Download x-intercept

Algebra-2
Lesson 4-3A
(Intercept Form)
Quiz 4-1, 4-2
1. What is the vertex of:
f ( x)  2 x  4 x  6
2
2. What is the vertex of:
f ( x)  ( x  7)  6
2
4-3A
Intercept Form
y  ax  bx  c
2
Standard Form:
y  2 x 2  12 x  1
Axis of symmetry:
Vertex:
(1)
b
 (12)
x
x
x 3
2a
2(2)
b
x
2a
x 3
y  2(3) 2  12(3)  1
y  17
(2) “2nd” “calculate” “min/max”
x-intercepts:
“2nd” “calculate” “zero”
Vertex Form:
Vertex:
y  a ( x  h)  k
2
(h, k)
(1)
y  2( x  1)2  3
(1,  3)
(2) “2nd” “calculate” “min/max”
Axis of symmetry:
x-intercepts:
xh
x  1
“2nd” “calculate” “zero”
Vocabulary
Intercept Form:
y  a( x  p)( x  q)
y  4( x  1)( x  3)
Intercept form
Graph the following
on your calculator:
y  ( x  1)( x  2)
x = -1
x = +2
What are the x-intercepts?
y=0
Vocabulary
Opens up
if positive
Intercept Form:
y  3( x  2)( x  4)
Opens
down
‘x-intercepts are:
‘-2’ and ‘-4’
‘x-intercepts are:
‘p’ and ‘q’
y  a( x  p)( x  q)
y  ( x  1)( x  3)
‘x-intercepts are:
‘+1’ and ‘+3’
Intercept form
Why do the intercept
have the opposite sign?
y  ( x  1)( x  2)
x = -1 x = +2
0  ( x  1)( x  2)
(x + 1) equals some number.
(x – 2) equals another number.
These two numbers multiplied together equal 0.
(x + 1) = 0
x = -1
(x – 2) = 0
x = +2
y=0
Vocabulary
Zero Product Property: If the product of 2 numbers equals 0,
A*B=0
then either:
A=0
and/or
0  ( x  2)( x  3)
Then by the zero product property:
x20
x  2
x 3  0
x 3
B = 0.
Your turn:
y  a( x  p)( x  q)
Which direction does it open and what are the
x-intercepts of the the following parabolas:
1.
y  5( x  3)( x  4)
2.
y  2( x  7)( x  6)
3.
y  ( x  2)( x  5)
y  a( x  p)( x  q)
Finding the vertex:
If you know the x-intercepts, how do you find the axis of
symmetry?
Half way between the x-intercepts.
y  2( x  4)( x  6)
x-intercepts are: 4, 6
Axis of symmetry is:
x=5
If you know the axis of symmetry, how do you find the
x-coordinate of the vertex?
Same as the axis of symmetry x = 5
If you know the x-coordinate of the vertex, how do you
find the y-coordinate?
y  2(( )  3)(( )  4) y  2(2)(1) The vertex is:
y  2((5)  3)((5)  4)
y4
(5, 4)
Your turn:
y  a( x  p)( x  q)
Find the vertex of the parabola:
4.
y  ( x  2)( x  6)
5.
y  2( x  2)( x  4)
6.
y  ( x  3)( x  5)
Vocabulary
Monomial: an expression with one term.
2x
Binomial: expression with two unlike terms.
x 1
The sum (or difference) of 2 unlike monomials.
Vocabulary
Trinomial: expression with three unlike terms.
The sum of 3 unlike monomials
x  3x  2
2
Or the product of 2 binomials.
( x  2)( x  1)
Intercept form is the product of 2 binomials!!
y  ( x  p)( x  q)
y  ( x  2)( x  1)
Product of Two Binomials
Know how to multiply two binomials
(x – 5)(x + 1)
Distributive Property (two times)
x(x + 1) – 5(x + 1)
x  x  5x  5
2
 x2  4x  5
Product of Two Binomials
Know how to multiply two binomials
(x – 3)(x + 2)
Distributive Property (two times)
x(x + 2) – 3(x + 2)
x  2 x  3x  6
2
 x2  x  6
Your turn:
Multiply the following binomials:
7.
8.
9.
( x  2)( x  6)
( x  2)( x  4)
( x  3)( x  5)
Taught to here as 4-3A
Your turn:
Multiply the following binomials:
1.
( x  2)( x  6)
2.
( x  2)( x  4)
3.
( x  3)( x  5)
I call this method the “smiley face”.
You have learned it as FOIL.
Smiley Face
x
2
 4x  2x  8
 x  2x  8
2
(x – 4)(x + 2) = ?
Left-most term
 left “eyebrow”
“nose and chin”
combine to form
the middle term.
right-most term 
right “eyebrow”
Your turn:
Multiply the following binomials:
4.
( x  1)( x  7)
5.
( x  3)( x  2)
6.
( x  3)( x  3)
Convert
Intercept Form to Standard Form
y  a( x  p)( x  q)
y  ax  bx  c
2
Just multiply the binomials.
y  ( x  1)( x  7)
y  x( x  7)  1( x  7)
y  x  7x  x  7
2
y  x  8x  7
2
Vocabulary
To Factor: split a binomial, trinomial (or any
“nomial”) into its original factors.
Standard form:
Factored form:
y  ax  bx  c
y  a( x  p)( x  q)
y  x  2x 1
y  ( x  2)( x  1)
2
2
Intercept form is a standard form that has been factored.
Factoring Quadratic expressions:
x  5x  x  5
2
(x – 5)(x + 1)
x  4x  5
2
(_ + _)(_ + _)
 x  4x  5
2
Factoring Quadratic expressions:
x  5x  x  5
2
 x2  4x  5
(x – 5)(x + 1) = ?
x2  4x  5
(x + _)(x + _)
-1, 5
1, -5
-1, 5
5, -1
-5, 1
1, -5
Factoring Quadratic expressions:
2

