Question

graph the function
$f(x) = 4sqrt{x} + 2$

choose the correct graph on the right.

\\(\circ\\) a.

\\(\circ\\) b.

\\(\boldsymbol{\odot}\\) c.

\\(\circ\\) d.

Explanation:

Step1: Analyze the parent function

The parent function is \( y = \sqrt{x} \), which has a domain \( x \geq 0 \) and starts at \( (0,0) \), increasing slowly then more steeply.

Step2: Analyze the transformations

The function \( f(x) = 4\sqrt{x} + 2 \) has two transformations:

• Vertical stretch by a factor of 4: This makes the graph steeper than \( y = \sqrt{x} \).
• Vertical shift up by 2 units: This moves the starting point from \( (0,0) \) to \( (0, 0 + 2) = (0, 2) \).

Step3: Evaluate the options

• Option A: The graph is horizontal, which doesn't match the square - root shape. Eliminate A.
• Option B: The starting point seems to be around \( (0, - 4) \) or lower, not \( (0,2) \). Eliminate B.
• Option C: The starting point is not at \( (0,2) \) and the steepness is not consistent with a vertical stretch of 4. Eliminate C.
• Option D: The graph starts at \( (0,2) \), has the shape of a square - root function (increasing, starting at the y - axis), and the steepness is consistent with a vertical stretch of 4.

Answer:

D