Math 2 Warm Up
Simplify each expression:
1.
2.
3.
4.
5.
6.
7.
8.
9.
2x2 – 4x(3x – 5)
3x(x – 2)
(x – 2)(x + 5)
(-4x + 3)(2x – 7)
3x(2x – 7) + 6x(4x + 5)
x(1 – x) – (1 – 2x2)
(5x + 3) 2
-7x(5x2 – 4x)
(4x – 5) (-2x2 + 3x – 9)
Unit 5: “Quadratic Functions”
Lesson 1 - Properties of Quadratics
Objective: To find the vertex & axis of symmetry of a
quadratic function then graph the function.
quadratic function – is a function that can be written in
the standard form:
y = ax2 + bx + c, where a ≠ 0.
Examples:
y = 5x2
y = -2x2 + 3x
y = x2 – x – 3
Properties of Quadratics
parabola – the graph of a quadratic equation. It is
in the form of a “U” which opens either
upward or downward.
vertex – the maximum or minimum point of a parabola.
Properties of Quadratics
axis of symmetry – the
line passing through the
vertex about which the
parabola is symmetric (the
same on both sides).
Properties of Quadratics
Find the coordinates of the vertex, the equation for the
axis of symmetry of each parabola. Find the
coordinates points corresponding to P and Q.
Graphing a Quadratic Equation
y = ax2 + bx + c
1) Direction of the parabola?
If a is positive,
then the graph opens up.
If a is negative,
then the graph opens down.
Graphing a Quadratic Equation
y = ax2 + bx + c
2) Find the vertex and axis of symmetry.
−𝐛
𝟐𝐚
The x-coordinate of the vertex is 𝐱 =
(also the equation for the axis of symmetry).
Substitute the value of x into the quadratic
equation and solve for the y-coordinate.
Write vertex as an ordered pair (x , y).
Graphing a Quadratic Equation
y = ax2 + bx + c
3) Table of Values.
Choose two values for x that are one side of
the vertex (either right or left). Substitute
those values into the quadratic equation to find
y values. Graph the two points.
Graph the reflection of the two points on the
other side of the parabola (same y-values and
same distance away from the axis of
symmetry).
Find the vertex and axis of symmetry of the following
quadratic equation. Then, make a table of values and
graph the parabola.
y = 2x2 + 4x + 3
Direction: _____
Vertex: ______
Axis: _______
Find the vertex and axis of symmetry of the following
quadratic equation. Then, make a table of values and
graph the parabola.
y = – x2 + 3x – 1
Direction: _____
Vertex: ______
Axis: _______
Find the vertex and axis of symmetry of the following
quadratic equation. Then, make a table of values and
graph the parabola.
y=–
𝟏 2
x
𝟐
+ 2x + 5
Direction: _____
Vertex: ______
Axis: _______
Find the vertex and axis of symmetry of the following
quadratic equation. Then, make a table of values and
graph the parabola.
y = 3x2 – 4
Direction: _____
Vertex: ______
Axis: _______
Apply!
The number of widgets the Woodget Company sells can
be modeled by the equation -5p2 + 10p + 100, where p
is the selling price of a widget. What price for a widget
will maximize the company’s revenue? What is the
maximum revenue?
End of Day 1
Math 2
Unit 5
Lesson 2
Unit 5:"Quadratic Functions"
Title: Translating Quadratic Functions
Objective: To use the vertex form of a
quadratic function.
y = a(x – h)2 + k
where (h, k) is the vertex.
Example 1: Graphing from Vertex Form
y = 2(x – 1) 2 + 2
Direction: _____
Vertex: ______
Axis: _______
Example 2: Graphing from Vertex Form
y = (x + 3) 2 – 1
Direction: _____
Vertex: ______
Axis: _______
Example 3: Graphing from Vertex Form
y=
−1
2
(x – 3) 2 – 2
Direction: _____
Vertex: ______
Axis: _______
Example 4: Write quadratic equation in vertex form.
Example 5: Write quadratic equation in vertex form.
Example 6: Converting Standard Form to Vertex Form.
Step 1: Find the Vertex
x = -b =
y = x2 - 4x + 6
2a
y
=
Step 2: Substitute into Vertex Form:
Example 7: Converting Standard Form to Vertex Form.
Step 1: Find the Vertex
x = -b =
y = 6x2 – 10
2a
y
=
Step 2: Substitute into Vertex Form:
Example 8: Converting Vertex Form to Standard Form.
Step 1: Square the Binomial.
y = 2(x – 1) 2 + 2
Step 2: Simplify to
Example 9: Converting Vertex Form to Standard Form.
Step 1: Square the Binomial.
y=
−1
2
(x – 3) 2 – 2
Step 2: Simplify to
Honors Math 2 Assignment:
In the Algebra 2 textbook:
pp. 251-253 #3, 6, 9, 17-20, 25, 27, 31, 34, 52, 54
End of Day 2
Factoring Quadratic Expressions
Objective: To find common factors and binomial factors
of quadratic expressions.
factor – if two or more polynomials are multiplied together,
then each polynomial is a factor of the product.
