Question

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

eq - 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")