Quadratic Functions Lesson 3.1

Applications of Parabolas Today we look at functions which describe parabolas. Solar rays reflect off a parabolic mirror and focus at a point This could make a good solar powered cooker (also wireless antenna reflector) YouTube view

Finding Zeros Often with quadratic functions     f(x) = a*x2 + bx + c   we speak of “finding the zeros” This means we wish to find all possible values of x for which    a*x2 + bx + c = 0

Finding Zeros Another way to say this is that we are seeking the x-axis intercepts This is shown on the graph below Here we see two zeros – what other possibilities exist?

Factoring Given the function x2 - 2x - 8 = 0  Factor the left side of the equation    (x - 4)(x + 2) = 0 We know that if the product of two numbers   a * b = 0     then either ... a = 0     or b = 0 Thus either x - 4 = 0    ==> x = 4     or x + 2 = 0    ==> x = -2

Warning!! Problem ... many (most) quadratic functions are NOT easily factored!!   Example:

The Quadratic Formula  It is possible to create two functions on your calculator to use the quadratic formula. quad1 (a,b,c)           which uses the    -b + ... quad2 (a,b,c)           which uses the    -b -

Click to view Spreadsheet Solution The Quadratic Formula Try it for the quadratic functions 4x2 - 7x + 3 = 0                           6x2 - 2x + 5 = 0 Click to view Spreadsheet Solution

The Quadratic Formula 4x2 - 7x + 3 = 0  

The Quadratic Formula Why does the second function give "non-real result?“ 6x2 - 2x + 5 = 0

Concavity and Quadratic Functions Quadratic function graphs as a parabola Will be either concave up Or Concave Down

Applications Consider a ball thrown into the air It's height (in feet) given by h(t) = 80t – 16t 2 Evaluate and interpret h(2) Solve the equation h(t) = 80 Interpret the solution Illustrate solution on a graph of h(t)

Quadratic Regression Our calculators will determine quadratic modeling functions Enter numbers into Data Matrix Choose F5, Calc Then choose Quadratic Regression Set x, y as before Send results to Y= screen

WindSurfer Curve

Assignment Lesson 3.1 Page 108 Exercises 1 – 35 Odd