Find the vertical asymptotes of the function.y = x2 + 37x − 2x2 x = (smaller value)x = (larger value)

To find the vertical asymptotes of the function $y = \frac{x^2 + 37x - 2x^2}{x}$, we need to follow these steps:

1. ### Break Down the Problem

First, rewrite the function in a simpler form and identify where the denominator is equal to zero, as these points will indicate potential vertical asymptotes.

2. ### Relevant Concepts

Vertical asymptotes occur where the denominator of a rational function is zero, provided the numerator is not also zero at those points.

3. ### Simplify the Function

The given function can be written as: $y = \frac{x^2 + 37x - 2x^2}{x} = \frac{-x^2 + 37x}{x}$ We can simplify this to: $y = -x + 37 \quad \text{for} \quad x \neq 0$

4. ### Identify the Denominator

The next step is to identify where the original denominator ($x$) is equal to zero: $x = 0$ Now we must check if there is a vertical asymptote at $x = 0$ by verifying the numerator at $x = 0$: $-0 + 37 = 37 \neq 0$ Since the numerator is not zero when $x$ approaches zero, $x = 0$ is indeed a vertical asymptote.

Final Answer

The vertical asymptotes of the function are: $x = 0$ There are no larger or smaller values, as there is only one vertical asymptote in this case.