Question

the graph of $y = \sqrt3{x}$ is shown below.

graph of $y = \sqrt3{x}$

which of the following is the graph of $y = 3\sqrt3{x} + 2$?

choose 1 answer:

a graph a
b graph b
c graph c
d graph d

Explanation:

Step1: Analyze vertical stretch

The original function is \( y = \sqrt[4]{x} \). The transformed function is \( y = 3\sqrt[4]{x}+2 \). The coefficient \( 3 \) in front of \( \sqrt[4]{x} \) means a vertical stretch by a factor of \( 3 \). This makes the graph steeper than the original.

Step2: Analyze vertical shift

The \( +2 \) at the end means a vertical shift up by 2 units. So the entire graph of \( y = 3\sqrt[4]{x} \) (after stretching) is shifted up 2 units.

Step3: Evaluate options

• Option A: The graph doesn't seem to have the correct vertical shift or stretch.
• Option B: The vertical shift and stretch don't match (it looks shifted down or has wrong steepness).
• Option C: The graph has a vertical shift up (since it's higher than original) and the steepness from the vertical stretch (steeper than original \( y=\sqrt[4]{x} \)) matches \( y = 3\sqrt[4]{x}+2 \).
• Option D: The vertical shift and stretch don't match (wrong steepness and shift).

Answer:

C