Question

this graph shows both a reflection and a translation. what function is being graphed?

\\(y = -\sqrt3{x}\\)
\\(y = \sqrt3{x - 2} - 1\\)
\\(y = -\sqrt3{x - 2} - 1\\)
\\(y = -\sqrt3{x + 2} - 1\\)

Explanation:

Identify the parent function and inflection point

Using the Cube Root Function knowledge point
The parent function is \(y = \sqrt[3]{x}\), which has its inflection point (center of symmetry) at \((0,0)\). Looking at the given graph, the inflection point has been shifted to \((-2, -1)\).

Determine the translations

Using the Horizontal Translation and Vertical Translation knowledge points
A shift of the inflection point from \((0,0)\) to \((h, k) = (-2, -1)\) corresponds to:

• A horizontal translation left by 2 units: \(x \to x + 2\)
• A vertical translation down by 1 unit: \(y \to y - 1\)

This gives the form:

$$y = a\sqrt[3]{x + 2} - 1$$

Determine the reflection

The standard cube root function \(y = \sqrt[3]{x}\) increases from left to right. The graphed function decreases from left to right, which indicates a vertical reflection across the \(x\)-axis. This means the leading coefficient \(a\) must be negative:

$$y = -\sqrt[3]{x + 2} - 1$$

Verify with a point on the graph

Using the Function Evaluation knowledge point
Let's test the point \((-2, -1)\):

$$y = -\sqrt[3]{-2 + 2} - 1 = -1$$

This matches the inflection point on the graph.
Let's test another clear point from the graph, such as \((-3, 0)\):

$$y = -\sqrt[3]{-3 + 2} - 1 = -\sqrt[3]{-1} - 1 = -(-1) - 1 = 0$$

The graph clearly passes through \((-3, 0)\), confirming the equation is:

$$y = -\sqrt[3]{x + 2} - 1$$

Answer:

• \(y = -\sqrt[3]{x}\)
• \(y = \sqrt[3]{x - 2} - 1\)
• \(y = -\sqrt[3]{x - 2} - 1\)
• \(y = -\sqrt[3]{x + 2} - 1\) (Correct answer)