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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
HW: WS
Warm Up – EOC
Graph Quadratics – Standard Form
HW: WS 19 20 21 22 23
Student
Holiday
Warm Up – EOC
Graph
Quadratics – Day 2
HW: WS
Quiz
Graph
Quadratics – Vertex Form
HW: WS
Warm Up – EOC
Vertex Form
– Day 2
HW: WS
Warm Up – EOC
Vertex Form
– Day 3
HW: WS 26 27 28 March 1 2
Quiz
Simplify
Radicals
HW: WS
Warm Up – EOC
Simplify
Radicals - Day 2
HW: WS
Warm Up – EOC
Square Root
Method
HW: WS
Quiz
Completing
the Square
HW: WS
Warm Up – EOC Completing
the Square - Day 2
HW: WS
5 6 7 8 9
Quiz
Quadratic
Formula
HW: WS
Warm Up – EOC
Quadratic
Formula - Day 2
HW: WS
Warm Up – EOC
Quadratic
Formula - Day 3
HW: WS
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
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
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)
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
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.)