(2x + 7)(3x – 5) = 6x2 + 11x – 35
FACTORS
PRODUCT
(2x – 5)(3x + 7) = 6x2 – x – 35
FACTORS
PRODUCT
“factoring a polynomial” – reverses the multiplication!
Finding Greatest Common Factor
greatest common factor (GCF) – the greatest of the
common factors of two or more monomials.
𝟐
𝟕𝒙 + 𝟐𝟏
𝟒𝒙𝟐 + 20x − 12
𝟗𝒙𝟐 − 24x
Finding Binomial Factors
𝒙𝟐 + 14x + 40
Finding Binomial Factors
𝒙𝟐 + 12x + 32
Finding Binomial Factors
𝒙𝟐 − 11x + 24
Finding Binomial Factors
𝟐
𝒙 − 17x + 72
Finding Binomial Factors
𝒙𝟐 − 14x − 32
Finding Binomial Factors
𝟐
𝒙 + 3x − 28
Finding Binomial Factors
𝟐𝒙𝟐 + 11x + 12
Finding Binomial Factors
𝟔𝒙𝟐 − 31x + 35
Finding Binomial Factors
𝟏𝟐𝒙𝟐 + 32x − 35
Finding Binomial Factors
𝟐
𝟑𝒙 − 16x − 12
Finding Binomial Factors*
𝟏𝟎𝒙𝟐 + 35x − 45
Finding Binomial Factors*
𝟗𝒙𝟐 + 42x + 𝟒𝟗
Finding Binomial Factors*
𝟐𝟓𝒙𝟐 − 90x + 𝟖𝟏
Factoring Special Expressions*
𝟒𝒙𝟐 − 49
𝟐𝟓𝒙𝟐 − 9
𝟑𝒙𝟐 − 192
𝟗𝒙𝟐 − 36
Honors Math 2 Assignment
In the Algebra 2 textbook,
pp. 259-260
#1, 5, 6, 7-45 odd, 48, 54
End of Day 3
Factor.
𝟏𝟎𝒙𝟐 + 35x − 45
𝟗𝒙𝟐 − 36
𝟑𝒙𝟐 − 16x − 12
Solving Quadratics Equations:
Factoring and Square Roots
Objective: To solve quadratic equations by
factoring and by finding the square root.
Solve by Factoring
𝒙𝟐 + 7x − 18 = 0
Solve by Factoring
𝟑𝒙𝟐 − 20x − 7 = 0
Solve by Factoring
𝟖𝒙𝟐 − 5 = 6x
Solve by Factoring
𝟔𝒙𝟐 = 𝟒𝟏𝐱 − 𝟔𝟑
Solve by Factoring*
𝟒𝒙𝟐 + 16x = 10x +𝟒𝟎
Solve by Factoring*
𝟏𝟔𝒙𝟐 − 𝟖𝒙 = 𝟎
Solve Using Square Roots
Quadratic equations in the form 𝒂𝒙𝟐 = 𝒄 can
be solved by finding square roots.
𝟑𝒙𝟐 = 243
Solve Using Square Roots
𝟓𝒙𝟐 − 200 = 0
Solve Using Square Roots*
𝟒𝒙𝟐 − 25 = 0
Honors Math 2 Assignment
In the Algebra 2 textbook,
p. 266
#1-19
End of Day 4
Complex Numbers
Math 2 Warm Up
In the Algebra 2 Practice Workbook,
Practice 5-5 (p. 64)
#1, 10, 13, 19, 25, 31,
40, 46, 55, 61, 71, 73
Unit 4, Lesson 5:
Complex Numbers
Objective: To define imaginary and complex
numbers and to perform operations on complex
numbers
Introducing Imaginary Numbers
Find the solutions to the following equation:
Introducing Imaginary Numbers
Now find the solutions to this equation:
Imaginary numbers offer solutions to this problem!
i1 = i
i2 = -1
i3 = -i
i4 = 1
Simplifying Complex Numbers
21
i
Adding/Subtracting Complex Numbers
(8 + 3i) – (2 + 4i)
7 – (3 + 2i)
(4 - 6i) + (4 + 3i)
Multiplying Complex Numbers
(12i)(7i)
(6 - 5i)(4 - 3i)
(3 - 7i)(2 - 4i)
(4 - 9i)(4 + 3i)
So, now we can finally find ALL solutions to this equation!
Complex Solutions
3x² + 48 = 0
8x² + 2 = 0
-5x² - 150 = 0
9x² + 54 = 0
Math 2 Assignment
In the Algebra 2 Textbook,
Pgs. 274-275
#s 1-17 odd, 29-39 odd, 41-46
End of Day 5
Completing the Square
1.) Move the constant to opposite side of the
equation as the terms with variables in them.
2.) Take half of the coefficient with the x-term
and square it
3.) Add the number found in step 2 to both sides
of the equation.
4.) Factor side with variables into a perfect
square.
5.) Square root both sides (put + in front of
square root on side with only constant)
6.) Solve for x.
Solve the following,
using completing the square
1.) x2 – 3x – 28 = 0
2.) x2 – 3x = 4
3.) x2 + 6x + 9 = 0
If a ≠ 1, then divide all the term by “a”.
