2.1 Quadratic Functions Completing the square Write Quadratic in Vertex form

Definition of a Polynomial Function F(x) = a n x n + a n-1 x n-1 + a n-2 x n a 1 x 1 + a 0 a n, a n-1,… are numbers In “x n ” n is the exponents going down in degree. This would be a polynomial function of x with degree n.

Linear functions are first degree Quadratic functions are second degree f(x) = ax 2 + bx + c Would be a graph of a Parabola A parabola is Axis of symmetry symmetric; “x = h” Where h is from the vertex (h, k) If a > 0, then the parabola opens up If a < 0, then the parabola opens downward

What equation do you think of when you hear Quadratic ?

This equation? Do you remember how to use it?

Standard form of a Quadratic equation f(x) = a(x – h) 2 + k How would you graph the equation f(x) = 2(x + 3) The vertex has moved off the origin 3 units to the left and 1 unit up. Since a = 2,the parabola opens up and gets skinny. The vertex is at (-3, 1)

f(x) = 2(x + 3) 2 + 1

Finding the vertex when it is not in Standard form f(x) = x x – 9 We need to complete the square to find the standard form Take half of b squared and add and subtract to the function. f(x) = x x + (6) 2 + ( -36) – 9 Factor the first three terms, then add the last terms f(x) = (x + 6) 2 – 45vertex ( - 6, -45)

Find the standard form of f(x) = x x + 8

Write the equation with the vertex ( - 4, 8) and the point (1, 4) Start with the standard equation f(x) = a(x – h) 2 + k from the vertex h = - 4, k = 8 from the point x = 1, f(x) = 4 4 = a(1 – (- 4)) = a(5) = 25a = 25a -4/25 = a Rewrite with a, h and kf(x) = -4/25 (x +4) 2 + 8

Find the vertex from f(x) = ax 2 + bx + c To find “h” we use “b” and “a” To find “k” we place h back in the equation.

Lets find the vertex of the equation f(x) = 4x 2 + 3x – 8 Now let h = - ⅜ k = 4(- ⅜) 2 + 3( ⅜) – 8k = -137/16 Vertex is ( - ⅜, - 137/16 )

Homework Page 116 – 117 #3, 11, 17, 19, 25, 31, 45, 59, 63

Homework Page 116 #22, 33, 51, 64, 73, 77, 80, 83, 89, 91, 104