Chapter 4 ReviewChapter 4 Review

What shape does a What shape does a Quadratic FunctionQuadratic Function make make when it is graphed?when it is graphed?

1.1.

2. 2. What does it mean to What does it mean to solvesolve a quadratic equation? a quadratic equation?

Parabola—a “u-shaped curve Parabola—a “u-shaped curve

3. 3. Name the 3 methods you can use to solve a quadraticName the 3 methods you can use to solve a quadratic equation.equation.

Find x-intercepts Find x-intercepts

GraphGraphFactorFactorQuadratic formulaQuadratic formula

Power of the calculator baby!Power of the calculator baby! Graph!Graph!

Chapter 4: Quadratic Chapter 4: Quadratic Functions Functions

An infinite number of points that fall along a An infinite number of points that fall along a curved shape called a curved shape called a parabola.parabola.

2xy Two variable equationTwo variable equation::

(when the ‘x’ variable has an (when the ‘x’ variable has an exponent = twoexponent = two) )

Your Turn:

4.4. Describe the transformation to the parent Describe the transformation to the parent function:function:

2xy

3)5( 2 xy

5.5. Describe the transformation to the parent Describe the transformation to the parent function:function:

2xy

2)1(2 xy

translated up 3 translated up 3 translated left 5translated left 5

2 times as steep 2 times as steep translated right 1translated right 1

6.6. Describe the transformation to the parent Describe the transformation to the parent function:function:

2xy

4)3(2

1 2 xyReflected across x-axis Reflected across x-axis

1/2 as steep 1/2 as steep translated up 4 translated up 4 translated left 3translated left 3

Vertex FormVertex Form: : khxay 2)(

Vertex is Vertex is (h, k)(h, k)What is the vertex?What is the vertex?

khxay 2)(

2)3(4 2 xy

-h = -3-h = -3

h = 3h = 3

k = 2k = 2

Vertex is Vertex is

(3, 2)(3, 2)

Think transformation:Think transformation: Vertex (0,0) has been Vertex (0,0) has been

moved right 3 and up 2. moved right 3 and up 2.

New vertex:New vertex: (3, 2) (3, 2)

VertexVertexkhxay 2)(

4)2( 2 xy

6)3(2 2 xy

2)7(3 2 xy

Your turn: Your turn: What is the vertex ?What is the vertex ?

8. 8.

9. 9.

7. 7.

Axis of symmetryAxis of symmetry

11 55

Vertical Line !!!Vertical Line !!!

3x

The axis of symmetry is The axis of symmetry is halfwayhalfway in between the two x-intercepts. in between the two x-intercepts.

Axis of SymmetryAxis of Symmetry

5 ,1x

Your turn: Your turn: What is the axis of symmetry What is the axis of symmetry (the two x-intercepts are given below)?(the two x-intercepts are given below)?

11. 11.

12. 12.

10. 10. 4 ,2x

7 ,2x

Standard FormStandard Form: :

Axis of symmetryAxis of symmetry: :

cbxaxy 2

1122 2 xxy

a

bx

2

)2(2

)12(x 3x

(1) “2(1) “2ndnd” “calculate” “min/max”” “calculate” “min/max”

““22ndnd” “calculate” “zero”” “calculate” “zero”

X-value of the vertex tells you the equation of the X-value of the vertex tells you the equation of the axis of symmetry. axis of symmetry.

Power of the calculator baby!Power of the calculator baby!Graph!Graph!

Finding the VertexFinding the Vertex 1122 2 xxyOpen bottom of Open bottom of window further window further down down y min y min more negativemore negative

22ndnd, calculate:, calculate:

Left bound,Left bound,Right bound,Right bound,guessguess

Vertex: (3, -17) Vertex: (3, -17)

Axis: x = 3Axis: x = 3

Standard FormStandard Form: :

Axis of symmetryAxis of symmetry: :

cbxaxy 2

1122 2 xxy

a

bx

2

)2(2

)12(x 3x

X-value of the vertex tells you the equation of the X-value of the vertex tells you the equation of the axis of symmetry. axis of symmetry.

Power of the calculator baby!Power of the calculator baby! Graph!Graph!

Standard FormStandard Form: :

Intercept FormIntercept Form: :

cbxaxy 2 132 2 xxy

))(( qxpxay )3)(1( xxy

Your turn: Your turn:

13. 13. Find the vertex of the following Find the vertex of the following parabola:parabola: 132 2 xxy

14. 14. Does the following function Does the following function have a minimum or a maximum? have a minimum or a maximum?

)3)(1( xxy

15. 15. What is the minimum/maximum What is the minimum/maximum value of the following function? value of the following function?

)3)(1( xxy

VocabularyVocabulary

Intercept FormIntercept Form: : ))(( qxpxay

)3)(1( xxy

Opens up Opens up if positiveif positive

‘‘x-intercepts are:x-intercepts are:‘‘pp’ and ‘’ and ‘qq’’

‘‘x-intercepts are:x-intercepts are:‘‘+1+1’ and ‘’ and ‘+3+3’’

)4)(2(3 xxy

‘‘x-intercepts are:x-intercepts are:‘‘-2-2’ and ‘’ and ‘-4-4’’

Opens Opens downdown

Your turnYour turn: : What are the solutions to the following equations?What are the solutions to the following equations?

)3)(63( xxy

16. 16.

17. 17.

18. 18.