x

4
x

5
(x – 5)(x + 1) = ?
x  4x  5
2
(x + _)(x + _)
(x – 5)(x + 1)
-1, 5
1, -5
(x – 1)(x + 5)
(x – 5)(x + 1)
Factoring
x  bx  c
(x  m)(x  n)
2
c = mn
b=n+m
x  (m  n) x  mn
2
x  5x  6
2
(x + 3)(x + 2)
What 2 numbers when
multiplied equal 6 and when
added equal 5?
Factoring
x  bx  c
x  (m  n) x  mn
(x  m)(x  n)
2
2
2
x  4x  5
(x – 5)(x + 1)
What 2 numbers when
multiplied equal -5 and when
added equal -4?
Factoring
x  6x  8
2
What 2 numbers when
multiplied equal 8 and when
added equal -6?
(x – 2)(x – 4)
Your Turn:
Factor:
7.
x  4x  3
8.
x  2x 1
9.
2
2
x  6x  9
2
They come in 4 types:
Both positive
1st Negative, 2nd Positive
x  4x  3
x  6x  5
(x + 3)(x + 1)
(x – 1)(x – 5)
2
Both negative
2
1st Positive, 2nd Negative
x  2x  8
x  6 x  16
(x – 4)(x + 2)
(x + 8)(x – 2)
2
2
Your Turn:
10.
11.
Factor:
x  6x  5
2
x  6 x  16
2
12.
x  2x  8
13.
x  4 x  12
2
2
Vocabulary
Solution (of a quadratic equation): The input values that
result in the function equaling zero.
If the parabola crosses the x-axis, these are the x-intercepts.
0  AB
If A= 5, what must B equal?
If B = -2, what must A equal?
Zero Product Property
Zero product property: if the product of two factors
equals zero, then either:
(a) One of the two factors must equal zero, or
(b) both of the factors equal zero.
f ( x)  ax  bx  c
2
Solve by factoring
y  x  3x  2
2
(1) factor the quadratic equation.
(2) set y = 0
y  ( x  2)( x  1)
0  ( x  2)( x  1)
(3) Use “zero product property” to find the x-intercepts
( x  2)  0 and
x  2
( x  1)  0
x  1
f ( x)  ax  bx  c
2
Solve by factoring
y  x  5x  6
2
(1) factor the quadratic equation.
(2) set y = 0
y  ( x  2)( x  3)
0  ( x  2)( x  3)
(3) Use “zero product property” to find the x-intercepts
( x  2)  0 and
x2
( x  3)  0
x 3
Your Turn:
Solve by factoring:
2
14. y  x  9 x  14
15.
y  x  7x  8
16.
y  x  8 x  16
2
2
What if it’s not in standard form?
2
x  17  7  11x
Re-arrange into standard form.
x  11x  24  0
2
( x  3)( x  8)  0
3 + 8 = 11
x = -3
3 * 8 = 24
x = -8
Your Turn:
17.
18.
Solve by factoring:
2x  x  3  x  6x  9
2
2
3x  2 x  10  2 x  8 x  6
2
2
What if the coefficient of ‘x’ ≠ 1?
Solve by factoring:
0  (2 x  4)(9 x  3)
Use “zero product property” to find the x-intercepts
2x  4  0 and 9x  3  0
9x  3
2x  4
3
x
x2
9
1
x
3
Your Turn:
Solve
19.
20.
y  (2 x  4)( x  14)
0  ( x  7)(3x  2)
Your turn:
Multiply the binomials:
21. (2x – 1)(x + 3)
22. (x + 5)(x – 5)
Factor the quadratic expressions:
2
2
26.
x  100
25.
4 x  64
26.
x  2x  8
2
27.
x 4
4
Special Products
Product of a sum and a difference.
(x + 2)(x – 2)
“conjugate pairs”
(x + 2)(x – 2)
x  2x  2x  4
2
“nose and chin”
are additive inverses
of each other.
“The difference
of 2 squares.”
x 4
2
( x)  (2)
2
2
Your turn:
Multiply the following
conjugate pairs:
13.
14.
(x – 3)(x + 3)
(x – 4)(x + 4)
“The difference
of 2 squares.”
 x 9
2
 x  16
“The difference of 2 squares”
factors as conjugate pairs.
2
Your Turn:
Factor:
15.
x 2  36
16.
x  49
2
Special Products
( x  2)
Square of a sum.
2
2
x  2x  2x  2
2
(x + 2)(x + 2)
x  4x  4
2
Special Products
( x  3)
Square of a sum.
x  3x  3x  3
2
(x + 3)(x + 3)
2
2
 x  6x  9
2
Special Products
Square of a difference.
x  4x  4x  4
2
( x  4)
2
2
(x - 4)(x - 4)  x  8 x  16
2
Special Products
Square of a difference.
x  3x  3x  4
2
( x  3)
2
2
(x - 3)(x - 3)  x  6 x  9
2
Your Turn:
Simplify (multiply out)
17.
( x  4)
18.
( x  6)
2
2
We now have all the tools to “solve by factoring”
Vocabulary
Quadratic Equation:
f ( x)  ax  bx  c
2
f ( x)  x  x  6
2
Root of an equation: the x-value where the graph
crosses the x-axis (y = 0).
Zero of a function: same as root
Solution of a function: same as both root and zero of the function.
x-intercept: same as all 3 above.