algebra-2 lesson 4-3b (solving intercept form). quiz 4-1, 4-2 1. what is the vertex of: 2. what is...
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Algebra-2Algebra-2Lesson 4-3B Lesson 4-3B
(Solving Intercept Form)(Solving Intercept Form)
Quiz 4-1, 4-2Quiz 4-1, 4-2
642)( 2 xxxf1. What is the vertex of:1. What is the vertex of:
2. What is the vertex of:2. What is the vertex of:
6)7()( 2 xxf
Solving Intercept Solving Intercept FormForm
4-3B4-3B
Standard FormStandard Form: :
Axis of symmetryAxis of symmetry: :
cbxaxy 2
1122 2 xxy
a
bx
2
1)3(12)3(2 2 y
)2(2
)12(x 3x
VertexVertex: : a
bx
2
3x
x-interceptsx-intercepts: :
17y
(1)(1)
(2) “2(2) “2ndnd” “calculate” “min/max”” “calculate” “min/max”
““22ndnd” “calculate” “zero”” “calculate” “zero”
Axis of symmetryAxis of symmetry: : 1x
3) ,1( VertexVertex: : k) ,(h
hx
x-interceptsx-intercepts: :
(1)(1)
(2) “2(2) “2ndnd” “calculate” “min/max”” “calculate” “min/max”
““22ndnd” “calculate” “zero”” “calculate” “zero”
Vertex FormVertex Form: : khxay 2)( 3)1(2 2 xy
Axis of symmetryAxis of symmetry: :
1x
50) ,1(
VertexVertex: : 2
qpx
qpx ,x-interceptsx-intercepts: : (1)(1)
(2) “2(2) “2ndnd” “calculate” “min/max”” “calculate” “min/max”
(2) “2(2) “2ndnd” “calculate” “zero”” “calculate” “zero”
Intercept FormIntercept Form: : ))(( qxpxay )6)(4(2 xxy
6 ,4 x
]6)1][(4)1[(2 y2
64 x
)5)(5(2 y
(1)(1)
2
qpx
1x
Your turn:Your turn:Find the vertex for the following:Find the vertex for the following:
)6)(2( xxy1.1.
2.2. )4)(2( xxy
Product of Two BinomialsProduct of Two BinomialsKnow how to multiply two binomialsKnow how to multiply two binomials
(x – 5)(x + 1)(x – 5)(x + 1)
2x 5x5x
542 xx
x(x + 1) – 5(x + 1)x(x + 1) – 5(x + 1)
Distributive Property (two times)Distributive Property (two times)
Your turn:Your turn:Multiply the following binomials:Multiply the following binomials:
)6)(2( xx3.3.
4.4. )4)(2( xx
5.5. )5)(3( xx
Smiley FaceSmiley FaceI call this method the “smiley face”.I call this method the “smiley face”.
(x – 4)(x + 2) = ?(x – 4)(x + 2) = ?Left-most term Left-most term
left “eyebrow” left “eyebrow”
2x 8x4 x2 822 xx
right-most term right-most term right “eyebrow” right “eyebrow”
““nose and mouth” nose and mouth” combine to form combine to form the middle term.the middle term.
You have learned it as FOIL.You have learned it as FOIL.
Your turn:Your turn:Multiply the following binomials:Multiply the following binomials:
)7)(1( xx6.6.
7.7. )2)(3( xx
8.8. )3)(3( xx
Convert Convert Intercept Form to Standard Intercept Form to Standard FormForm
))(( qxpxay cbxaxy 2
Just multiply the binomials.Just multiply the binomials.
782 xxy)7)(1( xxy
)7(1)7( xxxy
772 xxxy
But why would you want to? But why would you want to? (intercept form gives more information)(intercept form gives more information)
VocabularyVocabulary
To FactorTo Factor: split a binomial, trinomial (or any: split a binomial, trinomial (or any “ “nomial”) into its original factors.nomial”) into its original factors.
