https://file.aiquickdraw.com/mathbot/file/1730436897054fogha50m.jpg

To determine the equation for this graph, let's analyze its features:

1. The graph shows a curve that decreases sharply on the left side and approaches the y-axis as \( x \to 0^+ \), but does not cross it. This suggests a vertical asymptote at \( x = 0 \).
2. Additionally, the curve seems to get closer to the x-axis as \( x \to +\infty \), indicating a horizontal asymptote at \( y = 0 \).

These properties are consistent with the graph of a **reciprocal function** of the form:
\[
f(x) = \frac{k}{x}
\]
where \( k \) is a constant that affects the steepness of the curve.

Looking at the behavior of the graph, specifically the values it reaches, it appears that when \( x = 1 \), \( f(x) \approx 8 \). This implies that the constant \( k \) can be estimated by setting \( f(1) = 8 \):
\[
f(1) = \frac{k}{1} = 8 \Rightarrow k = 8
\]
So, the equation for this graph is likely:
\[
f(x) = \frac{8}{x}
\]

Would you like a more detailed explanation or have any questions about this solution?

Here are some related questions to expand your understanding:

1. How do vertical and horizontal asymptotes help determine the form of a function?
2. What other functions could have similar asymptotic behavior?
3. How would the graph change if \( k \) was negative instead of positive?
4. What if the function had an added constant, like \( f(x) = \frac{8}{x} + c \)?
5. How would the graph look if the function was \( f(x) = \frac{8}{x^2} \) instead?

**Tip:** Always look for asymptotes in graphs, as they can provide crucial hints about the form of the equation.

Find an equation for the graph sketched below.

Discover how to find the equation for a graph with vertical and horizontal asymptotes. Learn how to identify a reciprocal function, set constants based on graph values, and understand asymptotic behavior.

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