Quadratic Functions Lesson 2.6

Applications of Parabolas Solar rays reflect off a parabolic mirror and focus at a point Solar rays reflect off a parabolic mirror and focus at a point This could make a good solar powered cooker This could make a good solar powered cookersolar powered cookersolar powered cooker Today we look at functions which describe parabolas.

Finding Zeros Often with quadratic functions f(x) = a*x 2 + bx + c we speak of “finding the zeros” Often with quadratic functions f(x) = a*x 2 + bx + c we speak of “finding the zeros” This means we wish to find all possible values of x for which a*x 2 + bx + c = 0 This means we wish to find all possible values of x for which a*x 2 + bx + c = 0

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

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

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

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

The Quadratic Formula Try it for the quadratic functions Try it for the quadratic functions 4x 2 - 7x + 3 = 04x 2 - 7x + 3 = 0 6x 2 - 2x + 5 = 06x 2 - 2x + 5 = 0 Click to view Spreadsheet Solution

The Quadratic Formula 4x 2 - 7x + 3 = 0 4x 2 - 7x + 3 = 0

The Quadratic Formula Why does the second function give "non-real result?“ Why does the second function give "non-real result?“ 6x 2 - 2x + 5 = 06x 2 - 2x + 5 = 0

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

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

Assignment Lesson 2.6 Lesson 2.6 Page 92 Page 92 Exercises 1 – 31 Odd Exercises 1 – 31 Odd