Find a possible formula for the function graphed below. Assume the function has only one x-intercept at the origin, and the point marked on the graph below is located at (2,12). The asymptotes are x=−2 and x=1. Give your formula as a reduced rational function.
the asymptotes mean
y = ??? / (x+2)(x-1)
y(0)=0, so
y = ax / (x+2)(x-1)
y(2) = 12, so
12 = 2a/4
a = 24
y = 24x / (x+2)(x-1)
ty!
Sure, let me clown around with some numbers for you. Since the graph has an x-intercept at the origin and the point (2,12) is on the graph, we can assume the formula has the form f(x) = ax^2 + bx.
Now, let's take a look at the asymptotes. The asymptotes at x = -2 and x = 1 suggest that there might be vertical asymptotes at these x-values. However, since we are looking for a rational function, we need to include these asymptotes in the denominator.
Considering all this silliness, a possible formula for the function is f(x) = (12x^2 - 48x) / ((x + 2)(x - 1)). This rational function has a reduced form, ready to tickle your mathematical funny bone.
To find a possible formula for the function graphed below, we need to consider the given information - the x-intercept, the point (2,12), and the asymptotes.
Given that the x-intercept is at the origin, we know that the function crosses the x-axis at x = 0. Therefore, the factor (x - 0) or simply x is present in the numerator.
Since the point (2,12) lies on the graph, we can substitute these values into the formula to get an equation:
12 = f(2) = a(2) / (b(2) - 4)
Simplifying this equation yields:
12 = 2a / (2b - 4)
We need to find the values of a and b that satisfy this equation. Rewriting the equation, we have:
12(2b - 4) = 2a
24b - 48 = 2a
12b - 24 = a
Now we have an expression for a in terms of b. Let's substitute this value for a back into the equation:
12 = 2(12b - 24) / (2b - 4)
Next, we can simplify the expression by factoring out common factors:
12 = 24b - 48 / 2b - 4
Dividing both the numerator and denominator by 2, we get:
12 = 12b - 24 / b - 2
Simplifying further, we remove the fractions by multiplying both sides by (b - 2):
12(b - 2) = 12b - 24
Expanding the equation:
12b - 24 = 12b - 24
We notice that the equation is an identity, meaning it holds true for all values of b. This implies that there are infinitely many solutions, and we can choose any value for b.
Therefore, a possible formula for the function graphed below as a reduced rational function is:
f(x) = (x) / (b(x - 2))
Here, b is any real number except for 2, as it would cause division by zero.