Question

which system is represented in the graph?
$y > x^2 - 2x - 3$
$y > x + 3$
$y < x^2 - 2x - 3$
$y < x + 3$
$y \geq x^2 - 2x - 3$
$y \leq x + 3$
$y > x^2 - 2x - 3$
$y < x + 3$

Explanation:

To solve this, we analyze the boundary lines and the shaded regions:

Step 1: Analyze the parabola (\( y = x^2 - 2x - 3 \))

• The parabola \( y = x^2 - 2x - 3 \) has a dashed line (so the inequality is strict: \( > \) or \( < \), not \( \geq \) or \( \leq \)).
• The shaded region is above the parabola, so the inequality is \( y > x^2 - 2x - 3 \).

Step 2: Analyze the line (\( y = x + 3 \))

• The line \( y = x + 3 \) has a dashed line (so the inequality is strict: \( > \) or \( < \), not \( \geq \) or \( \leq \)).
• The shaded region is below the line, so the inequality is \( y < x + 3 \).

Step 3: Match with the options

Only the last option (\( y > x^2 - 2x - 3 \) and \( y < x + 3 \)) matches both conditions (dashed lines, above the parabola, below the line).

Answer:

\( y > x^2 - 2x - 3 \)
\( y < x + 3 \) (the fourth option, with the circle next to \( y > x^2 - 2x - 3 \) and \( y < x + 3 \))