October 3 rd copyright2009merrydavidson Happy Summer Birthday to: Libby Harper.
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Transcript of October 3 rd copyright2009merrydavidson Happy Summer Birthday to: Libby Harper.
Quadratic Functions
GENERAL form:
If a>0 it opens UPIf a<0 it opens DOWN
STANDARD or VERTEX form:
where is the vertex.
2( )f x ax bx c
2( ) ( )f x a x h k
0a , ,a b c
( , )h k
2x2 + 3x - 5
2( ) 2( 3) 1f x x
Is an
element of
f(x) = a(x – h)2 + k
Practice finding the vertex and axis of symmetry…..
y = 2(x – 3)2 + 5
y = -3(x + 4)2 - 1
(3, 5)
x = 3(-4, -1)
x = -4
standard/vertex
form
f(x) = ax2 + bx + c
Practice finding the vertex and axis of symmetry….. Vertex: (-b/2a,f(-b/2a))
y = 2x2 + 8x - 3
y = -3x2 – 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)2 + 5
y = -(x + 4)2 - 1
(3, 5) x = 3
(-4, -1) x = -4
Changing to Standard Form by “Completing the Square”
f(x) = x2 + 8x + 11Step 1: Group the 1st 2 terms
Step 2: Add & subtract blanks
Step 3: ½ the middle term squared
Step 4: factor
Step 5: simplify
f(x) = (x2 + 8x) + 11
f(x) = (x2 + 8x + _) + 11 - _42 42
f(x) = (x + 4)2 + 11 - 16
f(x) = (x + 4)2 - 5
f(x) = 2x2 + 8x + 7Step 1: Group the 1st 2 terms
Step 2: Factor out the 2
f(x) = (2x2 + 8x) + 7
f(x) = 2(x2 + 4x) + 7
f(x) = 2(x2 + 4x + _) + 7 - _(2)22 22
f(x) = 2(x + 2)2 + 7 - 8
f(x) = 2(x + 2)2 - 1
f(x) = -x2 - 4x + 21
f(x) = -(x2 + 4x + _) + 21 - _(-1)22 22
f(x) = -(x + 2)2 + 21 + 4
f(x) = -(x + 2)2 + 25
Find the x-intercepts of a quadratic function
f(x) = x2 - 6x + 8
Step 1: Set = 0Step 2: Factor
Step 3: Set each
factor = 0
Step 4: solve each partStep 5: intercepts are points
0 = x2 - 6x + 80 = (x – 4)(x – 2)
x– 4=0 x – 2=0
x = 4 x = 2(4,0)(2,0)
General Form
f(x) = ax2 + bx + c
Standard/vertex form
f(x) = a(x-h)2 +k
Vertex: (-b/2a,f(-b/2a)) (h, k)
AOS: x = -b/2a x = 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 +k3 = a(2 - h)2 +k3 = a(2 - 4)2 + -5 solve for “a”
a = 2
f(x) = 2(x – 4)2 - 5
3 = 4a - 5 8 = 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?
214
8y x x
Sketch a graph.
What are you
trying to find?
The y value of the vertex
Minimizing Cost
A local newspaper has daily production costs of
C = 55,000 – 108x + 0.06x2 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