Final answer:
To translate the graph of y = x^2 to y = (x + 5)^2 - 2, shift the original graph 5 units left and 2 units down, resulting in a new vertex at (-5, -2).
Step-by-step explanation:
Graph Translation
To translate the graph of y = x^2 to the graph of y = (x + 5)^2 - 2, we perform two main transformations on the original parabola:
- A horizontal shift to the left by 5 units. This is because adding a positive number inside the function's parentheses (to the x-variable) moves the graph in the opposite direction.
- A vertical shift downwards by 2 units. This is due to the subtraction of 2 from the entire function, which lowers the graph by 2 units along the y-axis.
With these transformations, the vertex of the original parabola, which was at the origin (0, 0), is now moved to the point (-5, -2). The shape of the curve does not change because the coefficient of the x^2 term remains 1.