February 14, 2018 March 9, 2018 - s3.amazonaws.com quadratic graph has 3 important characteristics – its vertex, its axis of symmetry, and its zeros.

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Coach Schmidt:

***Tutorials: Tuesday - Friday from 7:35 – 7:55 AM***

Ms. Martinez:

***Tutorials: Monday, Wednesday, Friday from 7:35 – 7:55 AM***

Mr. Landrum:

***Tutorials: Tuesday - Friday from 7:35 – 7:55 AM***

MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY February 12 13 14 15 16

Warm Up – EOC

Quadratic Features

HW: WS

Warm Up – EOC

Quadratic Features

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Warm Up – EOC

Graph Quadratics – Standard Form

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Student

Holiday

Warm Up – EOC

Graph

Quadratics – Day 2

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Quiz

Graph

Quadratics – Vertex Form

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Warm Up – EOC

Vertex Form

– Day 2

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Vertex Form

– Day 3

HW: WS 26 27 28 March 1 2

Quiz

Simplify

Radicals

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Warm Up – EOC

Simplify

Radicals - Day 2

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Warm Up – EOC

Square Root

Method

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Quiz

Completing

the Square

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Warm Up – EOC Completing

the Square - Day 2

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5 6 7 8 9

Quiz

Quadratic

Formula

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Quadratic

Formula - Day 2

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Warm Up – EOC

Quadratic

Formula - Day 3

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Warm Up – EOC

REVIEW!!

HW: REVIEW!!

TEST

(Turn in Review)

5th Six Weeks 2017-18 Unit 9 – Quadratic Functions February 14, 2018 – March 9, 2018

1

NOTES

CHARACTERISTICS OF QUADRATIC GRAPHS

A quadratic equation is any equation that has a degree (highest exponent) of _____.

Examples of quadratic equations are:

y = x2 y = 2x2 – 1 y = ½ x2 – 4x + 3 y = -x2 + 5x

Just like linear functions, quadratic functions have a parent function as well.

The quadratic parent function is _______.

The graph of any quadratic equation is called a

_________ and looks like a “U”.

The graph can open _____ or open _________.

Notice that it sits at the __________.

A quadratic graph has 3 important characteristics – its vertex, its axis of symmetry, and its

zeros. These characteristics have all been labeled on the graph at the bottom of the page.

1. The VERTEX of a quadratic graph is the lowest (_________) or highest

(__________) point of the graph – it is the turning point of the graph. The vertex

of the quadratic parent function is (0, 0) since that is the lowest point of the graph.

2. The AXIS OF SYMMETRY of a quadratic graph is the vertical line that cuts the

parabola in _____ – so it is the vertical line that goes through the vertex. The axis

of symmetry for quadratic parent function is x = 0 (remember that all vertical

lines are written x = number).

3. The ZEROS ( ________, _________) of a quadratic graph is the point or points

where the graph __________ or __________ the x-axis. A quadratic graph can

have one, two, or no zeros. Since the quadratic parent graph touches the x-axis

only once, it has only one zero at (0, 0).

2

Look at the three graphs below and see if you can label the vertex, axis of symmetry, and

zeros for each parabola. The answers are given below and are labeled for you at the

bottom of the page so you can check your answers.

1. 2. 3.

Graph 1:

Vertex: _____ because that is the highest point of the graph since the parabola opens

down.

Axis of symmetry: _____ because that is the vertical line that cuts the parabola in half

and goes through the vertex.

Zeros: _____ and _____ since those are the two points where the parabola crosses/touches

the x-axis.

Graph 2:

Vertex: _____ because that is the lowest point of the graph since the parabola opens up.

Axis of symmetry: _____ because that is the vertical line that cuts the parabola in half

and goes through the vertex.

Zeros: _____ only since that is the one point where the parabola crosses/touches the x-

axis.

Graph 3:

Vertex: ______ or ______ because that is the lowest point of the graph since the parabola

opens up.

Axis of symmetry: ______ or ______ because that is the vertical line that cuts the

parabola in half and goes through the vertex.

Zeros: _____ since the parabola does not cross/touch the x-axis.

3

If A > 1, then the parabola is ______________.

