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Activity 27:

Activity 27:. Quadratic Functions; (Section 3.5, pp. 259-266) Maxima and Minima. Graphing Quadratic Functions Using the Standard Form:. A quadratic function is a function f of the form f (x) = ax 2 + bx + c,

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Activity 27:

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  1. Activity 27: Quadratic Functions; (Section 3.5, pp. 259-266) Maxima and Minima

  2. Graphing Quadratic Functions Using the Standard Form: A quadratic function is a function f of the form f(x) = ax2 + bx + c, where a, b, and c are real numbers and a is not 0. The graph of any quadratic function is a parabola; it can be obtained from the graph of f(x) = x2 by the methods described in Activity 26. Indeed, by completing the square a quadratic function f(x) = ax2 + bx + c can be expressed in the standard form f(x) = a(x − h)2 + k. The graph of f is a parabola with vertex (h, k); the parabola opens upward if a > 0, or downward if a < 0.

  3. Example 1: Express the parabola y = x2 − 4x + 3 in standard form and sketch its graph. In particular, state the coordinates of its vertex and its intercepts. From the graph we see that the y – intercept is: x – intercepts are:

  4. Example 2: Express the parabola y = −2x2 − x + 3 in standard form and sketch its graph. In particular, state the coordinates of its vertex and its intercepts.

  5. Example 4: Find the maximum or minimum value of the function:

  6. Example 5: Find a function of the form f(x) = ax2 + bx + c whose graph is a parabola with vertex (−1, 2) and that passes through the point (4,16).

  7. Example 6 (Path of a Ball): • A ball is thrown across a playing field. Its path is given by the equation • y = −0.005x2 + x + 5, where x is the distance the ball has traveled horizontally, and y is its height above ground level, both measured in feet. • What is the maximum height attained by the ball? • (b) How far has it traveled horizontally when it hits the ground?

  8. Example 7 (Pharmaceuticals): When a certain drug is taken orally, the concentration of the drug in the patient’s bloodstream after t minutes is given by C(t) = 0.06t − 0.0002t2, where 0 ≤ t ≤ 240 and the concentration is measured in mg/L. When is the maximum serum concentration reached, and what is that maximum concentration?

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