This activity helped me in understanding that certain modifications to the original function can affect the slope of the tangent line. Reflecting a function across the x-axis by adding a negative, and achieving vertical/horizontal stretch/compression changes the slope of the tangent line. However, shifting the function up, down, to the left, or to the right by adding or subtracting a constant to the original function doesn't affect the slope of the tangent line. For example, after we know that the derivative of f(x)=x^(1/2) is f'(x)=1/(2(x)^(1/2)), we can find the derivative of functions that are slightly altered. This is shown in the picture.
We experimented with square root functions, however, I believe that it works for most functions with tangent lines. For example, if we tried using the same techniques for a quardratic function, x^2, it follows the same rules. If f(x)=x^2 is the original function, the slope of the tangent line at x=1 would be the same if f'(x)=(x^2)+2. Just because the new function shifts 2 to the right, it doesn't change the slope. However, if f'(x)=-x^2, the slope would be the same except it'd be negative. If f'(x)=2x^2, the graph would be vertically stretched by a factor of 2. Therefore, the slope would also be changed by a factor of 2. That's why I came to the conclusion that shifting the function does not affect the slope, but, vertically/horizontally compressing/stretching or reflecting the function across the x-axis does.
We experimented with square root functions, however, I believe that it works for most functions with tangent lines. For example, if we tried using the same techniques for a quardratic function, x^2, it follows the same rules. If f(x)=x^2 is the original function, the slope of the tangent line at x=1 would be the same if f'(x)=(x^2)+2. Just because the new function shifts 2 to the right, it doesn't change the slope. However, if f'(x)=-x^2, the slope would be the same except it'd be negative. If f'(x)=2x^2, the graph would be vertically stretched by a factor of 2. Therefore, the slope would also be changed by a factor of 2. That's why I came to the conclusion that shifting the function does not affect the slope, but, vertically/horizontally compressing/stretching or reflecting the function across the x-axis does.