Activity A: t th izm r : Quadratics and . Select Quadratic function and Show function. cubics - Turn off Show derivative. 1. Set a to 1, b to 0, and c to 2 to graph f(x) = x2 2. Take a look at its graph in the Gizmo. A. Where is the \"slope" of the graph of f(x) = x2 2: positive? negative? B. The \"slope" of a curve at a point is the slope of the line tangent to the curve at that point. (This is usually called the derivative.) Select Show tangent line. Drag the red point along the parabola, and watch the blue tangent line as you do. Where is the slope of the tangent line zero? Explain why this makes sense. 2. Graph x) = 0.5x2 4. Select Show derivative. Drag the red point. The y-values on the light blue line give you the slope of the dark blue tangent line at the current x-value. A. What does the light blue line tell you about the derivative of f(x) = 0.5x2 4? B. Vary a, b, and c. What type of function is the derivative of a quadratic? C. Vary c. Will c have any effect on the derivative? Regardless of the function will a vertical shift have an effect on the derivative? D. Set b and c to 0. Vary a. In general. what is the derivative of f(x) = axz? This is an example of the powerruie: the derivative of x) = x" is for) = nxi-'l. E. Vary a, b, and c. Look for a pattern in how these values affect the derivative. In general, what is the derivative of f(x) = ax2 + bx + c? f'(x) = (Activity A continued on next page)