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

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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

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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

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