Question

consider the table representing a rational function.

which equation could represent a vertical asymptote of the graph of the function?

• (x = 0)
• (y = 0)
• (x = -1)
• (y = -1)

Explanation:

⚡ Using what you learned: Identifying Restrictions and Asymptotes

Step 1: Analyze the behavior near the undefined value

A vertical asymptote occurs where a rational function's output approaches positive or negative infinity as the input \( x \) approaches a specific constant value from either side.

Looking at the table:

• As \( x \) approaches \(-1\) from the left (values like \(-1.1\), \(-1.009\)), the function values \( f(x) \) decrease rapidly toward negative infinity:

$$ f(-1.1) = -10, \quad f(-1.009) = -111.\overline{1} $$

• At \( x = -1 \), the function is undefined.
• As \( x \) approaches \(-1\) from the right (values like \(-0.99\), \(-0.9\)), the function values \( f(x) \) increase rapidly toward positive infinity:

$$ f(-0.99) = 100, \quad f(-0.9) = 10 $$

Step 2: Identify the vertical asymptote equation

Because the function values blow up to \(\pm\infty\) as \( x \) approaches \(-1\), there is a vertical asymptote at this boundary.

Vertical lines on a coordinate plane are always written in the form:

$$ x = c $$

Since the behavior occurs at \( x = -1 \), the equation of the vertical asymptote is:

$$ x = -1 $$

Answer:

\( x = -1 \)