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