5.1 Quadratic Functions
The graph of a quadratic function is a parabola, as shown below.
Standard Form:
f ( x) = ax 2 + bx + c
⎛ b
⎛ −b ⎞ ⎞
vertex: ⎜ − , f ⎜ ⎟ ⎟
⎝ 2a ⎝ 2a ⎠ ⎠
a<0
graph opens down
a>0
graph opens up
b
axis of symmetry: x = −
2a
Larson Text, Ch 5, p. 249
Graph each function given in Standard Form:
1. y = x 2 + 4x + 3
2. y = x 2 + 6x + 11
a=
a=
b=
b=
−
b
=
2a
⎛ b⎞
f ⎜− ⎟ =
⎝ 2a ⎠
−
b
=
2a
⎛ b⎞
f ⎜− ⎟ =
⎝ 2a ⎠
(x, y) =
(x, y) =
Vertex Form:
f ( x) = a( x − h) 2 + k
vertex: (h, k)
axis of symmetry: x = h
a<0
graph opens down
a>0
graph opens up
Graph each function given in Vertex Form.
1.
y = −(x − 1)2 + 4
2.
(h, k) =
(h, k) =
(x, y) =
(x, y) =
a=
a=
y = (x + 3)2 − 2
Intercept Form:
f (x) = a(x − r1 )(x − r2 )
x-intercepts: (r1 , 0) and (r2 , 0)
⎛r +r
⎛ r + r ⎞⎞
vertex: ⎜ 1 2 , f ⎜ 1 2 ⎟ ⎟
⎝ 2 ⎠⎠
⎝ 2
a<0
a>0
graph opens down
graph opens up
Graph each function in Intercept Form.
1.
f (x) = (x − 2)(x + 4)
2.
f (x) = −2(x + 1)(x − 5)
r1 =
r1 =
r2 =
r2 =
vertex =
vertex =
a=
a=
5.2 Solving Quadratic Equations by Factoring
Zero Product Property
Let A and B be real numbers or algebraic expressions. If AB = 0, then A = 0 or B =
0.
Solve each equation by using the Zero Product Property:
1.
0 = (x − 1)(x − 3)
2.
0 = x(x + 4)
3.
0 = x2 − 9
4.
0 = x 2 + 3x − 18
5.
2x 2 − 17x + 45 = 3x − 5
6.
4x 2 + 12x − 7 = 0
7.
0 = 6x 2 − 16x + 8
8.
125x 2 − 5 = 0
5.3 Solving Quadratic Equations by Finding Square Roots
Simplify expressions using properties of square roots:
1)
48
2)
90
3)
3
16
4)
25
3
5)
6 i 10
6)
5
3
125
Solve the quadratic equation by finding square roots.
7)
2x 2 + 1 = 17
8)
x 2 − 9 = 16
9)
4x 2 + 7 = 23
10)
5(x − 1)2 = 50
11)
1
(x + 8)2 = 14
2
5.5 Completing the Square
Warm Up:
1.) f ( x) = x 2 − 4 x − 12
Vertex: ____________________
x intercepts: ________________
2.) y = ( x − 2) 2 − 16
Vertex: ______________
X intercepts: __________________
Standard Form:________________
3.) Factor the Perfect Square Trinomials.
25
A.) x 2 − 8 x + 16 =
B.) x 2 + 5 x +
=
4
C.) x 2 − 7 x +
49
=
4
Can you see any pattern on how the second term in factored form is
related to the middle term of the original quadratic?
Find the value of c that makes the quadratic equation a perfect square
trinomial. Then write the quadratic in vertex form.
10
1.) y = x 2 − 14 x + c
2.) y = x 2 + x + c
3
Completing the Square to Graph a Quadratic Function
Rewrite the equation in vertex form by completing the square. Find the vertex. Then solve for
the x intercepts. Verify your x intercepts on the calculator. Graph the parabola.
1.) y = x 2 + 10 x − 3
Vertex Form :______________
Vertex:_________________
X intercepts: _________________
__________________
2.) y = x 2 + 6 x − 8
Vertex Form: _________________
Vertex:__________________
X intercepts : _____________
______________
3.) y = −x 2 + 4x − 1
Vertex Form : _________________
Vertex:__________________
X intercepts : _____________
_____________
4.) y = 2 x 2 − 12 x + 14
Vertex Form : _________________
Vertex:__________________
X intercepts : _____________
_____________
5.) y = 4x 2 − 6x + 1
Vertex Form : _________________
Vertex:__________________
X intercepts : _____________
_____________
6.) y = −3x 2 − 6x − 8
Vertex Form : _________________
Vertex:__________________
X intercepts : _____________
_____________
Completing the Square to solve a Quadratic Equation
Solve the following equations by completing the square.
1.)
x 2 − 12x = −28
2.)
x 2 + 3x − 1 = 0
3.)
−3x 2 + 24x = 27
4.)
4x 2 − 40x − 8 = 0
5.)
3x 2 − 26x + 2 = 5x 2 + 1
6.)
2x 2 + 3x + 1 = 0
7.)
−4x 2 − 2x = −5
8.)
−3x 2 + 5x = −7