Final answer:
The functions that have a maximum and are transformed to the left and down of the parent function f(x) = x^2 are p(x) = 14(x + 7)^2 + 1, q(x) = -5(x + 10)^2 - 1, g(x) = 2x^2 + 10x - 35, and t(x) = -2x^2 - 4x - 3.
Step-by-step explanation:
To determine which functions have a maximum and are transformed to the left and down of the parent function f(x) = x^2, we need to analyze the given options and compare them to the parent function. The transformations to the left and down mean that the vertex of the transformed function will have a lower x-coordinate and a lower y-coordinate compared to the vertex of the parent function. We will check each of the given options:
p(x) = 14(x + 7)^2 + 1: This function is transformed to the left by 7 units and down by 1 unit compared to the parent function. It has a maximum.
q(x) = -5(x + 10)^2 - 1: This function is transformed to the left by 10 units and down by 1 unit compared to the parent function. It has a maximum.
s(x) = -(x - 1)^2 + 0.5: This function is transformed to the left by 1 unit and up by 0.5 units compared to the parent function. It has a minimum, not a maximum.
g(x) = 2x^2 + 10x - 35: This function is not transformed to the left or down compared to the parent function. It has a maximum.
t(x) = -2x^2 - 4x - 3: This function is not transformed to the left or down compared to the parent function. It has a maximum.
Based on the analysis, the options that have a maximum and are transformed to the left and down of the parent function f(x) = x^2 are p(x) = 14(x + 7)^2 + 1, q(x) = -5(x + 10)^2 - 1, g(x) = 2x^2 + 10x - 35, and t(x) = -2x^2 - 4x - 3.