122 xxy
cbxaxy 2Standard form:Standard form: Factored form:Factored form:
))(( qxpxay
)1)(2( xxy
Intercept formIntercept form is a is a standard formstandard form that has been that has been factoredfactored..
Factoring Quadratic Factoring Quadratic expressions:expressions:
(x – 5)(x + 1)(x – 5)(x + 1)
2x 5x5 x 542 xx
542 xx
(_ + _)(_ + _)(_ + _)(_ + _)
Factoring Quadratic Factoring Quadratic expressions:expressions:
(x – 5)(x + 1) = ?(x – 5)(x + 1) = ?
2x 5x5 x 542 xx
542 xx
(x + _)(x + _)(x + _)(x + _)
-1, 5-1, 55, -15, -1-5, 1-5, 11, -51, -5
-1, 5-1, 51, -51, -5
Factoring Quadratic Factoring Quadratic expressions:expressions:
(x – 5)(x + 1) = ?(x – 5)(x + 1) = ? 542 xx
542 xx
(x + _)(x + _)(x + _)(x + _)-1, 5-1, 51, -51, -5
(x – 1)(x + 5)(x – 1)(x + 5)
(x – 5)(x + 1)(x – 5)(x + 1)
(x – 5)(x + 1)(x – 5)(x + 1)
652 xx
(x (x mm)(x )(x nn)) c = mn c = mn
cbxx 2
(x + 3)(x + 2)(x + 3)(x + 2)
FactoringFactoring
What 2 numbers when What 2 numbers when multiplied equal 6 multiplied equal 6 and when and when added equal 5added equal 5??
b = n + m b = n + m
mnxnmx )(2
542 xx
(x (x mm)(x )(x nn))
cbxx 2
(x – 5)(x + 1)(x – 5)(x + 1)
FactoringFactoring
mnxnmx )(2
What 2 numbers when What 2 numbers when multiplied equal -5 multiplied equal -5 and when and when added equal -4added equal -4??
862 xx
(x – 2)(x – 4)(x – 2)(x – 4)
FactoringFactoring
What 2 numbers when What 2 numbers when multiplied equal 8 multiplied equal 8 and when and when added equal -6added equal -6??
Your Turn:Your Turn: Factor:Factor:
7. 7.
8.8.
9. 9.
122 xx
962 xx
342 xx
They come in 4 types:They come in 4 types:
342 xx(x + 3)(x + 1)(x + 3)(x + 1)
Both positiveBoth positive 11stst Negative, 2 Negative, 2ndnd Positive Positive
562 xx
1662 xx
(x – 1)(x – 5)(x – 1)(x – 5)
11stst Positive, 2 Positive, 2ndnd Negative Negative
(x + 8)(x – 2)(x + 8)(x – 2)
Both negativeBoth negative
822 xx(x – 4)(x + 2)(x – 4)(x + 2)
Your Turn:Your Turn: Factor:Factor:
10. 10.
11.11.
562 xx
1662 xx
12. 12.
13.13.
822 xx
1242 xx
VocabularyVocabulary
SolutionSolution (of a quadratic equation): The input values that (of a quadratic equation): The input values that result in the function equaling result in the function equaling zerozero..
If the parabola crosses the x-axis, these are the x-intercepts.If the parabola crosses the x-axis, these are the x-intercepts.
Solve by factoring:Solve by factoring:
Factor:Factor:
442 xxySet y = 0Set y = 0
440 2 xx
)2)(2(0 xxUse zero Use zero product product property to property to solve.solve. 2 ,2x
Your Turn:Your Turn:
Solve by factoringSolve by factoring::
14. 14.
15.15.
16.16.
1492 xxy
872 xxy
1662 xxy
What if it’s not in standard What if it’s not in standard form?form?
xx 117172 Re-arrange into standard form.Re-arrange into standard form.
024112 xx
0)8)(3( xx
3 + 8 = 113 + 8 = 11 3 * 8 = 243 * 8 = 24
x = -3x = -3 x = -8x = -8
Your Turn:Your Turn: Solve by factoring:Solve by factoring:
17. 17.
