To understand the translation from the graph of [tex]\( y = 6x^2 \)[/tex] to the graph of [tex]\( y = 6(x + 1)^2 \)[/tex], let's analyze the transformation step-by-step.
1. Identify the original function: The original quadratic function given is [tex]\( y = 6x^2 \)[/tex]. This is a parabola centered at the origin [tex]\((0, 0)\)[/tex] and opens upwards.
2. Understand the transformation: The transformed function is [tex]\( y = 6(x + 1)^2 \)[/tex]. To see how this changes the graph, we need to analyze the [tex]\( (x + 1)^2 \)[/tex] term.
3. Effect of the [tex]\( (x + 1)^2 \)[/tex] term:
- In the function [tex]\( y = 6(x + 1)^2 \)[/tex], the [tex]\( +1 \)[/tex] inside the square term means that the graph of the function is moved horizontally.
- Specifically, the [tex]\( (x + 1) \)[/tex] means every [tex]\( x \)[/tex] value is replaced by [tex]\( x + 1 \)[/tex], which effectively shifts the graph to the left by 1 unit.
4. Visualization:
- For the original equation [tex]\( y = 6x^2 \)[/tex], the vertex is at [tex]\( (0, 0) \)[/tex].
- For the transformed equation [tex]\( y = 6(x + 1)^2 \)[/tex], the new vertex is at [tex]\( (-1, 0) \)[/tex].
Therefore, the correct phrase that best describes the translation is:
1 unit left
This matches our detailed step-by-step analysis.
