Definition of a Polynomial Function in x of degree n

Polynomial functions are classified by degree Polynomial degree name

Definition of a quadratic function f (x) = ax 2 + 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. x y axis f (x) = ax 2 + bx + c vertex

The leading coefficient of ax 2 + bx + c is a. When the leading coefficient is positive, the parabola opens upward and the vertex is a minimum. When the leading coefficient is negative, the parabola opens downward and the vertex is a maximum. x y f(x) = ax 2 + bx + c a > 0 opens upward vertex minimum x y f(x) = ax 2 + bx + c a < 0 opens downward vertex maximum

Example: f(x) = (x –3) Graph f (x) = (x – 3) and find the vertex and axis. 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).

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

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

Vertex and x-Intercepts 4. Given: f (x) = –x 2 + 6x + 7. Find: a. the vertex b. x-intercepts c. then graph

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

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

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

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

If a > 0, it opens up -> Minimum If a Maximum Standard Form Vertex Form

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?

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.

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 Using this model, determine the year in which the number of grants awarded was greatest.

11. The width of a rectangular park is 5 m shorter than its length. If the area of the park is 300 m 2, find the length and the width.

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:

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