definition of a polynomial function in x of degree n
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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