If 0 < A < 1, then the parabola is ______________.

“C” will translate the parabola ___________ or ____________.

“-A” will ________ parabola ________.

Directions: Compare the following graphs to the Quadratic parent function graph. Choose all the letters that apply.

A) Reflected B) Narrower C) Wider

D) Translated Up E) Translated Down

1. 𝒚 = 𝟓𝒙𝟐 2. 𝒚 =𝟏

𝟑𝒙𝟐 + 𝟏𝟏

3. 𝒚 = −𝒙𝟐 − 𝟗 4. 𝒚 = −𝟕𝒙𝟐

6. For the equation y = ax2 + c, the graph intersects the y-axis above the origin if c is ____. A. Positive B. Negative C. Zero

7. Describe the appearance of the quadratic function 𝒚 = −𝟒𝒙𝟐 − 𝟕.

A. Opens upward, shifted down, wide graph B. Opens downward, shifted down, wide graph C. Opens downward, shifted up, narrow graph D. Opens downward, shifted down, narrow graph

8. How can the graph of y = x2 + 6 be obtained from the graph of y = x2 - 8? A. Move the graph of y = x2 – 8 up 6 B. Move the graph of y = x2 – 8 down 8 C. Move the graph of y = x2 – 8 down 14 D. Move the graph of y = x2 – 8 up 14

4

Graphing Quadratics from Standard Form

Quadratic Parent Function…y = ____

Must rewrite in standard form (y = ax2 + bx + c)

1. Find __________________ 2

bx

a

2. Then, find _______ (x, y) 3. Y-intercept is ( , ) 4. Factor to solve and find _____________. 5. Then graph…remember, if “a” is negative,

parabola opens _______!!

Ex: 2 6 8y x x

AOS = Up or Down?

Vertex – Max or Min?

Vertex =

(x, y)

Y-Intercept =

X-Intercepts =

5

Ex: 2 4 3y x x

AOS = Up or Down?

Vertex – Max or Min?

Vertex =

(x, y)

Y-Intercept =

X-Intercepts =

Ex: 2 4y x

AOS = Up or Down?

Vertex – Max or Min?

Vertex =

(x, y)

Y-Intercept =

X-Intercepts =

EX: Compare 𝑦 = −5𝑥2 + 8 to the quadratic parent function.

6

Graphing Quadratics from Vertex Form

Quadratic Parent Function…f(x) = ____

Vertex Form: 2( ) ( )f x a x h k

a =

h =

k =

-a = -f(x) =

-x = f(-x) =

Ex: 2( 2) 3y x

Vertex = Up or Down?

(x, y) Vertex – Max or Min?

AOS =

Y-Intercept =

(0, y)

X-Intercepts =

7

Ex: 2( ) ( 3) 4f x x

Vertex = Up or Down?

(x, y) Vertex – Max or Min?

AOS =

Y-Intercept =

(0, y)

X-Intercepts =

Ex: 2( ) 2( 4) 2f x x

Vertex = Up or Down?

(x, y) Vertex – Max or Min?

AOS =

Y-Intercept =

(0, y)

X-Intercepts =

8

Converting Quadratics from

Standard form to Vertex form and Vice-Versa

Standard form: Ax2 + Bx + C = 0

Ex: x2 – 2x + 6 = 0

Vertex form: 2( ) ( )f x a x h k

Ex: What if Vertex is (3, 4) and A=-2?

f(x)=

To convert to Vertex form…find the (x,y) or the (h,k)

*Find AOS from standard form 2

bx

a

*Sub this back into Standard form and find “y”

*Then sub the “x” in for “h”, the “y” in for “k”, and use the “a”

Convert these into Vertex Form:

Ex: x2 + 5x + 4 = 0 Ex: 2x2 - 12x + 10 = 0

9

Convert these into Standard Form:

Ex: y = (x + 2) 2 – 5 Ex: f(x) = -2(x – 4) 2 + 2

10

NOTES (day 1)

Simplifying Radicals

List the following perfect squares.

12 = 112 = 22 = 122 = 32 = 132 = 42 = 142 = 52 = 152 = 62 = 162 = 72 = 172 = 82 = 182 = 92 = 192 = 102 = 202 =

*Memorize these perfect squares.

