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Chapter 5 Notes
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5.1 Graphing Quadratic Functions
* Quadratic Function: y  ax 2  bx  c
(standard form)
* Parabola: u-shaped graph.
* a  0 : opens up.
* a  0 : opens down.
* Vertex: lowest or highest point.
* x-coordinate of the vertex: 
b
2a
* Axis of Symmetry: vertical line that cuts the parabola into 2 congruent parts
b
x
2a
Ex. 1 Write the quadratic function in standard form.
A) y 
1
 x  6  x  4 
2
B) y   4  x  7   2
2
* Vertex Form: y  a  x  h   k
2
* Vertex =  h , k 
* Axis of Symmetry: x  h
* Intercept Form: y  a  x  p  x  q 
* x – intercepts are p and q .
1
* Axis of Symmetry is
way between  p , 0  and  q ,0 .
2
Ex. 2 Graph
A) y   x 2  x 12
What form are we in?
B) y  2  x  3 x 1
What form are we in?
C) y  2x  x  4
What form are we in?
Ex. 3 Graph
A) y  2  x 1  3
What form are we in?
B) y  4  x 1 x 1
What form are we in?
2
5.2 Solving Quadratic Equations by Factoring
* Binomial: 2 terms ( x  5 or y 2  3 )
* Trinomial: 3 terms ( x 2  x  2 )
Ex. 1 Factor x 2  2 x  48
* Special Factoring Patterns:
* Difference of 2 Squares:
a2  b2   a  b  a  b 
* Perfect Square Trinomial:
a 2  2ab  b 2   a  b 
a 2  2ab  b 2   a  b 
Ex. 2 Factor
A) 4 y 2  4 y  3
B) 16 y 2  225
C) 4 x 2  12 x  9
D) 36w2  60w  25
E) 3v 2  18v
F) 12 x 2  3x  3
G) 14 x 2  2 x  12
H) 4k 2  36
2
2
* Zero Product Property: Let a and b be real numbers If ab  0, then a  0 or b  0 .
Ex. 3 Solve
A) 9 x 2  12 x  4  0
B) 3x  6  x 2  10
Ex. 4 A painter is making a rectangular canvas for her next painting. She wants the
length to be 4 ft more than twice the width. The area of the canvas must be 30 ft 2 . Find
the length and width of the canvas.
* Recall: x – intercept of y  a  x  p  x  q  are p and q .
* p and q are also called zeros of the function (since y = 0).
The zeros, the solutions to the equation when y = 0, the x-intercepts, and the roots are all
the same.
Ex. 5 Find the zeros of y  x 2  8x 15
5.3 Solving Quadratic Equations by Finding Square Roots
* Radical Sign:
* Radicand: what is beneath the radical (x) =
* Radical: expression x
* Product Property:
ab 
a
b
x
* Quotient Property:
Ex. 1 Simplify
A)
500
(rationalizing the denominator)
25
C)
3
E)
8
2
B) 3 12
D)
2
11
F)
6
6
8
Ex. 2 Solve
A) 3  5 x 2   9
B) 3  x  2   21
2
a

b
a
b
5.4 Complex Numbers
* x 2   1  no real solution.
* Imaginary Unit: i
where i  1
*
so, i 2  1
1  (1)(1)  1
of a negative number:
1.
r  i r
example:
5  1  5  1  5  i 5 or
9  3i
2.
i r 
2
 r
example:
i 7 
2
 i2
 7
2
 i2 7   7
Ex. 1 Solve
A) x 2   11
B)  x 2  4  14
C) x 2  26   10
D) 6  x  5   120
* Complex Numbers: of the form:
2
 a  bi 
where
a is the real part and bi is the imaginary part.

3
,  , 17 .
7
Imaginary Numbers  a  bi  where b  0 : examples: 14  21i , 3  5i .

Pure Imaginary Numbers

Real Numbers  a  0i  : examples: 1 , 13 ,
 0  bi  where b  0 :
examples: 7i ,  25i .
Ex. 2 Plot the complex numbers on a complex plane.
A) 4  i
B) 3
C) 1  2i
D)  i
Ex. 3 Adding and Subtracting complex numbers (standard form).
A)
 1  2i   3  3i 
B)
 2  3i   3  7i 
C) 2i   3  i    2  3i 
Ex. 4 Multiplying complex numbers.
A) i  3  i 
B)
 2  3i  6  2i 
C) 1  2i 1  2i 
* Complex Conjugates: where the product is a real number:
 a  bi  a  bi 
Ex. 5 Dividing complex numbers.
A)
2  7i
1 i
B)
3  11i
1  2i
* Absolute Value of a Complex Number: z
z  a 2  b2
Ex. 6 Find the absolute value.
A) 2  5i
B) 6i
C) 5  3i
5.5 Completing the Square
* Completing the Square:
x 2  bx  c
(another way to solve)
2
b
b
x  bx     c   
2
 2
2
2
: where the left side is a perfect square trinomial
Ex. 1 Find c to make it a perfect square trinomial.
A) x 2  3x  c
B) x2  8x  c
Ex. 2 Solve by completing the square.
A) x 2  6 x  8  0
B) 5x 2  10 x  30  0
* Standard Form: y  ax 2  bx  c
* Vertex Form: y  a  x  h   k where  h , k  is the vertex.
2
Ex. 3 Write the equation in vertex form. What is the vertex?
A) y  x 2  6 x  16
B) y   2 x 2  2 x  7
5.6 The Quadratic Formula and Discriminant
* Quadratic Formula: ax 2  bx  c  0 where
x
b  b2  4ac
2a
(another way to solve)
Ex. 1 Solve by using the quadratic formula.
A) 3x 2  8 x  35
B) 12 x  5  2 x 2  13
C) 2 x 2   2 x  3
b  b2  4ac
2a
2
* if b  4ac  0 : then it has 2 real solutions.
* if b 2  4ac  0 : then it has 1 real solution.
* if b 2  4ac  0 : then is has 2 imaginary solutions.
* Discriminant: b 2  4ac from
Ex. 2 Find the discriminant and tell how many solutions the equation has.
A) 9 x 2  6 x  1  0
C) 9 x 2  6 x  5  0
B) 9 x 2  6 x  4  0
5.7 Graphing and Solving Quadratic Inequalities
* Graph y  ax 2  bx  c
*  ,  : dashed parabola
*  ,  : solid parabola
*  ,  : shade down (less than)
*  ,  : shade up (greater than)
Ex. 1 Graph y  2 x 2  5x  3
Ex. 2 Solve the inequality algebraically
A) 2 x2  x  3
B) 3x 2  11x  4
Ex. 3 Graph the system of inequalities
y   x2  9
y  x2  5x  6
Chapter 5 Extension
Ex. 1 Graph
  x2 , x  0
f  x  
3x  2, x  0
Ex. 2 Graph
 x , x   2

f  x    x2 ,  2  x  2
 x , x  2

Ex. 3 Graph
3x  2, x  0
f  x   2
 x  2, x  0
Ex. 4 Graph
2 x  1, x  0

f  x     x2 , 0  x  2
 x, x  2
