Question

this graph shows both a reflection and a translation. what function is being graphed?
$y = \sqrt3{x - 2} - 1$
$y = -\sqrt3{x}$
$y = -\sqrt3{x - 2} - 1$
$y = -\sqrt3{x + 2} - 1$

Explanation:

Step1: Recall parent cube root function

The parent function is \( y = \sqrt[3]{x} \), which passes through \((0,0)\), \((1,1)\), \((-1,-1)\). A reflection over the x - axis would make it \( y=-\sqrt[3]{x} \), which passes through \((0,0)\), \((1, - 1)\), \((-1,1)\).

Step2: Analyze translations

For a cube root function \( y = a\sqrt[3]{x - h}+k \), \( h \) is the horizontal shift (right if \( h>0 \), left if \( h < 0 \)) and \( k \) is the vertical shift (up if \( k>0 \), down if \( k < 0 \)).

Looking at the graph, let's find a key point. The parent \( y = \sqrt[3]{x} \) has a point at \((0,0)\). The reflected parent \( y=-\sqrt[3]{x} \) also has \((0,0)\). Now, let's check the shifted points.

Let's analyze the options:

- Option 1: \( y=\sqrt[3]{x - 2}-1 \). This is not reflected (no negative sign in front of the cube root), so it has the same general shape as \( y = \sqrt[3]{x} \), which is increasing. Our graph is decreasing, so eliminate this.

- Option 2: \( y =-\sqrt[3]{x} \). This is a reflection of the parent, but no horizontal or vertical shift ( \( h = 0,k = 0 \) ). The graph in the image seems to be shifted. Let's check the key point. If \( x=-2 \), for \( y =-\sqrt[3]{x + 2}-1 \), when \( x=-2 \), \( y=-\sqrt[3]{0}-1=- 1 \)? Wait, no, let's check the graph's key point. Let's find a point on the graph. From the graph, when \( x=-2 \), let's see the y - value. Wait, let's check the last option: \( y=-\sqrt[3]{x + 2}-1 \).

For \( y=-\sqrt[3]{x + 2}-1 \), when \( x=-2 \), \( y=-\sqrt[3]{0}-1=-1 \)? Wait, no, let's re - express the horizontal shift. The formula \( y = a\sqrt[3]{x - h}+k \), so for \( y=-\sqrt[3]{x+2}-1 \), \( h=-2 \) (since \( x - h=x + 2\Rightarrow h=-2 \)), so it's a shift left by 2 units and down by 1 unit, and reflected over the x - axis.

Let's check the behavior. The parent \( y = \sqrt[3]{x} \) is increasing. After reflection \( y=-\sqrt[3]{x} \) is decreasing. Then, shifting left 2 (\( x\to x + 2\)) and down 1 (\( + k=-1\)) gives \( y=-\sqrt[3]{x + 2}-1 \).

Let's check a point. Let's take \( x=-2 \), then \( y=-\sqrt[3]{-2 + 2}-1=-\sqrt[3]{0}-1=-1 \). Let's see the graph: when \( x=-2 \), what's the y - value? From the graph, the curve passes near \( x=-2 \), let's check another point. If \( x = 0 \), for \( y=-\sqrt[3]{0 + 2}-1\approx-\sqrt[3]{2}-1\approx - 1.26-1=-2.26 \). For \( y=-\sqrt[3]{x} \), when \( x = 0 \), \( y = 0 \), which does not match the graph. For \( y=-\sqrt[3]{x-2}-1 \), when \( x = 2 \), \( y=-\sqrt[3]{0}-1=-1 \). The graph in the image seems to have a shift left (since the "center" of the cube root graph, the point where the curve changes direction, is at \( x=-2,y=-1 \) maybe? Wait, let's re - evaluate.

The general form of the cube root function transformation:

The parent function \( y=\sqrt[3]{x} \) has its inflection point (the point where the concavity changes) at \((0,0)\). After reflection \( y =-\sqrt[3]{x} \) also at \((0,0)\). For a horizontal shift \( h \) and vertical shift \( k \), the inflection point moves to \((h,k)\).

Looking at the graph, the inflection point (the point where the curve bends) seems to be at \((-2,-1)\). So \( h=-2 \) (since \( x - h=x+2\Rightarrow h = - 2 \)) and \( k=-1 \), and it's reflected (so \( a=-1 \)). So the function is \( y=-\sqrt[3]{x+2}-1 \), which is \( y =-\sqrt[3]{x+2}-1=-\sqrt[3]{x-(-2)}-1 \).

Let's check the options:

- Option 1: \( y=\sqrt[3]{x - 2}-1 \): No reflection, incorrect.

- Option 2: \( y =-\sqrt[3]{x} \): No horizontal or vertical shift, incorrect.

- Option 3: \( y=-\sqrt[3]{x - 2}-1 \): Horizontal shift right 2, but our inflection point is at \( x=-2 \), so shift left 2, incorrect.

- Option 4: \( y=-\sqrt[3]{x + 2}-1 \): Reflection (negative sign), horizontal shift left 2 (\( x+2\)), vertical shift down 1 (\(-1\)), which matches the inflection point at \((-2,-1)\) and the decreasing shape (due to the negative sign).

Answer:

\( y =-\sqrt[3]{x + 2}-1 \) (the fourth option)