which is the graph of $y = \\sqrt3{x + 1} - 2$?

Explanation:

Step1: Analyze the parent function

The parent function is ( y = \sqrt[3]{x} ), which has a point at ((0,0)) and is symmetric about the origin, with a cubic - like shape (increasing throughout the domain ((-\infty,\infty))).

Step2: Analyze the horizontal shift

For the function ( y=\sqrt[3]{x + 1}), we use the transformation rule for horizontal shifts. The general form for a horizontal shift of a function ( y = f(x)) to ( y=f(x - h)) (shift right by (h) units) or (y = f(x+h)) (shift left by (h) units). Here, (h = 1), so the graph of (y=\sqrt[3]{x}) is shifted left by 1 unit. So the point ((0,0)) on the parent function will be shifted to ((- 1,0)) for (y=\sqrt[3]{x + 1}).

Step3: Analyze the vertical shift

For the function (y=\sqrt[3]{x + 1}-2), we use the transformation rule for vertical shifts. The general form for a vertical shift of a function (y = f(x)) to (y=f(x)+k) (shift up by (k) units) or (y = f(x)-k) (shift down by (k) units). Here, (k = 2), so the graph of (y=\sqrt[3]{x + 1}) is shifted down by 2 units. The point ((-1,0)) on (y=\sqrt[3]{x + 1}) will be shifted to ((-1,-2)) for (y=\sqrt[3]{x + 1}-2).

Step4: Analyze the shape and domain/range

The function (y=\sqrt[3]{x+1}-2) has the same general shape as the cube - root function (increasing throughout the domain ((-\infty,\infty)) and range ((-\infty,\infty))). We can also check some other points. For example, when (x = 0), (y=\sqrt[3]{0 + 1}-2=1 - 2=-1). When (x=7), (y=\sqrt[3]{7 + 1}-2=\sqrt[3]{8}-2 = 2-2 = 0). When (x=-2), (y=\sqrt[3]{-2 + 1}-2=\sqrt[3]{-1}-2=-1 - 2=-3).

To identify the correct graph, we look for a cubic - root - shaped graph (increasing, passing through ((-1,-2)), ((0,-1)), ((7,0)), ((-2,-3)) etc.).

Answer:

To identify the graph of (y=\sqrt[3]{x + 1}-2), we analyze the transformations of the parent function (y = \sqrt[3]{x}):

1. Horizontal Shift: The (x+1) inside the cube - root shifts the graph of (y=\sqrt[3]{x}) left by 1 unit. The key point ((0,0)) on (y = \sqrt[3]{x}) moves to ((-1,0)) for (y=\sqrt[3]{x + 1}).
2. Vertical Shift: The (-2) outside the cube - root shifts the graph of (y=\sqrt[3]{x + 1}) down by 2 units. The key point ((-1,0)) on (y=\sqrt[3]{x + 1}) moves to ((-1,-2)) for (y=\sqrt[3]{x + 1}-2).
3. Shape and Other Points: The function is increasing for all real numbers (since the derivative (y^\prime=\frac{1}{3(x + 1)^{\frac{2}{3}}}) is non - negative for all (x\neq - 1) and defined at (x=-1) with (y^\prime=\infty) (vertical tangent)). We can also check additional points: when (x = 0), (y=-1); when (x = 7), (y = 0); when (x=-2), (y=-3). The correct graph will be the one that shows a cubic - root - shaped curve passing through ((-1,-2)), ((0,-1)), ((7,0)) and is increasing throughout the domain ((-\infty,\infty)).

(If we assume there are three graphs, for example, and the correct one has the key point ((-1,-2)) and the correct shape, we can say: "The graph with the point ((-1,-2)) and an increasing cubic - root - like shape (passing through ((0,-1)) and ((7,0)) etc.) is the correct one. If the options are labeled as Graph A, Graph B, Graph C, and Graph B has the correct key points and shape, then the answer is B. Graph B")