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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 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 ­intercept is . [why?] 2. This particular function may be factored relatively nicely and we see that 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 . or 3. By completing the square, we can write the function as . [Can you do this?] Thus the vertex of the parabola is located at the point .
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 decreasing? b) Where is ? 5. Now consider the function : First note: Sketch of 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 for all values of , and it is equal to
zero only at . (We call this a double root, or a root of multiplicity two.) The local minimum still occurs at . The function is still decreasing on then interval . Problem 1 where on the intervals Sketch a cubic function (third degree polynomial function Then determine a formula for your function. and SKETCH: Formula: _____________________________________________________ Problem 2
Sketch a cubic function (third degree polynomial function with two distinct zeros at and and has a
local maximum located at . Then determine a formula for your function. [Hint: you will have one double zero] SKETCH: Formula: _____________________________________________________ . Problem 3 Find the formula for the quadratic function whose graph has a vertex of Step 1:
Use the coordinates of the vertex to write in the form: and passes through the point where is not known. Step 2:
Use the coordinates of the second point to solve for the
leading coefficient, . ___________________________________________ Problem 4
Sketch the graph of a polynomial
function with the following
properties:
Increasing on Decreasing on Relative maximum at Relative maximum at ­intercepts at ­intercept at .

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