If the parent cube root function, \( f(x) \), is transformed to \( 2 f(x+5) \) what will be the effect on the graph?

To analyze the transformation of the parent cube root function \( f(x) = \sqrt[3]{x} \) to \( 2 f(x+5) \), we will break down the transformations step by step. 1. **Understanding the Parent Function**: - The parent function is \( f(x) = \sqrt[3]{x} \). This function has a characteristic shape that passes through the origin (0,0) and is symmetric about the origin. 2. **Transformation \( f(x+5) \)**: - The expression \( f(x+5) \) indicates a horizontal shift of the graph to the left by 5 units. This means that every point on the graph of \( f(x) \) will move 5 units to the left. For example, the point (0,0) will move to (-5,0). 3. **Transformation \( 2 f(x+5) \)**: - The multiplication by 2 in \( 2 f(x+5) \) indicates a vertical stretch of the graph by a factor of 2. This means that for every y-coordinate of the points on the graph of \( f(x+5) \), we will multiply it by 2. For example, if a point on \( f(x+5) \) is (-5,0), it will remain at (-5,0) after the stretch, but if there was a point at (-4, y), it will move to (-4, 2y). ### Summary of Effects on the Graph: - **Horizontal Shift**: The graph shifts 5 units to the left. - **Vertical Stretch**: The graph is stretched vertically by a factor of 2. Thus, the overall effect on the graph of the parent cube root function \( f(x) \) when transformed to \( 2 f(x+5) \) is a leftward shift of 5 units and a vertical stretch by a factor of 2.