Upload
others
View
8
Download
0
Embed Size (px)
Citation preview
College Prep Unit 9: Quadratic Functions
Ms. Talhami 1
College Prep Unit 9: Quadratic Functions
Name_________________
College Prep Unit 9: Quadratic Functions
Ms. Talhami 2
Helpful Vocabulary Word Definition/Explanation Examples/Helpful Tips
College Prep Unit 9: Quadratic Functions
Ms. Talhami 3
What is a Quadratic Function? Basic Form Standard Form
What does the graph of a quadratic function look like? This shape is called a _______________.
Axis of Symmetry (Line)
Vertex (Turning Point)
College Prep Unit 9: Quadratic Functions
Ms. Talhami 4
For each of the following parabolas, find the axis of symmetry and the vertex.
AOS:__________ Vertex:__________
AOS:__________ Vertex:__________
AOS:__________ Vertex:__________
AOS:__________ Vertex:__________
AOS:__________ Vertex:__________
AOS:__________ Vertex:__________
College Prep Unit 9: Quadratic Functions
Ms. Talhami 5
Standard Form vs Vertex Form Standard Form Vertex Form
How does changing the value of “a” change the graph?
Therefore as |𝑎| increases, the graph becomes _______________.
Therefore as |𝑎| decreases, the graph becomes _______________.
And if 𝑎 is negative, the graph ________________________________________. How does changing the value of “c” (which is “k” in vertex form) change the graph?
Therefore if 𝑐 is positive, the graph _______________ 𝑐 units.
Therefore if 𝑐 is negative, the graph _______________ 𝑐 units.
Parent Function
𝑦 = 𝑥!
𝑦 = 2𝑥!
𝑦 =12𝑥!
Parent Function
𝑦 = 𝑥!
𝑦 = 𝑥! + 3
𝑦 = 𝑥! − 2
College Prep Unit 9: Quadratic Functions
Ms. Talhami 6
How does changing the value of “h” change the graph?
Therefore if ℎ is positive, the graph _______________ ℎ units.
Therefore if ℎ is negative, the graph _______________ ℎ units. Do not use a calculator. Graph the following. Describe the transformations. You must plot and state the 3 “key” points, wherever they end up after transformation. 1. 𝑓(𝑥) = −(𝑥 + 1)! + 4 2. 𝑦 = (𝑥 − 3)!
3. 𝑓(𝑥) = −(𝑥 + 4)! − 2 4. 𝑦 = 2𝑥! − 5
Parent Function
𝑦 = 𝑥!
𝑦 = (𝑥 − 2)!
𝑦 = (𝑥 + 4)!
College Prep Unit 9: Quadratic Functions
Ms. Talhami 7
5. 𝑓(𝑥) = "!(𝑥 − 2)! 6. 𝑦 = −3(𝑥 − 1)! + 6
Write the quadratic equation, in vertex form for each graph. 7. ____________________ 8. ____________________
9. ____________________ 10. ____________________
College Prep Unit 9: Quadratic Functions
Ms. Talhami 8
11. ____________________ 12. ____________________
How to Graph Using the Axis of Symmetry, the Vertex, and the Intercepts
Steps to Sketch the Graph the Quadratic Function 𝑦 = 𝑎𝑥! + 𝑏𝑥 + 𝑐 1. Determinewhethertheparabolaopensupwardordownward.
If𝑎 > 0,itopensupward.If𝑎 < 0,itopensdownward.
2. Graphtheaxisofsymmetry,𝑥 = − !"#
3. Plotthevertex,$− !"#, 𝑓 '− !
"#()
