Question

1. which is the correct graph of $y = \frac{2x - 3}{x - 2}$ (two graphs are provided as options)

Explanation:

To determine the correct graph of \( y = \frac{2x - 3}{x - 2} \), we analyze its key features:

Step 1: Identify Vertical Asymptote

The denominator \( x - 2 = 0 \) when \( x = 2 \). Thus, there is a vertical asymptote at \( x = 2 \).

Step 2: Identify Horizontal Asymptote

For a rational function \( \frac{ax + b}{cx + d} \), the horizontal asymptote is \( y = \frac{a}{c} \) (if degrees of numerator and denominator are equal). Here, \( a = 2 \), \( c = 1 \), so horizontal asymptote is \( y = 2 \).

Step 3: Analyze Behavior Around Vertical Asymptote

• For \( x < 2 \) (left of \( x = 2 \)): Test \( x = 0 \): \( y = \frac{2(0) - 3}{0 - 2} = \frac{-3}{-2} = 1.5 \) (positive, near \( y = 2 \) as \( x \to -\infty \)).
• For \( x > 2 \) (right of \( x = 2 \)): Test \( x = 3 \): \( y = \frac{2(3) - 3}{3 - 2} = \frac{3}{1} = 3 \) (positive, approaching \( y = 2 \) as \( x \to \infty \)).

Step 4: Compare with Graphs

• The first graph has a vertical asymptote at \( x = 2 \), horizontal asymptote \( y = 2 \), and behavior consistent with \( y = \frac{2x - 3}{x - 2} \) (left branch near \( y = 2 \), right branch approaching \( y = 2 \)).
• The second graph does not match the horizontal asymptote \( y = 2 \) (its horizontal asymptote appears to be \( y = 2 \) but the left branch’s intercept and behavior differ).

Answer:

The first graph (with vertical asymptote at \( x = 2 \), horizontal asymptote \( y = 2 \), and branches consistent with the function’s behavior) is correct.