Question

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

Explanation:

Step1: Find the parent function

The parent function of \( y = \sqrt[3]{x + 1}-2 \) is \( y=\sqrt[3]{x} \), which has a point at \((0,0)\) and is symmetric about the origin, increasing throughout its domain.

Step2: Analyze transformations

• Horizontal shift: The \( x + 1 \) inside the cube root means a shift left by 1 unit. So the point \((0,0)\) on the parent function moves to \((-1,0)\) after this shift.
• Vertical shift: The \(-2\) outside the cube root means a shift down by 2 units. So the point \((-1,0)\) from the horizontal shift moves to \((-1, -2)\) after this shift.

Step3: Analyze the graphs

• The first graph has a y - intercept around \( y = 3 \) and passes through \((-8,0)\), which doesn't match our key point \((-1,-2)\).
• The second graph (the lower one) should have a point around \((-1,-2)\) (since when \( x=-1 \), \( y=\sqrt[3]{-1 + 1}-2=\sqrt[3]{0}-2=-2 \)). Also, the shape of the cube - root function (increasing, with the characteristic S - shape) shifted left 1 and down 2 should match the lower graph.

Answer:

The Middle Graph (assuming the second graph is the middle one in the original problem's context, or more precisely, the graph with the curve passing near \((-1,-2)\) and having the correct shape of a shifted cube - root function)