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Quadratic Functions
Chapter 7
Vertex Form
• Vertex (h, k)
khxaxf 2)()(
)8,5(
8)5(2 2 x
)2,7(
2)7(3 2
x
)6,0(
65 2 x
)0,0(
3 2x
)0,9(
)9(2 2x
Vertex Form
• a > 0, opens upward
• a < 0, opens downward
• the larger│a│is the narrower the parabola
• the closer a is to zero the wider the parabola
khxaxf 2)()(
Stretching the Unit Quadratic
2)( xxf 22)( xxf
2
2
1)( xxf
Reflecting Across the x-axis
2)( xxf
2)( xxf
Translating Graphs Up/Down
2)( xxf
2)( 2 xxf
2)( 2 xxf
Translating Graphs Right/Left
2)( xxf
2)3()( xxf
2)4()( xxf
Graphing a Quadratic Function
• First graph vertex
• Find a point
1)3(2)( xxf
)1,2(
1)2(
12)2(
1)1(2)2(
1)1(2)2(
1)32(2)2(2
2
f
f
f
f
f
)1,3(
• Draw axis of symmetry through vertex
• Reflect point over axis
Graphing a Quadratic Function1)3(2)( xxf
)1,4(
3x
Finding a Quadratic Model
• Create a scattergram• Select a vertex (Doesn’t have to be data
point)• Select non-vertex point• Plug vertex in for h and k, and the nonvertex
point for x and f(x)/y into a standard equation
• Solve for a• Then substitute a into the standard equation
Graph Quadratic Model
• Pick vertex– (70, 5)
• Pick point– (40, 9)
x f(x)
1930 (30) 12
1940 (40) 9
1950 (50) 7
1960 (60) 6
1970 (70) 5
1980 (80) 6
1990 (90) 7
2000 (100) 10
900
4
9004
5590059
5)30(9
5)7040(9
5)70(
)(
)()(
2
2
2
2
2
a
a
a
a
a
xay
khxay
khxaxf5)70(
900
4)( 2 xxf
7.2 Graphing Quadratics in Standard Form
Quadratic in Standard Form
• Find y-intercept (0, c)
• Find symmetric point
• Use midpoint formula of the x-coordinates of the symmetric points to find the x-coordinate of the vertex
• Plug x-coordinate of the vertex into equation for x
cbxaxxf 2)(
Graphing Quadratics
• Y-intercept– (0, 7)
• Symmetry Point
76)( 2 xxxf
)7,6)(7,0(
6,0
06,0
)6(0
60
77677
767
2
2
2
xorx
xorx
xx
xx
xx
xx
Graphing Quadratics
• (0, 7) (6, 7)• Midpoint
76)( 2 xxxf
32
)6(0
16)3(
7189)3(
7)3(6)3()3( 2
f
f
f
Vertex formula
• vertex formula
x-coordinate
• y-coordinate
cbxaxxf 2)(
a
bx
2
a
bf
2
a
bf
a
b
2,
2
Vertex Formula
3
23
3
11
3
12
3
2
3
1
43
12
9
13
43
12
3
13
3
1
3
1
)3(2
2
423)(
2
2
f
x
xxxf
3
23,
3
1
Maximum/Minimum
• For a quadratic function with vertex (h, k)
• If a > 0, then the parabola opens upward and the vertex is the minimum point (k minimum value)
• If a < 0, then the parabola opens downward and vertex is the maximum point (k maximum value)
cbxaxxf 2)(
Maximum Value Model
• A person plans to use 200 feet of fencing and a side of her house to enclose a rectangular garden. What dimensions of the rectangle would give the maximum area? What is the area?
22200
)2200(
2200
2002
wwA
wwA
wl
lwA
lw
100
100200
)50(2200
2200
504
200
)2(2
200
l
l
l
wl
w
Maximum area would be 50 x 100 = 5000
7.3 Square Root Property
Product/Quotient Property for Square Roots
• For a ≥ 0 and b ≥ 0,
• For a ≥ 0 and b > 0,
• Write radicand as product of largest perfect-square and another number
• Apply the product/quotient property for square roots
baab
b
a
b
a
Simplifying Radical Expressions
• No radicand can be a fraction
• No radicand can have perfect-square factors other than one
• No denominator can have a radical expression
Examples
52
54
54
20
53
59
59
45
5
3
25
3
25
3
6
14
2
2
23
7
29
7
18
7
Square Root Property• Let k be a nonnegative constant. Then,
is equivalent to kx 2 kx
3
3
5
15
5
5
785
2
2
2
x
x
x
x
24
173
8
173
8
173
8
17)3( 2
x
x
x
x
4
3412
4
34
4
12
4
343
2
2
22
173
x
x
x
x
Imaginary Numbers
• Imaginary unit, (i), is the number whose square is -1.
• Square root of negative number– If n is a positive real number,
12 i 1i
nin
Complex Numbers
• A complex number is a number in the form
• Examples
• Imaginary number is a complex number, where a and b are real numbers and b ≠ 0
bia
i73 i35 ii 330 606 i
Solving with Negative Square Roots
ix
ix
x
x
6
36
36
362
24
216
32
322
ix
ix
x
x
7.4 Completing the Square
Perfect Square Trinomial
• For perfect square trinomial in the form
dividing by b by 2 and squaring the result gives c:
cbxx 2
cbxx 2
cb
2
2 c
c
cxx
16
9
4
3
2
1
2
3
2
3
2
2
2
Examples
34
34
3)4(
3)4(8
32
88
38
2
22
22
2
x
x
x
xx
xx
xx
5
2
5
1
5
2
5
2
10
2
5
2
5
2
2
1
5
2
5
2
5
2
5
2
225
0225
22
22
22
2
2
2
xx
xx
xx
xx
xx
xx
5
101
5
10
5
1
5
5
5
2
5
1
5
2
5
1
5
2
5
12
x
x
x
x
x
7.5 Quadratic Formula
Quadratic Formula
• The solutions of a quadratic equation in the form are given by the quadratic formula:
02 cbxax
a
acbbx
2
42
Determining the Number of Real-Number Solutions
• The discriminant is and can be used to determine the number of real solutions
• If the discriminant > 0, there are two real-number solutions
• If the discriminant = 0, there in one real-number solution
• If the discriminant < 0, there are two imaginary-number solutions (no real)
acb 42
Quadratic Formula
8
1284
8
144164
)4(2
)9)(4(4)4()4(
944
2
2
xx
22
18
284
8
2644
i
i
i
Examples
0169 2 xx
0
3636
)1)(9(462
0852 2 xx
39
6425
)8)(2(4)5( 2
One real-number solution Two imaginary-number solutions
Intersections with y = n lines/points at a certain height
12
02432
y
xx
2
573
2
4893
)1(2
)12)(1(4)3()3(
0123
12243
2
2
2
xx
xx
Note if the discriminant is < 0, then there are no intersections