1. Quadratic Functions and Polynomials
Section 2.1
Quadratic Functions
and Polynomials

2. EXAMPLE
t (sec)
d (feet) 1 16 2 64 3
144 4
256 5
400
Suppose a baseball is dropped from the top of the Empire State Building. The table to the right shows (based on measurements) the distance d (in feet) the baseball has fallen after t seconds.

3. A FORMULA FOR DROPPED BASEBALL
t (sec)
d (feet) 1 16 2 64 3
144 4
256 5
400
Note that d is not a linear functions of t.
The distance d is given by the formula
d = 16t2.

4. QUADRATIC FUNCTIONS
Definition: A quadratic function is one of the form
f (x) = ax2 + bx + c
with a ≠ 0, so its formula involves a square term as well as linear and constant terms.

5. GRAPHS OF QUADRATIC FUNCTONS
Graph the following quadratic functions on your calculator.
f (x) = x2
f (x) = 3x2
f (x) = 6x2
f (x) = 0.5x2
f (x) = 0.1x2
f (x) = -x2
f (x) = -2x2
f (x) = -0.75x2
f (x) = x2 – x – 6
f (x) = – 2x2 + 3x + 1

6. PARABOLAS
The graph of a quadratic function is a parabola. The low point (or high point) is called the vertex. The vertex is also the place where the graph changes direction.
The quadratic function
f (x) = ax2 + bx + c
has its vertex at the x-value
To find the y-coordinate, substitute this value in for x.

7. POWER FUNCTIONS Definition: A power function is one of the form
f (x) = xk
where k is a given constant, which can either be positive or negative, either an integer or a fraction.

8. GRAPHS OF POWER FUNCTIONS
Graph the following power functions.
f (x) = x2
f (x) = x4
f (x) = x6
f (x) = x
f (x) = x3
f (x) = x5
f (x) = x1/2
f (x) = x1/3

9. POLYNOMIALS
Definition: A polynomial function (or simply a polynomial) is a sum of constant multiples of the power functions
1, x, x2, x3, xn, . . .
The degree of the polynomial is equals the exponent of the highest power appearing among its terms.

10. THE “END BEHAVIOR” OF A POLYNOMIAL
The end behavior of a graph is what the shape of the graph is on the extreme left and extreme right.
The end behavior of a polynomial is the same the end behavior of the term with the highest power.

11. TURNING POINTS OF POLYNOMIALS
In general, the graph of a polynomial can have as many
degree - 1
turning points (or “bends”).
In other words, if the degree of the polynomial is n, the graph can have as many as n - 1 turning points (or “bends”).

12. SOLVING POLYNOMIAL EQUATIONS
Principle: Graphic Solution of Equations
The solutions of the equation
f (x) = 0
are precisely the x-intercepts of the graph
y = f (x).
Recall, x-intercepts are the places where the graph crosses (or touches) the x-axis.