In the activity that we worked on, "Tangent Lines and Derivatives: Reprise," we compared the tangent lines and slopes of very similar functions. Each function that we investigated showed a transformation from the base function, sqrt(x), which has a tangent line of y=0.5x+0.5 at the point x=1. The second function, -sqrt(x), was the base function reflected on the x-axis. The equation of the tangent line of this function at x=1 was y=-0.5x-0.5. In this first example of a simple transformation, we can see that both the function and the tangent line were multiplied by -1. Another example in the activity of a transformation that affects the slope and equation of the tangent line is the function that was a vertical stretch of factor 2 of the base function. The tangent line for this function was equal to the base function's tangent line multiplied by 2. The activity helped us to learn and recognize how function transformations affect the equation of the tangent line and the derivative. These examples show that the derivative of a transformed function is equal to the transformation of the tangent line and derivative equations. After completing the activity, I learned that stretch and reflection transformations affect the slope, but vertical shifts do not. This connects to my learning last week that there can be multiple functions with the same derivative, and both concepts make a lot more sense after completing this activity.
This activity involved investigation of relationships between functions and their derivatives and tangent lines when they are transformed. Although only square root functions were used in this activity, the lessons learned in this activity apply to all types of functions. The ways in which specific transformations affect the shape and location of a function's graph are the same no matter the type of function, as are the ways in which transformations affect the derivatives and tangent lines of a function. For example, the derivative of the quadratic function, f(x)=x^2 is f'(x)=2x. When this function undergoes the transformation of a vertical stretch by a factor of 2 (f(x)=2x^2), the derivative is also multiplied by 2 (f'(x)=2x -> f'(x)=4x). This rule applies to all types of functions.
This activity involved investigation of relationships between functions and their derivatives and tangent lines when they are transformed. Although only square root functions were used in this activity, the lessons learned in this activity apply to all types of functions. The ways in which specific transformations affect the shape and location of a function's graph are the same no matter the type of function, as are the ways in which transformations affect the derivatives and tangent lines of a function. For example, the derivative of the quadratic function, f(x)=x^2 is f'(x)=2x. When this function undergoes the transformation of a vertical stretch by a factor of 2 (f(x)=2x^2), the derivative is also multiplied by 2 (f'(x)=2x -> f'(x)=4x). This rule applies to all types of functions.