18.18.
9632 22 xxxx
6821023 22 xxxx
What if the coefficient of ‘x’ What if the coefficient of ‘x’ ≠ 1?≠ 1?
)39)(42(0 xxSolve by factoring:Solve by factoring:
42 x
Use “zero product property” to find the x-interceptsUse “zero product property” to find the x-intercepts
39 x039 and 042 xx
2x 9
3x
3
1x
Your Turn:Your Turn:
Solve Solve
19. 19.
20.20.
)14)(42( xxy
)23)(7(0 xx
Special ProductsSpecial ProductsProduct of a Product of a sumsum and a and a difference.difference.
(x + 2)(x – 2)(x + 2)(x – 2)
““conjugate pairs”conjugate pairs” 2x
(x + 2)(x – 2)(x + 2)(x – 2)
x2 4x2““nose and chin” nose and chin”
are additive are additive inversesinverses of each other.of each other.
2x 4““The difference The difference of 2 squares.”of 2 squares.”
22 )2()( x
Your turn:Your turn:
Multiply the following Multiply the following conjugate pairs:conjugate pairs:
21. 21. (x – 3)(x + 3)(x – 3)(x + 3)
22. 22. (x – 4)(x + 4)(x – 4)(x + 4)
““The difference The difference of 2 squares.”of 2 squares.”
92 x
162 x
““The difference of 2 squares” The difference of 2 squares” factors as conjugate pairs.factors as conjugate pairs.
Your Turn:Your Turn:
SolveSolve::23. 23.
23.23.
362 xy
148 2 x
Special ProductsSpecial ProductsSquare of a Square of a sumsum..
(x + 2)(x + 2)(x + 2)(x + 2)
2)2( x2x 22xx 22
442 xx
Special ProductsSpecial ProductsSquare of a Square of a differencedifference..
(x - 4)(x - 4)(x - 4)(x - 4)
2)4( x2x 24x4 x4
1682 xx
Your Turn:Your Turn: SimplifySimplify (multiply out) (multiply out)
24. 24.
25.25.
2)4( x
2)6( x
Your turn:Your turn:Solve by factoringSolve by factoring
26.26.
xxy 644 2
xxy 1002
27. 27.
VocabularyVocabularyQuadratic EquationQuadratic Equation: :
cbxaxxf 2)(
6)( 2 xxxf
Root of an equationRoot of an equation: the x-value where the graph : the x-value where the graph crosses the x-axis (y = 0).crosses the x-axis (y = 0).
Zero of a functionZero of a function: same as root: same as root
Solution of a functionSolution of a function: same as both root and zero of the function.: same as both root and zero of the function.
x-interceptx-intercept: same as all 3 above.: same as all 3 above.
Zero Product PropertyZero Product Property
AB0If A= 5, what must B equal?If A= 5, what must B equal? If B = -2, what must A equal?If B = -2, what must A equal?
Zero product propertyZero product property: if the product of two factors : if the product of two factors equals zero, then either:equals zero, then either:(a)(a)One of the two factors must equal zero, or One of the two factors must equal zero, or (b)(b)both of the factors equal zero.both of the factors equal zero.
Solve by factoringSolve by factoring cbxaxxf 2)(
232 xxy
(1) factor the quadratic equation.(1) factor the quadratic equation. )1)(2( xxy
2x
(2) set y = 0(2) set y = 0
(3) Use “zero product property” to find the x-intercepts(3) Use “zero product property” to find the x-intercepts
)1)(2(0 xx
1x0)1( and 0)2( xx
Solve by factoringSolve by factoring cbxaxxf 2)(
652 xxy
(1) factor the quadratic equation.(1) factor the quadratic equation. )3)(2( xxy
2x
(2) set y = 0(2) set y = 0
(3) Use “zero product property” to find the x-intercepts(3) Use “zero product property” to find the x-intercepts
)3)(2(0 xx
3x
0)3( and 0)2( xx