CHAPTER 2. Polynomials and Rational functions

Transcription

1 CHAPTER 2 Polynomials and Rational functions

2 Section 2.1 (e-book 3.1) Quadratic Functions Definition 1: A quadratic function is a function which can be written in the form (General Form) Example 1: Determine which of the following are quadratic functions. If a function is quadratic, identify a, b, and c. Properties of a Quadratic Function: 1) The domain is 2) Graph: a (vertical) parabola 3) Opens 4) Vertex: with and

3 5) The parabola is symmetric with respect to the vertical line. This line is called the axis of symmetry. 44 6) The range of the function is if the parabola opens up, or if the parabola opens down. 7) The y-intercept. 8) (Important)The solutions of equation are called the zeros of the quadratic function. A quadratic function has exactly two zeros. The zeros are either both real numbers or both complex numbers. a) If the zeros are real, then they are just the x-intercepts of the function. In this case the zeros are either distinct (different) or the same. If they are the same, then the parabola would be tangent to x-axis at that zero. b) If the zeros are complex then the graph does not cross x-axis, i.e., there are no x- intercepts

4 45 Example 2: Check all 8 properties of quadratic functions for. Use your results to sketch the graph of the function. Verify your answers by using a graphing calculator.

5 Example 3: Repeat example 2 for. 46

6 Example 4: For functions and (of examples 2 and 3), choose the word maximum or minimum and fill in the blanks: 47 has a at has a at Theorem 1: A quadratic function, given in general form, can be written in the standard form (vertex form) of,where and are the coordinates of the vertex given by the formulas in property #4 of quadratic functions. Remark 2: The proof of this theorem uses the method of Completing Square and the details are omitted. Example 5: The following quadratic functions are given in standard forms. For each, determine the coordinates of the vertex and convert the function to its general form. a) b)

7 Example 6: Write the following quadratic functions in their standard forms (use formulas in property # 4.) a) 48 b) Example 7: Find the standard form of the quadratic function with vertex at and passing through point.

8 49 Applications Example 8: a) A baseball is hit at a height of 3 feet above the ground with an initial velocity of 100 ft/sec at an angle of 45 with respect to the ground. It can be shown that the path of the baseball is given by, where is the height of the ball and is the horizontal distance of the ball from the home plate. Find the maximum height of the ball. b) A rancher has 1500 feet of fencing to fence a rectangular pen, shown below, for his dogs. i. Find a function that represents the area of the pen in term of its width x. ii. Find the dimensions of the pen that maximizes the area of the pen.

9 50 Exercise: 1. Check all 8 properties of quadratic functions for. Use your results to sketch the graph of the function. Verify your answers by using a graphing calculator. 2. Repeat problem # 1 for. 3. A small theater has a seating capacity of When the ticket price is $20, the attendance is For each $1decease in price, attendance is increased by 100. a) Write the revenue R s a function of ticket price x. b) What ticket price will yield a maximum revenue? What is that maximum revenue? 4. A ball is thrown upward with an initial velocity of 30 ft/s. It can be shown that the height h of the ball as a function of time t seconds (after it is thrown) is given by. How many seconds after it is thrown is the ball at its maximum height? Find that maximum height as well. 5. The quadratic function is given in standard form. Determine the coordinates its vertex and convert it to its general form. 6. Write the quadratic function in its standard form. 7. Find the standard form of the quadratic function with vertex at and passing through point. 8. Find the standard form of the quadratic function whose graph is given below