To determine which graph represents the inequality [tex]\( y > \sqrt{4x} + 2 \)[/tex], let's carefully examine the inequality step by step.
1. Understanding the inequality [tex]\( y > \sqrt{4x} + 2 \)[/tex]:
- The expression [tex]\( \sqrt{4x} \)[/tex] is the square root function, multiplied by 2 inside.
- Adding 2 to [tex]\( \sqrt{4x} \)[/tex] shifts the graph of [tex]\( \sqrt{4x} \)[/tex] upward by 2 units.
2. Graphing the equality [tex]\( y = \sqrt{4x} + 2 \)[/tex]:
- This equation represents a curve that starts from [tex]\((0, 2)\)[/tex] because when [tex]\( x = 0 \)[/tex], [tex]\( y = \sqrt{4 \cdot 0} + 2 = 2 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] increases due to the [tex]\( \sqrt{4x} \)[/tex] term.
- The curve gets steeper as [tex]\( x \)[/tex] increases, resembling the right half of a parabola that is stretched and shifted upward.
3. Interpreting the inequality [tex]\( y > \sqrt{4x} + 2 \)[/tex]:
- This inequality implies that the region of interest is above the curve [tex]\( y = \sqrt{4x} + 2 \)[/tex].
- Any point in the graph that lies above this curve satisfies the inequality.
To determine which graph is correct, we need to identify a graph that:
- Contains the curve [tex]\( y = \sqrt{4x} + 2 \)[/tex].
- Has the region above this curve shaded, indicating the region where [tex]\( y \)[/tex] values are greater than [tex]\( \sqrt{4x} + 2 \)[/tex].
Given this understanding, the correct graph showing the region that satisfies the inequality [tex]\( y > \sqrt{4x} + 2 \)[/tex] is:
- Option A: Graph B
- Option B: Graph C
- Option C: Graph A
- Option D: Graph D
By analyzing the various attributes and matching them against the described characteristics, the correct option is:
[tex]\[ \boxed{C} \][/tex]