)4)(1(2 xxy

)2)(3(5 xxy

Solve by graphing:Solve by graphing:

Move the “cursor” on the Move the “cursor” on the graph to the graph to the leftleft of the of the left x-intercept. left x-intercept.

Move the “cursor” to the Move the “cursor” to the rightright side of the left x-intercept.side of the left x-intercept.

Find the Find the leftleft “x-intercept” by hitting: “x-intercept” by hitting: “ “22ndnd” + “calculate” + “2” (“zero”) ” + “calculate” + “2” (“zero”)

742 2 xxy

3)2( 2 xy19. 19. Solve the quadratic by graphing.Solve the quadratic by graphing.

20.20. Solve the quadratic by graphing.Solve the quadratic by graphing.

1522 xxy

VocabularyVocabularyTrinomialTrinomial: expression with : expression with three unlike termsthree unlike terms. .

The The sumsum of 3 unlike monomials of 3 unlike monomials

Or the Or the productproduct of 2 binomials of 2 binomials..

))(( qxpxy )1)(2( xxy

Intercept form Intercept form is the product of 2 binomials!!is the product of 2 binomials!!

232 xx

)1)(2( 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:

)6)(2( xx21.21.

22.22. )4)(2( xx

Factoring Quadratic Factoring Quadratic expressions:expressions:

(x + _)(x + _)(x + _)(x + _)

542 xx(_ + _)(_ + _)(_ + _)(_ + _)

Multiplied = -5Multiplied = -5

added = -4added = -4

(x + 1)(x – 5) (x + 1)(x – 5)

Your Turn:Your Turn: Factor:Factor:

23. 23.

24.24. 962 xx

1242 xx

25. 25. Solve the quadratic by factoring.Solve the quadratic by factoring.

442 xxy

1522 xxy

25. 25. Solve the quadratic by factoring.Solve the quadratic by factoring.

Do you remember how to solve the quadratic by Do you remember how to solve the quadratic by “extracting a square root”?“extracting a square root”?

4)3(2 2 xy

4)3(20 2 x

2)3(24 x2)3(2 x

Isolate the power, undo the power.Isolate the power, undo the power.

32 xi

2)3(2 x

xi 23

27. 27. Solve the quadratic.Solve the quadratic.

15)4(3 2 xy

How many solutions?How many solutions?

How many solutions?How many solutions?

How many solutions?How many solutions?

Remember this?Remember this?

i4

3

i

i

4

32 i3

2

i

i

22

28

Let’s learn the Let’s learn the easy wayeasy way to do this. to do this.

Your turn:Your turn:

i2

5

i

i

96

57

28. 28. SimplifySimplify

29. 29. SimplifySimplify

VocabularyVocabularyQuadratic FormulaQuadratic Formula: gives the solutions (x-intercepts) to : gives the solutions (x-intercepts) to ANYANY

quadratic equation in quadratic equation in standard formstandard form..

a

acbbx

2

42

cbxaxy 2 432 2 xxy a = ? b = ? c = ?a = ? b = ? c = ?

This formula is on your reference sheet.This formula is on your reference sheet.

The The DescriminantDescriminant is the is the radicandradicand of the quadratic of the quadratic formula.formula. a

acbbx

2

42

042 acb

042 acb

042 acb

TwoTwo real roots real roots

Parabola intersects the x-axis at 2 placesParabola intersects the x-axis at 2 places

oneone real root real root

Parabola intersects the the x-axis at 1 placeParabola intersects the the x-axis at 1 place

The original equation was a The original equation was a perfect square perfect square trinomialtrinomial

nono real roots real roots

Parabola does Parabola does notnot intersect the x-axis intersect the x-axis

What if you can’t remember what the What if you can’t remember what the Radicand is?Radicand is?““Find the radicand then give the number of type Find the radicand then give the number of type of the solutions for the quadratic equation.”of the solutions for the quadratic equation.”

6322 2 xx

a. 36, two reala. 36, two real

b. -39, one realb. -39, one real

c. -36, two imaginaryc. -36, two imaginary

The height (feet) of a falling object on earth can be modeled The height (feet) of a falling object on earth can be modeled with the equationwith the equation

The height (feet) of a falling object on earth can be modeled The height (feet) of a falling object on earth can be modeled with the equationwith the equation 2016)( 2 tth

30. 30. When will it hit the ground (h = 0)When will it hit the ground (h = 0) sec1.1t

ttth 10016)( 2

31. 31. When will it reach its maximum height? (h is a max)When will it reach its maximum height? (h is a max)

sec1.3t

Can’t figure it out? Can’t figure it out? try graphing try graphingFind the equation of a parabola that has x-intercepts of 4 and Find the equation of a parabola that has x-intercepts of 4 and 1 and passes through the point (3, -4).1 and passes through the point (3, -4).

a. a.

b.b.

c.c.

92 xy

)1)(4( xxy

)1)(4( xxy

)1)(4(2 xxyd.d.

These 3 points are on the parabola: (4, 0), (1, 0), and (3, -4)These 3 points are on the parabola: (4, 0), (1, 0), and (3, -4)

Your turn: Your turn: 32. 32. Find the equationFind the equation

Vertex: (3, -1)Vertex: (3, -1)Point: (x, y) = (2, -4)Point: (x, y) = (2, -4)

a.a.

1)3(3 2 xyb.b.

c.c.

d.d.

1)3(2 2 xy

1)3(2 2 xy

1)3(3 2 xy