Find a possible formula for the function graphed below. The x-intercepts are marked with points located at (5, 0) and (-6, 0), while the y-intercept is marked with a point located at \left( 0, \frac{3}{2} \right). The asymptotes are y = 1, x = -5, and x = 4. Give your formula as a reduced rational function.
The horizontal asymptote at y=1 indicates that the numerator and denominator are of the same degree.
The vertical asymptotes indicate that
y = p(x) / (x+5)(x-4)
for some quadratic polynomial p(x)
The x-intercepts then dictate that
y = a(x+6)(x-5) / (x+5)(x-4)
The y-intercept at (0,3/2) means that
a(6)(-5) / (5)(-4) = 3/2
a = 3/2 * 20/30 = 1
So, y = (x+6)(x-5) / (x+5)(x-4) = (x^2+x-30)/(x^2+x-20)
x-intercepts are at (5, 0) and (-6, 0)
y = (x-5)(x+6)
asymptotes are x = -5, and x = 4.
y = (x-5)(x+6) / (x+5)(x-4)
asymptote at y = 1
still good for that, since top and bottom are the same degree, with highest coefficients both = 1
y-intercept is (0,3/2)
so let's check what y is when x=0.
y = (-5)(6) / (5)(-4) = -30/-20 = 3/2
so our final function is
y = (x-5)(x+6) / (x+5)(x-4)
so -- extra credit. How would you change the y-intercept, without changing any of the other parts?
To find a possible formula for the given function, we can gather information from the graph, including the x-intercepts, y-intercept, and asymptotes.
First, let's consider the x-intercepts. We are given that the x-intercepts are located at (5, 0) and (-6, 0). This means that the function crosses the x-axis at these points.
Next, let's look at the y-intercept. We are given that the y-intercept is located at (0, 3/2). This means that the function crosses the y-axis at this point.
Finally, let's consider the asymptotes. We are given three asymptotes: y = 1, x = -5, and x = 4. Asymptotes are lines that the function approaches but does not cross.
Now, based on the given information, we can conclude that the function is a rational function, since it has the form of a fraction (as it crosses the x and y-axes).
A possible formula for the function can be:
f(x) = A(x - 5)(x + 6) / ((x - 4)(x + 5)(x + 6)) + 1
Where A is a constant that is yet to be determined.
To find the value of A and reduce the function to its simplest form, we can use the given y-intercept.
f(0) = A(0 - 5)(0 + 6) / ((0 - 4)(0 + 5)(0 + 6)) + 1 = 3/2
Simplifying this equation, we get:
-30A / (4*5*6) + 1 = 3/2
Now, we solve this equation for A:
-30A / 120 + 1 = 3/2
-30A + 120 = 180
-30A = 60
A = -2
Now that we have the value of A, we can substitute it back into the formula to get the final answer:
f(x) = -2(x - 5)(x + 6) / ((x - 4)(x + 5)(x + 6)) + 1
Therefore, a possible formula for the function is a reduced rational function:
f(x) = -2(x - 5) / (x - 4)(x + 5)