![Page 1: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/1.jpg)
Chapter 4 ReviewChapter 4 Review
![Page 2: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/2.jpg)
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!
![Page 3: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/3.jpg)
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) )
![Page 4: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/4.jpg)
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
![Page 5: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/5.jpg)
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)
![Page 6: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/6.jpg)
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.
![Page 7: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/7.jpg)
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.
![Page 8: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/8.jpg)
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
![Page 9: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/9.jpg)
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!
![Page 10: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/10.jpg)
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
![Page 11: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/11.jpg)
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!
![Page 12: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/12.jpg)
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
![Page 13: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/13.jpg)
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
![Page 14: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/14.jpg)
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
![Page 15: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/15.jpg)
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
![Page 16: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/16.jpg)
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
![Page 17: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/17.jpg)
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
![Page 18: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/18.jpg)
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.
![Page 19: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/19.jpg)
Your turn:Your turn:Multiply the following binomials:Multiply the following binomials:
)6)(2( xx21.21.
22.22. )4)(2( xx
![Page 20: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/20.jpg)
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)
![Page 21: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/21.jpg)
Your Turn:Your Turn: Factor:Factor:
23. 23.
24.24. 962 xx
1242 xx
![Page 22: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/22.jpg)
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.
![Page 23: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/23.jpg)
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
![Page 24: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/24.jpg)
27. 27. Solve the quadratic.Solve the quadratic.
15)4(3 2 xy
![Page 25: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/25.jpg)
How many solutions?How many solutions?
How many solutions?How many solutions?
How many solutions?How many solutions?
![Page 26: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/26.jpg)
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.
![Page 27: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/27.jpg)
Your turn:Your turn:
i2
5
i
i
96
57
28. 28. SimplifySimplify
29. 29. SimplifySimplify
![Page 28: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/28.jpg)
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.
![Page 29: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/29.jpg)
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
![Page 30: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/30.jpg)
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
![Page 31: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/31.jpg)
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
![Page 32: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/32.jpg)
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)
![Page 33: Chapter 4 Review. What shape does a Quadratic Function make What shape does a Quadratic Function make when it is graphed? when it is graphed?1. 2. What](https://reader035.vdocuments.site/reader035/viewer/2022062308/56649dda5503460f94ad0c14/html5/thumbnails/33.jpg)
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