Our teacher gaves us a rational graph and we are supposed to determine its equation, there is more than one answer. So the aymptotes are
vertical= x=-5 and x=2
horizontal asymptote= y=2
then there is a cubic function that crosses the x-axis at -2,0 and the y axis at 2,0
To determine the equation of the rational graph given the information about its asymptotes and the points it passes through, you can follow a step-by-step approach:
1. Vertical Asymptotes: The given vertical asymptotes are x = -5 and x = 2. This indicates that the rational function has factors of (x + 5) and (x - 2) in the denominator.
2. Horizontal Asymptote: The given horizontal asymptote is y = 2. This suggests that the degree of the numerator and denominator polynomials should be the same. In this case, both the numerator and denominator will have a degree of 3.
3. x-Intercepts: The rational graph crosses the x-axis at -2, 0. This means that the function has a factor of (x + 2) in its numerator.
4. y-Intercept: The rational graph crosses the y-axis at 2, 0. This implies that the numerator evaluates to 2 when x = 0.
With the information above, we can form an equation for the rational function:
Numerator = A(x + 2)(x - 2)(x - k), where A is a constant and k represents another x-intercept.
Denominator = B(x + 5)(x - 2)(x - k), where B is a constant.
By setting the numerator equal to 2 when x = 0, we can determine the value of A:
2 = A(0 + 2)(0 - 2)(0 - k)
2 = A(-4k)
Solving for A, we find that A = -1/(2k).
So far, our equation looks like this:
Numerator = -1/(2k)(x + 2)(x - 2)(x - k)
Denominator = B(x + 5)(x - 2)(x - k)
Now, let's substitute the values of the x-intercepts (-2, 0) into the equation to find the value of k.
For x = -2:
0 = -1/(2k)(-2 + 2)(-2 - 2)(-2 - k)
0 = -1/(2k)(-4)(-4)(-2 - k)
0 = 16/(2k)(2 + k)
Since the numerator is zero, this equation implies that either 16 = 0 or k = -2 or k = -1.
Since 16 ≠ 0, k can either be -2 or -1.
So, the two possible equations for the rational function are:
Numerator = -2(x + 2)(x - 2)(x + 2)
Denominator = B(x + 5)(x - 2)(x + 2)
or
Numerator = -1(x + 2)(x - 2)(x + 1)
Denominator = B(x + 5)(x - 2)(x + 1)
Remember, these are just two possible equations. There may be other solutions depending on the specific details of the graph.