1. Pre-Calculus Honors Day14 2.1 Quadratic Functions - How do you write quadratic functions in standard form? - How to use quadratic functions to model and solve real-life problems?

2. The simplest type of quadratic function is f(x) = x 2. Quadratic Functions! Parabolas! Opens UpwardOpens Downward Axis of Symmetry Axis of Symmetry Vertex: Minimum Vertex: Maximum

3. Quadratic Function: f(x) = ax 2 + bx +c; a ≠ 0 Characteristics of Parabolas Axis of symmetry: Vertex: substitute x value from axis of symmetry to find the y value of the vertex. (x, y) If a > 0 (a = positive), parabola opens upward If a < 0 (a = negative), parabola opens downward Y-intercept (0, c) X-intercepts: 0, 1, or 2, roots of solutions If b 2 - 4ac = 0; 1 root (vertex) If b 2 - 4ac > 0; 2 roots If b 2 - 4ac < 0; no root

4. Example 1: Find i) direction of opening ii) axis of symmetry iii) vertex iv) x and y – intercepts i) a = 1, a >0, Opens UP iii) Vertex: (-1, -9) iv) y-intercept (x = 0) x-intercept (y=0) solve! (0, -8) (-4, 0)(2, 0) ii) Axis of Symmetry:

5. Example 1: Find i) direction of opening ii) axis of symmetry iii) vertex iv) x and y – intercepts i) a = -2, a <0, Opens DOWN ii) Axis of Symmetry: iii) Vertex: (1, -5) iv) y-intercept (x = 0) x-intercept (y=0) solve! (0, -7) No Real Roots! No x-intercepts

6. The Standard Form of a Quadratic Function Axis of Symmetry: Vertical Line x = h Vertex: Point (h, k) a > 0: Parabola opens upward a < 0: Parabola opens downward

7. Example 2: Find i) direction of opening ii) axis of symmetry iii) vertex i) a = 1, a > 0, Opens UP i) a = -1/2, a < 0, Opens DOWN ii) x = -4 ii) x = 2 iii) (-4, -3) iii) (2, 1)

8. Applications: Many applications involve finding the maximum and minimum value of a quadratic function. 1.If a > 0, f has a minimum that occurs at 2.If a < 0, f has a maximum that occurs at

9. Example 4: The path of a baseball is given by the function f(x) = -0.0032x 2 + x + 3, where f(x) is the height of the baseball (in feet) and x is the horizontal distance from home plate (in feet). What is the maximum height reached by the baseball? Hint: Find the distance (x) first, then use that to find the height Height: 81.125 feet

10. Example5: The percent of income that Americans give to charities is related to their household income. For families with an annual income of $100,000 or less, the percent is approximately P = 0.0014x 2 – 0.1529x + 5.855, 5≤ x ≤ 100 where P is the percent of annual income given, and x is annual income (in thousands of dollars). What income level corresponds to the minimum percent of charitable contributions? Income Level: 54.6 = $54,600

11. Example 6 on your own! A textile manufacturer has daily production costs of C = 10,000-110x+0.45x 2 where C is the total cost in dollars and x is the number of units produced. How many units should be produced each day to yield a minimum cost?

12. Example 7: What is the largest rectangular area that can be enclosed with 400 feet of fencing? What are the dimensions of the rectangle? l =100ft w = 100ft A = 10,000 feet squared

13. Example 8: A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river. If the farmer doesn’t fence the side along the river, what is the biggest area that can be enclosed? A = 2,000,000 meters squared

14. Tonight’s Homework Pg 143 #1-8 all, 23-26, 35, 36, 72, 73