Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Section 3.3 Quadratic Functions and Their Properties
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Section 3.3
Quadratic Functions and Their Properties
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
2( ) 2 4 __ 5 2(__)f x x x
2( ) 4 52 4 2(4)f x x x
2( ) 2 2 3f x x
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Without graphing, locate the vertex and axis of symmetry of the parabola defined by . Does it open up or down? 22 3 2f x x x
3 7Vertex is , .4 8
3 32 2 2 4ba
23 3 3 72 3 24 4 4 8
f
3Axis of symmetry is .4
x
Because 2 0, the parabola opens up.a
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
2(a) Use the information from the previous example and the locations of the intercepts to graph ( ) 2 3 2.f x x x
3 7Vertex is , .4 8
3Axis of symmetry is .4
x
x
y
3 7,4 8
Since a = 2 > 0 the parabola opens up and therefore will have no x-intercepts.
2(0) 2 0 3 0 2 2 so the -intercept = 2.f y
0, 2 3 , 22
By symmetry, the point with the same
3 -value but to the right of the axis of 4
3 3 3symmetry is on the graph. 4 4 2
3so the point , 2 is on the graph.2
y
(b) Determine the domain and the range of .(c) Determine where is increasing and decreasing.
ff
x
y
3 7,4 8
The domain of f is the set of all real numbers.7Based on the graph, the range is the interval , .8
0, 2 3 , 22
The function is 3 from ,4
and3 from , .4
incr
decreasi
g
ng
easin
x
y
2(a) Graph ( ) 2 4 1 by determining whether the graph opens up or down and by finding its vertex, axis of symmetry, and and intercepts if any.
f x x xx y
h
b2a
4
2(2) 1
Since a = 2 > 0 the parabola opens up.
1, 3
-intercepts can be found when ( ) 0.x f x
21 2 1 4( 1) 1 3k f
Vertex = 1, 3
2(0) 2 0 4 0 1 1 so the -intercept = 1.f y
By symmetry, the point ( 2, 1) is also on the graph.
0, 1 2, 1
20 2 4 1 Use the quadratic formula to solve.x x
24 4 4(2)( 1) 4 24 2 6 =2(2) 4 2
x
-intercepts 0.22 and 2.22x Axis of symmetry: 1x
x
y
1, 3
0, 1 2, 1
(b) Determine the domain and the range of .(c) Determine where is increasing and decreasing.
ff
The domain of f is the set of all real numbers.
Based on the graph, the range is the interval 3, .
The function is from , 1
and from 1, .
inc
decreasing
reasing
x
y
21(a) Graph ( ) 2 2 by determining whether the graph opens up or down 2
and by finding its vertex, axis of symmetry, and - and -intercepts if any.f x x x
x y
h b
2a
2
2 12
2
Since a is negative, the parabola opens down.
2,0
As seen on the graph, the -intercept is 2.x
212 2 2( 2) 2 02
k f
Vertex = 2,0
21(0) 0 2 0 2 2 so the -intercept = 2.2
f y
By symmetry, the point ( 4, 2) is also on the graph. 0, 2 4, 2
Axis of symmetry: 2x
x
y
2,0
0, 2 4, 2
(b) Determine the domain and the range of .(c) Determine where is increasing and decreasing.
ff
The domain of f is the set of all real numbers.
Based on the graph, the range is the interval ,0 .
The function is from , 2
and from 2, .
dec
increasing
reasing
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Determine the quadratic function whose vertex is (–2, 3) and whose y-intercept is 1.
2 22 3f x a x h k a x
Using the fact that the y-intercept is 1: 1a 0 2 23
1 4 3a 12
a
21 2 32
f x x
x
y
21 2 12
f x x x
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
2Determine whether the quadratic function
4 5has a maximum or minimum value.
Then find the maximum or minimum value.
f x x x
Since a is negative, the graph of f opens down so the function will have a maximum value.
4 2
2 2 1bxa
2So the maximum value is 2 (2) 4(2) 5 9f
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.