the points (0,0) and (4, 8/9 (fraction) ); has the x-axis as a horizontal asymptote; and has 2 vertical asymptotes x=3 and x= -3. Show calculations.
I’m genuinely confused on how to do this question and tried everything :(
Let's start with 2 vertical asymptotes x=3 and x= -3
That means that the denominator must contain (x-3)(x+3)
Since it passes through (0,0), it would suggest that we have a factor of ax at the top
If the x-axis is a horizontal asymptote, then the numerator must be a degree lower than the
degree of the denominator, confirming my suggestion in the last sentence
We have the point (4, 8/9) to worry about
So how about starting with y = ax/((x-3)(x+3))
or y = ax/(x^2 - 9)
testing the point (0,0), yup that works
if x = 4, y = 8/9
8/9 = 4a/(16-9)
56 = 36a
a = 56/36 = 14/9
the function f(x) = 14x/(9(x^2 - 9)) or 14x/(9x^2 - 81) satisfies all your stated conditions
check:
https://www.wolframalpha.com/input/?i=graph+f%28x%29+%3D+14x%2F%289%28x%5E2+-+9%29%29+from+-8+to+8
YUP!!
Well, let's break it down step by step!
First, let's start with the equation of a rational function. It takes the form:
f(x) = (ax + b)/(cx + d)
We're given that the function passes through the points (0,0) and (4, 8/9). Let's use these points to find the values of a, b, c, and d.
When x = 0, y = 0, so we have:
0 = (0a + b)/(0c + d)
0 = b/d ... (1)
Next, when x = 4, y = 8/9, we have:
8/9 = (4a + b)/(4c + d)
8/9 = (4a + b)/(4c + d) ... (2)
We also know that the x-axis is a horizontal asymptote. This means that as x approaches positive or negative infinity, the function approaches the x-axis. As a result, the constant term in the denominator, d, must be zero.
Next, we know that there are two vertical asymptotes at x = 3 and x = -3. This means that the constants in front of x in the denominators, c and -c, must also be zero.
With these conditions in mind, we have:
b = 0 ... (from Equation 1)
8/9 = 4a/(4d) ... (from Equation 2)
c = 0
d = 0
From Equation 2, we can simplify it to:
8/9 = a/d
a = (8/9)d
Now our equation becomes:
f(x) = (ax + b)/(cx + d)
= ((8/9)d)x ... (substituting values of a, b, c, and d)
= (8/9)x
So, the equation of the function that satisfies all the given conditions is f(x) = (8/9)x. I hope that puts a smile on your face!
To find the equation of the function that satisfies the given conditions, we can follow these steps:
Step 1: Determine the vertical asymptotes.
Given that the function has two vertical asymptotes at x = 3 and x = -3, we can write the function in factored form as:
f(x) = A(x – 3)(x + 3)
Step 2: Determine the behavior as x approaches infinity.
Since the x-axis is a horizontal asymptote, as x approaches infinity, the function approaches the x-axis. This means that the degree of the numerator (A in this case) must be less than the degree of the denominator (2 in this case). Since there is no numerator given, we can write:
f(x) = A/(x² + 9)(x - 3)(x + 3)
At this point, we have determined the equation in a general form. Now, we need to determine the value of A.
Step 3: Use the given points to find the value of A.
Given the points (0, 0) and (4, 8/9), we can substitute these values into the equation to form two equations and solve for A.
For the point (0, 0):
0 = A/(0² + 9)(0 - 3)(0 + 3)
0 = A/27
For the point (4, 8/9):
8/9 = A/(4² + 9)(4 - 3)(4 + 3)
8/9 = A/325
Now, we can solve these two equations simultaneously to find the value of A.
0 = A/27 ----> Equation 1
8/9 = A/325 ----> Equation 2
To solve, multiply Equation 1 by 9 and Equation 2 by 325:
0 = 9A/27 ----> Equation 1
8 = 325A/325 ----> Equation 2
Simplify:
0 = 9A/27 ----> Equation 1
8 = A ----> Equation 2
Therefore, the value of A is 8.
Step 4: Substitute the value of A back into the equation.
Now that we have the value of A, we can substitute it back into the general form equation:
f(x) = 8/(x² + 9)(x - 3)(x + 3)
Thus, the equation of the function that passes through the points (0, 0) and (4, 8/9), has the x-axis as a horizontal asymptote, and has vertical asymptotes at x = 3 and x = -3 is:
f(x) = 8/(x² + 9)(x - 3)(x + 3)
Note: Make sure to check if the function satisfies any other given conditions or restrictions, if any, to ensure the accuracy of the solution.