Question

1. choose the correct graph of the function $y = 3sqrt3{x + 2} - 4$

Explanation:

Step1: Find the key point (when \( x=-2 \))

Substitute \( x = -2 \) into the function: \( y=3\sqrt[3]{-2 + 2}-4=3\sqrt[3]{0}-4 = 0 - 4=-4 \). So the graph should pass through \( (-2, -4) \).

Step2: Analyze the shape and other points

The parent function is \( y=\sqrt[3]{x} \), which is a cubic root function, symmetric about the origin, with a point at \( (0,0) \). Our function is a transformation: horizontal shift left 2 units, vertical stretch by 3, and vertical shift down 4 units. Let's check the second graph (the lower one): when \( x=-2 \), let's see the y - value. From the graph, around \( x = -2 \) (which is between -4 and 0 on x - axis), the y - value should be -4. The lower graph seems to have a point around \( x=-2,y = - 4\) (or near that), while the upper graph has a y - intercept at 8, which when \( x = 0 \), \( y=3\sqrt[3]{2}-4\approx3\times1.26 - 4=3.78 - 4=-0.22
eq8 \), so the upper graph is wrong. So the correct graph is the lower one (the second graph, or the middle - looking one here).

(Note: Since the original problem's graph display might have three options, but in the provided image, there are two visible. Assuming the second graph (the lower one) is the correct one. If there are three, we would re - evaluate, but based on the function analysis, the graph passing through \( (-2,-4) \) and having the shape of a cubic root function (with the transformation) is the lower graph here.)

Answer:

The Middle Graph (assuming the second graph is the middle one here, but actually, let's analyze the function \( y = 3\sqrt[3]{x + 2}-4 \))