Activity A: tth izm r : Quadratics and - Select Quadratic function and Show function. cubics o Turn off Show derivative. 1. Set a to 1, b to 0, and c to 2 to graph fix) = x2 2. Take a look at its graph in the Gizmo. A. Where is the \"slope" of the graph of fix) = x:2 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 f(x) = (1.5x2 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) = 0.5):2 4? B. Vary a, b, and c. What type of function is the derivative of a quadratic? C. Vary c. Will 0 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) = 3x2? This is an example of the power rule: the derivative of f(x) = x" is f(x) = nx("-1). 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) = 3x2 + bx + 0? f(x) = {Activity A continued on next page) Activity B: Get the Gizmo ready: Absolute value Turn off Show derivative. -2 2 functions . Select Absolute value function. Turn on Show tangent line. 1. Set a to 1 and b to -2 to graph f(x) = |x| - 2. (Notice that, for absolute value functions, the tangent line is an extension of one part of the graph. ) Drag the red point along the graph. A. What is the equation of the left half of the graph (where x 0)? C. What is the derivative (slope) of the left half? Of the right half? D. If the graph of a function has a break in it (a hole or discontinuity), or if it has a sharp turn (like a corner), then the derivative (f(x)) is not defined at that point. Where do you think f(x) for an absolute value function is undefined? E. Based on what you have seen; how would you write the derivative of f(x) = [x| - 2? *Hint: Write it as a piece wise derivative: f(x) =? if x #, this is the form we always use for derivative of the absolute value function f ( x ) = Select Show derivative to check. (The light blue graph shows f(x) at all x-values.) F. Vary a and b to see other absolute value functions. In general, what is the derivative of f(x) = alx| + b? f(x) =