Graphs of Quadratic Functions

1. Graphs of Quadratic Functions

2. Definition of a Polynomial Function in x of degree n

6. Polynomial functions are classified by degree Polynomialdegree name

7. y vertex x Definition of a quadratic function f(x) = ax2 + bx + c Where a, b, and c are real numbers and The graph of a quadratic function is a _____________ Every parabola is symmetrical about a line called the axis (of symmetry). The intersection point of the parabola and the axis is called the vertex of the parabola. f(x) = ax2 + bx + c axis

8. y x vertexminimum y x vertexmaximum Graphs of Quadratic Functions The leading coefficient of ax2 + bx + c is a. a > 0 opens upward When the leading coefficientis positive, the parabola opens upward and the vertex is a minimum. f(x) = ax2 + bx + c When the leading coefficient is negative, the parabola opens downwardand the vertex is a maximum. f(x) = ax2 + bx + c a < 0 opens downward

9. The vertex form for the equation of a quadratic function is:f(x) = a(x – h)2+ k (a is not 0) The graph is a parabola opening upward if a > 0 and opening downward if a < 0. The axis is x = h, and the vertex is (h, k). Example: f(x) = (x –3)2 + 2 1. Graph f(x) = (x –3)2 + 2 and find the vertex and axis.

10. Use the completing the square method to rewrite the function f(x) = 2x2 + 4x – 1 in vertex form and then find the equation of the axis of symmetry and vertex. Quadratic Function in Standard Form

11. a. find the axis and vertex by completing the square b. graph the parabola

12. 4. Given: f(x) = –x2 +6x + 7. Find: a. the vertex b. x-intercepts c. then graph Vertex and x-Intercepts

13. 5. Write the standard form of the equation of the parabola whose vertex is (1, 2) and that passes through the point (3,-6)

14. 6. Write an equation of the parabola below in vertex form.

15. Identifying the x-intercepts of a quadratic function 7. Find the x-intercepts of

16. Another way to find the Minimum and Maximum Values of Quadratic Functions is to use the formula below. Minimum and Maximum Values of Quadratic Functions

17. Standard Form Vertex Form If a < 0 , it opens down ->Maximum If a > 0 , it opens up ->Minimum

18. The Maximum Height of a Baseball 8. A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second . The path of the baseball is given by the function 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?

19. 9. A soft drink manufacturer has daily production costs of Where C is the total cost ( in dollars) and x is the number of units produced. Estimate numerically the number of units that should be produced each day to yeald a minimum cost.

20. 10. The numbers g of grants awarded from the National Endowment for the Humanities fund from 1999 to 2003 can be approximated by the model Where t represents the year, with t = 0 corresponding to 1990. Using this model, determine the year in which the number of grants awarded was greatest.

21. 11. The width of a rectangular park is 5 m shorter than its length. If the area of the park is 300 m2, find the length and the width.

22. 12. A basketball is thrown from the free throw line from a height of six feet. What is the maximum height of the ball if the path of the ball is:

23. Example: Maximum Area 13. A fence is to be built to form a rectangular corral along the side of a barn 65 feet long. If 120 feet of fencing are available, what are the dimensions of the corral of maximum area? barn corral x x 120 – 2x Let x represent the width of the corral