4. Determineanyx-interceptsandplotthecorrespondingpoints.Anx-interceptisasolutiontotheequation𝑎𝑥! + 𝑏𝑥 + 𝑐 = 0.
5. Determinethey-intercept,c,andplotthecorrespondingpoint.Thenusesymmetrytoplottheimageofthepoint(0, 𝑐).
6. Connectthepointswithasmoothcurve. Sketch the following graphs: 1. 𝑦 = 𝑥! − 2𝑥 − 3 2. 𝑦 = −2𝑥! + 2𝑥
College Prep Unit 9: Quadratic Functions
Ms. Talhami 9
3. 𝑦 = 3𝑥! − 2𝑥 − 1 4. 𝑦 = −2𝑥! − 4𝑥
Let’s Review Factoring Quadratics Solve the following by factoring (if factorable): 1. 𝑥! + 10𝑥 − 11 = 0 2. 𝑥! − 12𝑥 + 7 = 0 Standard Form and Perfect Square Trinomials
1. (x – 2)2 a = ______ b= ______ c= ______
2. (x + 5)2 a = ______ b= ______ c= ______
3. (x – 9)2 a = ______ b= ______ c= ______
Completing the Square
Determine the value of the constant term, c, to create a perfect square trinomial then write the trinomial in factored form. 1.
x2 + 4x + ___ Factored Form _____________
2. x2 + 10x + ___
Factored Form _____________
3. x2 + 14x + ___
Factored Form _____________
4. x2 – 12x + ___
Factored Form _____________
5. x2 – 8x + ___
Factored Form _____________
6. x2 – 2x + ___
Factored Form _____________
College Prep Unit 9: Quadratic Functions
Ms. Talhami 10
Using Completing the Square with Quadratic Equations to Rewrite from Standard Form to Vertex Form 1.
x2 + 6x + 3 = 0
2. x2 + 10x + 20 = 0
3. x2 – 8x – 3 = 0
How to Solve Quadratics (where 𝑎 = 1 and solutions are real numbers) by Completing the Square 1. 𝑥! + 10𝑥 − 11 = 0 2. 𝑥! − 12𝑥 + 7 = 0 3. 𝑥! + 14𝑥 − 51 = 0 4. 𝑥! = 2𝑥 + 3 5. 𝑥! + 14𝑥 = 48 6. −49 = −𝑥! + 6𝑥 7. 𝑥! − 48 = 14𝑥 8. 𝑥! + 6𝑥 − 49 = 0
College Prep Unit 9: Quadratic Functions
Ms. Talhami 11
How to Solve Quadratics (where 𝑎 ≠ 1 and solutions are imaginary) by Completing the Square 1. 5𝑥! + 20𝑥 − 60 = 0 2. 8𝑥! + 16𝑥 − 42 = 0 3. 𝑥! − 6𝑥 = −91 4. 2𝑥! − 3𝑥 − 11 = 0 5. 𝑥! + 6𝑥 + 41 = 0 6. 3𝑥! = −4 + 8𝑥 Another Method to Solving Quadratics If the quadratic equation is written in standard form, you can use the quadratic formula to solve for the roots.
𝑥 =−𝑏 ± √𝑏" − 4𝑎𝑐
2𝑎
Examples 1. 2𝑥! + 5𝑥 − 7 = 0 2. 4𝑥! − 8𝑥 + 13 = 0 3. 𝑥! + 4𝑥 − 14 = 0
College Prep Unit 9: Quadratic Functions
Ms. Talhami 12
Practice Solving Quadratics Using the Quadratic Formula
©n C2v0Z1q2v wKzu2t8az aSPopfptvwDaAruet FLKLfC2.S s KANltlH trIiAgPhKtJsI prgeFsXeQrJv9e8dM.E F fMOavdqe7 fwxintLhg DI0nIfgiRnui2tgeQ OAKlMgdecb0rBa9 01i.I Worksheet by Kuta Software LLC
Kuta Software - Infinite Algebra 1 Name___________________________________
Period____Date________________Using the Quadratic Formula
Solve each equation with the quadratic formula.
1)
m2 − 5
m − 14 = 0 2)
b2 − 4
b + 4 = 0
3)
2
m2 + 2
m − 12 = 0 4)
2
x2 − 3
x − 5 = 0
5)
x2 + 4
x + 3 = 0 6)
2
x2 + 3
x − 20 = 0
7)
4
b2 + 8
b + 7 = 4 8)
2
m2 − 7
m − 13 = −10
-1-