October 3 rd copyright2009merrydavidson Happy Summer Birthday to: Libby Harper

Quadratic Functions GENERAL form: If a>0 it opens UP If a<0 it opens DOWN STANDARD or VERTEX form: where is the vertex. 2x 2 + 3x - 5 Is an element of

All quadratic functions are symmetric over a line called the axis of symmetry. x = 3

f(x) = a(x – h) 2 + k Practice finding the vertex and axis of symmetry….. y = 2(x – 3) y = -3(x + 4) (3, 5) x = 3 (-4, -1) x = -4 standard/vertex form

f(x) = ax 2 + bx + c Practice finding the vertex and axis of symmetry….. Vertex: (-b/2a,f(-b/2a)) y = 2x 2 + 8x - 3 y = -3x 2 – 12x + 4 (-2, -11) x = -2 (- 2, 16) x = - 2 general form

Graph: Plot the vertex. Draw in the axis of symmetry. y = (x – 3) y = -(x + 4) (3, 5) x = 3 (-4, -1) x = -4

Graph: Plot the vertex. Draw in the axis of symmetry. y = 2(x – 3) (3, 5) x = 3 Stretch of 2

Changing to Standard Form by “Completing the Square” f(x) = x 2 + 8x + 11 Step 1:Group the 1 st 2 terms Step 2: Add & subtract blanks Step 3: ½ the middle term squared Step 4: factor Step 5: simplify f(x) = (x 2 + 8x) + 11 f(x) = (x 2 + 8x + _) _ f(x) = (x + 4) f(x) = (x + 4) 2 - 5

f(x) = 2x 2 + 8x + 7 Step 1:Group the 1 st 2 terms Step 2: Factor out the 2 f(x) = (2x 2 + 8x) + 7 f(x) = 2(x 2 + 4x) + 7 f(x) = 2(x 2 + 4x + _) _(2)2 2 f(x) = 2(x + 2) f(x) = 2(x + 2) 2 - 1

f(x) = -x 2 - 4x + 21 f(x) = -(x 2 + 4x + _) _(-1) 2 2 f(x) = -(x + 2) f(x) = -(x + 2)

Find the x-intercepts of a quadratic function f(x) = x 2 - 6x + 8 Step 1:Set = 0 Step 2: Factor Step 3: Set each factor = 0 Step 4: solve each part Step 5: intercepts are points 0 = x 2 - 6x = (x – 4)(x – 2) x– 4=0 x – 2=0 x = 4 x = 2 (4,0)(2,0)

General Form f(x) = ax 2 + bx + c Standard/vertex form f(x) = a(x-h) 2 +k Vertex: (-b/2a,f(-b/2a)) (h, k) AOS:x = -b/2ax = h Easier to find Easier to graph & x-intercepts find vertex

Find the equation of the quadratic function that goes through the point (2,3) with a vertex at (4,-5). y = a(x - h) 2 +k 3 = a(2 - h) 2 +k 3 = a(2 - 4) solve for “a” a = 2 f(x) = 2(x – 4) = 4a = 4a

Find the equation of the quadratic function that goes through the point (-2,-2) with a vertex at (-1,0). f(x) = -2(x + 1) 2

The height y (in feet) of a ball thrown by a child is given by where x is the horizontal distance (in feet) from where the ball is thrown. How high is the ball when it is at its maximum height? Sketch a graph. What are you trying to find? The y value of the vertex

(4, y) find y 6 ft high

Minimizing Cost A local newspaper has daily production costs of C = 55,000 – 108x x 2 where C = total cost in $ and x is the number of newspapers printed. How many newspapers should be printed each day to yield a minimum cost? Graph in calculator/play with window What are you looking for? X value of the vertex