Graphs of Quadratic Functions

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) = ax2 + bx + cWhere 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) = ax2 + bx + c

vertex

The leading coefficient of ax2 + 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) = ax2 + bx + ca > 0 opens upward

vertex minimum

x

y

f(x) = ax2 + bx + ca < 0 opens downward

vertex maximum

Graphs of Quadratic Functions

1. Graph f (x) = (x – 3)2 + 2 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).

2. 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.

a. find the axis and vertex by completing the square

b. graph the parabola

4. Given: f (x) = –x2 + 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 < 0 , it opens down -> 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 1990.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 m2, 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: 21

2 6.9

y x x

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

corralx x

120 – 2x

Let x represent the width of the corral