What is the change applied to the parent function f(x) = x^2 to obtain the function f(x) = x^2 + 7?

The change applied to the parent function f(x) = x^2 to obtain the function f(x) = x^2 + 7 is a vertical shift upward by 7 units. This means that for every x-value, the value of the original function (x squared) is increased by 7.

In mathematical terms, if you take the graph of the parent function f(x) = x^2 and shift every point on the graph 7 units up, you will get the graph of the new function f(x) = x^2 + 7.

Extra: When dealing with transformations of functions, especially in algebra, there are a few key types of transformations that can occur:

1. Vertical shifts: These occur when a constant is added or subtracted from the function, resulting in the graph moving up or down. In this case, adding 7 caused a vertical shift upwards.

2. Horizontal shifts: These happen when a constant is added or subtracted within the function argument (inside the parentheses). For example, f(x) = (x - 3)^2 moves the parent function f(x) = x^2 three units to the right.

3. Reflections: Reflecting about the x-axis involves multiplying the entire function by -1, turning f(x) into -f(x). For the parent function, this would look like -x^2. Reflecting about the y-axis means replacing x with -x within the function, so f(-x).

4. Stretching and Compressing: Multiplying the entire function by a number greater than 1 stretches the graph vertically, while a number between 0 and 1 compresses it. For horizontal stretching/compression, you divide x (or multiply within the function) by a constant.

The transformation you asked about (adding 7 to f(x) = x^2) does not change the shape of the graph but merely shifts the entire parabola up on the coordinate plane, indicating that every output value is 7 more than it would be for the corresponding input value in the original parent function. The vertex of the parabola, originally at (0,0), is now at (0,7).