POLYNOMIAL AND RATIONAL FUNCTIONS

We will look at:oQuadratic FunctionsoPolynomial Functions of a Higher DegreeoPolynomials and Synthetic DivisionoComplex NumbersoZeroes of Polynomial FunctionsoRational Functions

INTRODUCTION TO QUADRATIC FUNCTIONS A quadratic function is a function, or equation, that

features an x^2 term, and x term, and a constant. It is represented by:

f(x)=ax2+bx+c The graph a quadratic is a parabola. It resembles a “U”

shape. The difference between a quadratic function and a

polynomial function is that a quadratic has an x that is the second power, where as a in a polynomial the x is to any power.

An example of using quadratics in everyday life is the projectile motion function. You use the acceleration of gravity (-9.8m/sec sq. – attached to x2) and your initial velocity (which is attached to the x) and your initial height (which is your constant) It looks like -9.8x2+(initial velocity)x+initial height

CHARACTERISTICS OF QUADRATICS

Standard form: the standard form for a quadratic graph is f(x)=a(x-h)2+k “a” tells you whether the graph opens up or

down “h” is the x-coordinate of your vertex, or middle

of your parabola “k” is the y – coordinate of your vertex

The parent function, or typical reference graph, for quadratics is f(x)=x2. It looks like:

EXAMPLE OF QUADRATICS

Find the vertex of f(x)=3(x+5)2 – 3We know that f(x)=a(x-h)2 + kWe also know that our x coordinate is h,

and our y coordinate is k; so our vertex is (h,k)

From the equation we notice that h=-5 and k=-3

So our vertex is at (-5,-3)

EXAMPLES OF QUADRATICS

Graph f(x)=(x-2)2 + 3 Looking at the equation we find that our vertex is

at (2,3) We also know the graph of the parent function

f(x)=x2

So using this knowledge we can shift the vertex of the parent function 2 to the right, and 3 up

EXAMPLE OF PROJECTILE MOTION FUNCTION

A baseball player hits the ball with an initial velocity of 100 fps. The ball meets the bat a the height of 3 feet. The baseball’s path is represented by the function;

f(x)=-0.0032x2+x+3f(x) is the height of the baseball in feet and x is

the distance from the initial contact point of the bat and ball in feet.

SOLUTION

Since the function has a maximum at x=-b/2a, you can conclude that the baseball reaches its maximum height when the ball is x feet away from the initial point. X=-1/2(-0.0032) X=156.25 feet

Now to find the height at when x=156.25, you plug x back into your original function. f(156.25)=-0.0032(156.25)2+156.25+3 f(156.25)=81.125 feet

WORKS CITED

Hosteler, Robert P. Larson, Ron. Precalculus Sixth Edition. Boston, MA: Houghton Mifflin Company. 2004. Print.