Working with polynomial functions and their graphs (Note comments in brackets [?] should be pondered you need not submit a response.) Introduction, working with a quadratic function: Consider the function f (x) = x 2 6x + 5 1. By inspection, we see this is a quadratic function (or 2nd degree polynomial function) [why?] and its graph is a parabola that opens upwards. We also know the y intercept is y = 5 . [why?] 2. This particular function may be factored relatively nicely and we see that f (x) = Thus, the zeros of the function are x = 1 and x = 5 [why?] If you were to graph this function, the xintercepts are located at (1, 0) and 5, 0). Furthermore, the axis of symmetry is x = b = 2a 6 or x (x 1)(x 5) . = 3 2 3. By completing the square, we can write the function as f (x) = (x 3) 2 4 . [Can you do this?] Thus the vertex of the parabola is located at the point (3, 4) . 4. Using the work above, a short analysis (quick sketch of the function) will help answer many questions: Quick sketch based upon the prior information: We can answer such questions as: a) Where is f (x) decreasing? b) Where is f (x) < 0 ? 5. Now consider the function g(x) = f (x) + 4 : First note: g(x) 2 = (x = x = (x 3)(x 3) = (x 3) 2 Sketch of y 6x + 5) + 4 = 6x + 9 2 This represents a new function, which \"looks\" like the previous function, excepting that it has been shifted vertically upwards 4 units.[why?] We note that g(x) > 0 for all values of x , and it is equal to zero only at x = 3 . (We call this a double root, or a root of multiplicity two.) The function is still decreasing on then interval (, 3). The local minimum still occurs at x = 3 . = g(x) Problem 1 Sketch a cubic function (third degree polynomial function y Then determine a formula for your function. = p(x) where p(x) > 0 on the intervals (, 3) and (5, 8) . SKETCH: Formula: p(x) = _____________________________________________________ Problem 2 Sketch a cubic function (third degree polynomial function y = p(x) with two distinct zeros at x = 2 and x = 5 and has a local maximum located at x = 5 . Then determine a formula for your function. [Hint: you will have one double zero] SKETCH: Formula: p(x) = _____________________________________________________ Problem 3 Find the formula for the quadratic function whose graph has a vertex of (1, 2) and passes through the point (1, 6) . Step 1: Use the coordinates of the vertex to write p(x) in the form: p(x) = a(x h) 2 + k where a is not known. Step 2: Use the coordinates of the second point to solve for the leading coefficient, a. p(x) = ___________________________________________ Problem 4 Sketch the graph of a polynomial function with the following properties: Increasing on (, 1) Decreasing on (3, ) Relative maximum at x = 1 Relative maximum at x x intercepts at x = 1, 2, and 4 y intercept at y = 2 = 3