Simplify the following.

√64 √32 2√300

Simplify the following by using perfect squares or by factoring into

primes.

√50

3√24

11

√20

10√28

−3√200

12

NOTES (day 2)- Multiplying and Dividing Radicals

A radical expression is in simplest form if the following conditions are

true:

No perfect square factors other than 1 are in the radicand.

No fractions are in the radicand.

No radicals appear in the denominator of a fraction.

Multiplying Radicals: √𝑎 ∙ √𝑏 = √𝑎𝑏

Examples

a. √6 ∙ √3 b. √10 ∙ √15

c. 2√3 ∙ √8 d. √8 ∙ √5

e. 3(4√20)

Rationalizing the Denominator:

RULE: CANNOT have a RADICAL in the denominator! **THEREFORE, you need to Rationalize the Denominator.

13

The process of eliminating a radical from an expression’s denominator

by multiplying the expression by an appropriate value of 1.

Example:

5

√7=

5

√7 ∙

√7

√7=

5√7

√49=

𝟓√𝟕

𝟕

1.

1

√3 =

2. 2

√5 =

Appropriate value of 1

14

Solve Quadratics using the

Square Root Method

Remember…Quadratic Parent Function…y = ___

Solutions to quadratic equations are called:

1. _____________

2. ________

3. ________

Since they have x2, they have ___ solutions!

Use Square Root method if there is no “___”!!

Ex: 2 9x

Ex: 26 600x (Isolate x2)

15

Ex: 21

123

x

Ex: 22 5 27x

Ex: 23 48 0x

Ex: 2 5 0x

16

Solve Quadratics by

Completing the Square

Let’s start by creating a perfect square trinomial!

Recall: If we factor 2 6 9x x Why is this special?

So, if we take one-half of “b” and square it, what do we get?

Find the value of c that makes each trinomial a perfect square.

1. 𝑥2 + 10x + c 2. 𝑥2 + 14x + c

3. 𝑥2 – 4x + c 4. 𝑥2 – 8x + c

To solve by Completing the Square, we must create

___________ ____________ trinomials.

17

To complete the square for any quadratic equation of the

form 𝑥2 + bx + c:

Step 1 Move “c” to the ________ side.

Step 2 Find one-half of “b” and __________ it.

Step 3 Add the result of Step 2 to _______ sides of the equation.

Step 4 ____________ the new perfect square trinomial.

Step 5 Take the _________ ___________ of both sides

Step 6 Solve for “x” – you will have ____answers. Solve each equation by completing the square. Round to the nearest

tenth if necessary.

1. 𝑥2 – 4x + 3 = 0 2. 𝑥2 + 10x = –9

18

3. 𝑥2 – 8x – 9 = 0 4. 𝑥2 – 6x = 16

5. 𝑥2 – 4x – 5 = 0 6. 𝑥2 – 12x = 9

19

Solve Quadratics using the

Quadratic Formula

Example: Solve 4𝑥2 + 7𝑥 = 15

4𝑥2 + 7𝑥 − 15 = 0 Write in standard form

a = b = c = Identify a, b and c

𝑥 = − ±√ 2− 4( )( )

2• Substitute into formula

𝑥 = − ±√

Simplify

Some equations are ____ factorable.

In this case, you can always use the

___________________.

Given a quadratic equation: 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0

Then 𝑥 = −𝑏±√𝑏2− 4𝑎𝑐

2𝑎

It’s EASY!

20

𝑥 = −7 ± √289

8

𝑥 = −7+

8 𝑥 =

−7−

8

First solution 𝑥 = −7+17

8=

8=

Second Solution 𝑥 = −7−17

8=

Examples: Find the roots (zeros, solutions, x-ints)

a. 𝑥2 − 8𝑥 + 16 = 0

21

b. 4𝑧2 = 7𝑧 + 2

c. 2𝑥2 − 𝑥 = 5 (Find the exact roots and zeros, then

estimate to the nearest hundredth.)

22

Review:

Solutions are also called:

_________________

_________________

_________________

4 Methods to Solve Quadratic Equations:

1. ___________________________

2. ___________________________

3. ___________________________

4. ___________________________