Describe the single transformation that maps the graph of y=3-(1/4*x*2) onto the graph y=(1/2*x^2)-6.
To find the single transformation that maps the graph of y=3-(1/4*x^2) onto the graph y=(1/2*x^2)-6, we need to identify the transformations that were applied to the original function.
Let's start by analyzing the components of the functions:
Original Function: y = 3 - (1/4 * x^2)
New Function: y = (1/2 * x^2) - 6
1. Vertical Translation: The original function y = 3 - (1/4 * x^2) has a constant term of 3, which represents a vertical shift upwards by 3 units. On the new graph y = (1/2 * x^2) - 6, the constant term -6 represents a vertical shift downwards by 6 units. So, there is a vertical translation of 3 units upward followed by a vertical translation of 6 units downward.
2. Horizontal Scaling: The original function y = 3 - (1/4 * x^2) has a coefficient of 1/4 in front of the x^2 term. This means the graph is horizontally compressed by a factor of 4. On the new graph y = (1/2 * x^2) - 6, the coefficient of 1/2 in front of the x^2 term corresponds to a horizontal compression by a factor of 2. So, there is a horizontal compression by a factor of 4 followed by a horizontal compression by a factor of 2.
Therefore, the single transformation that maps the graph of y = 3 - (1/4 * x^2) onto the graph y = (1/2 * x^2) - 6 is a vertical translation of 3 units upward followed by a vertical translation of 6 units downward, and a horizontal compression by a factor of 4 followed by a horizontal compression by a factor of 2.