Question

graphing asymptotes for a rational functions
the same rational function are shown below.

on graph below draw the horizontal asymptote and write the equation for the horizontal asymptote underneath.
(f(x) = \frac{-2}{x - 3})

on graph below, draw the vertical asymptote and write the equation for the vertical asymptote underneath.
(f(x) = \frac{-2}{x - 3})

Explanation:

⚡ Using what you learned: Graphing Rational Functions

Step 1: Find the Horizontal Asymptote

For the rational function:

$$ f(x) = \frac{-2}{x - 3} $$

As \( x \to \infty \) or \( x \to -\infty \), the denominator becomes extremely large, causing the fraction to approach \( 0 \).

The equation of the horizontal asymptote is:

$$ y = 0 $$

To draw this on the left graph, draw a dashed horizontal line along the x-axis (\( y = 0 \)).

Step 2: Find the Vertical Asymptote

The vertical asymptote occurs where the denominator is equal to zero and the numerator is non-zero:

$$ x - 3 = 0 $$

$$ x = 3 $$

The equation of the vertical asymptote is:

$$ x = 3 $$

To draw this on the right graph, draw a dashed vertical line passing through \( 3 \) on the x-axis (\( x = 3 \)).

Answer:

• Left Graph (Horizontal Asymptote):
• Draw: A horizontal dashed line along the x-axis (\( y = 0 \)).
• Equation: \( y = 0 \)

• Right Graph (Vertical Asymptote):
• Draw: A vertical dashed line passing through \( x = 3 \).
• Equation: \( x = 3 \)