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Polynomial and Rational Functions. We will look at: Quadratic Functions Polynomial Functions of a Higher Degree Polynomials and Synthetic Division Complex Numbers Zeroes of Polynomial Functions Rational Functions. Introduction to Quadratic Functions.
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Polynomial and Rational Functions We will look at: Quadratic Functions Polynomial Functions of a Higher Degree Polynomials and Synthetic Division Complex Numbers Zeroes of Polynomial Functions Rational 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 – 3 • We know that f(x)=a(x-h)2 + k • We 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+3 f(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.