Final Answer:
Translations, reflections, and dilations for the functions based on f(x) = x^2:
1. f(x) = -(x-1)^3 + 1: This function represents a reflection across the x-axis, a horizontal translation right by 1 unit, and a vertical translation up by 1 unit.
2. f(x) = 1/4 |2x+3| - 1: This function involves a vertical dilation by a factor of 1/4, a horizontal compression by a factor of 1/2, a horizontal translation left by 3/2 units, and a vertical translation down by 1 unit.
3. f(x) = 1/(1/3)x + 5: This function indicates a vertical dilation by a factor of 3, and a vertical translation up by 5 units.
Step-by-step explanation:
1. For f(x) = -(x-1)^3 + 1, start with the parent function f(x) = x^2. The negative sign reflects the graph across the x-axis. The (x - 1)^3 part causes a horizontal shift to the right by 1 unit, and the +1 at the end shifts the graph vertically upwards by 1 unit. No dilation is present.
2. Analyzing f(x) = 1/4 |2x+3| - 1, the 1/4 outside the absolute value results in a vertical compression by a factor of 1/4. The 2x inside the absolute value causes a horizontal compression by a factor of 1/2. The +3 within the absolute value shifts the graph left by 3/2 units. The -1 outside the absolute value shifts the graph down by 1 unit.
3. Looking at f(x) = 1/(1/3)x + 5, the 1/(1/3) = 3 represents a vertical dilation by a factor of 3. The +5 at the end shifts the graph vertically upward by 5 units. No other transformations such as reflections or translations are present.
These transformations alter the parent function f(x) = x^2, changing its shape, position, and size on the coordinate plane, showcasing how different modifications affect the graph of the quadratic function.