To determine the graph of the function \( g(x) = -2f(x) \), we need to understand how the given transformations affect the parent function \( f(x) = \sqrt[3]{x} \).
1. Parent Function: The parent function is \( f(x) = \sqrt[3]{x} \), which is the cube root function. It has a characteristic shape with points such as \((1, 1)\), \((0, 0)\), and \((-1, -1)\).
2. Transformation: The transformed function is given by \( g(x) = -2 \cdot \sqrt[3]{x} \).
- Vertical Stretch and Compression: By multiplying by 2, the function is stretched vertically by a factor of 2. This means that each \( y \)-value of the original function is multiplied by 2.
- Reflection: The negative sign in front of the \( 2 \) indicates that the function is reflected over the \( x \)-axis. Thus, every \( y \)-value of the function is negated.
Step-by-Step Transformation:
1. Vertical Stretch by 2: Take each \( y \)-value of \( f(x) \) and multiply it by 2.
- For example, if \( f(1) = 1 \), then \( g(1) = -2 \times 1 = -2 \).
- Similarly, if \( f(-1) = -1 \), then \( g(-1) = -2 \times (-1) = 2 \).
2. Reflection Over the \( x \)-Axis: Reflect the results from the vertical stretch over the \( x \)-axis by taking their negative values.
- The point \((1, 2)\) (from vertical stretch) becomes \((1, -2)\).
- The point \((-1, -2)\) (from vertical stretch) becomes \((-1, 2)\).
So, the final transformed function \( g(x) = -2 \cdot \sqrt[3]{x} \) will have the following characteristics:
- It will pass through the same \( x \)-intercept \((0, 0)\) as the parent function.
- The points on the graph will be \( (1, -2) \) and \( (-1, 2) \) after applying the described transformations.
To select the correct graph without the visual options, you should look for a graph that reflects these transformations: vertical stretch by a factor of 2, and reflection over the \( x \)-axis based on the cube root function's typical shape. A proper graph should depict this specific pattern:
- It will be vertically stretched and flipped upside down relative to the parent function's graph.