1.) 2x2 + 6x = -6
2.) 3x2 – 12x + 7 = 0
3.) 5x2 + 20x + -50
Math 2 Assignment
In the Algebra 2 Textbook,
Pgs 281-283
# 15 – 25, 37, 39, 51-53
End of Day 6
Solve using Completing the square
x2 + 4x = 21
x2 – 8x – 33 = 0
4x2 + 4x = 3
Solving Quadratic Equations:
Quadratic Formula
Objective: To solve quadratic equations using the
Quadratic Formula.
𝟐𝒙𝟐 + 5x − 𝟖 = 0
Not every quadratic equation can be solved by
factoring or by taking the square root!
Solve using Quadratic Formula
𝟐𝒙𝟐 + 5x − 8 = 0
Solve using Quadratic Formula
𝟑𝒙𝟐 + 23x + 40 = 0
Solve using Quadratic Formula
𝟗𝒙𝟐 + 𝟔𝐱 − 𝟏 = 𝟎
Solve using Quadratic Formula*
𝟒𝒙𝟐 − 𝟖𝒙 = − 𝟏𝟎
Solve using Quadratic Formula
𝟐𝟓𝒙𝟐 − 𝟑𝟎𝒙 + 𝟏𝟐 = 𝟎
Solve using Quadratic Formula
𝟑𝒙𝟐 − 𝟐𝒙 + 𝟒 = 𝟎
Solve using Quadratic Formula
𝟐𝒙𝟐 = -6x – 7
Honors Math 2 Assignment
In the Algebra 2 textbook,
pp. 289-290
#1, 2, 22-30
Solve
𝒙𝟐 + 𝟒𝒙 = 41
{-8.71, 4.71}
𝟐𝒙𝟐 = -6x – 7
No Solution
End of Day 7
Solving Quadratic Equations:
Graphing
Objective: To solve quadratic equations and systems
that contain a quadratic equation by graphing.
When the graph of a function intersects the x-axis,
the y-value of the function is 0.
Therefore, the solutions of the quadratic equation
ax2 + bx + c = 0 are the x-intercepts of the graph.
Also known as the “zeros of the function” or the
“roots of the function”.
Solve Quadratic Equations by Graphing
Solution
Solution
Solve Quadratic Equations by Graphing
Step 1: Quadratic equation must equal 0!
ax2 + bx + c = 0
Step 2: Press [Y=]. Enter the quadratic equation in
Y1. Enter 0 in Y2. Press [Graph].
MAKE SURE BOTH X-INTERCEPTS ARE ON SCREEN!
ZOOM IF NEEDED!
Step 3: Find the intersection of ax2 + bx + c and 0.
Press [2nd] [Trace]. Select [5: Intersection].
Press [Enter] 2 times for 1st and 2nd curve.
Move cursor to one of the x-intercepts then press
[Enter] for the 3rd time.
Repeat Step 3 for the second x-intercept!
Solve by Graphing
𝒙𝟐 + 6x + 4 = 0
Solve by Graphing
𝟐𝒙𝟐 + 4x – 7 = 0
Solve by Graphing
𝟑𝒙𝟐 + 5x = 20
Solve by Graphing
𝟓𝒙𝟐 + 𝟕 = 19x
Solve by Graphing
𝒙𝟐 = -2x + 7
Solve by Graphing
−𝟑𝒙𝟐 + 2x – 6 = 0
Solve by Graphing
𝒙𝟐 + 𝟖𝒙 + 16 = 0
End of Day 8
Solving Systems of Equations
Solve a System with a Quadratic Equation
𝒚 = 𝒙𝟐 + x − 𝟐
𝒚 = −𝒙 + 𝟑
Solve a System with a Quadratic Equation
𝒚 = 𝟐𝒙𝟐 + x
𝟒
𝒚= 𝒙+𝟒
𝟑
Solve a System with a Quadratic Equation
𝒚 = 𝒙𝟐 + 𝟒𝒙 + 𝟕
𝒚 = −𝟐𝒙
Solve a System with a Quadratic Equation
𝒚 = 𝒙𝟐 −𝟔x + 𝟏𝟎
𝒚=𝟏
Solve a System with Quadratic Equations
𝒚 = 𝒙𝟐 − 𝟔𝒙 + 𝟓
𝒚 = −𝟐𝒙𝟐 + 𝟓𝒙
Solve a System with Quadratic Equations
𝒚 = 𝒙𝟐 + 𝟕𝒙
𝟏 𝟐
𝐲 = 𝒙 − 𝟓𝒙 − 𝟗
𝟒
𝒚 = 𝒙𝟐 − 𝟔𝒙 + 𝟓
𝒚 = −𝟐𝒙𝟐 + 𝟓𝒙
Honors Math 2 Assignment
In the Algebra 2 textbook,
pp. 266-267
#20-31, 35, 54-56
Solve each quadratic equation or
system by graphing.
Modeling Data with Quadratic Equations
Objective: To model a set of data with a quadratic
function.
Graph:
(-3, 7), (-2, 2), (0, -2)
(3, 7), (1, -1), (2, 2)
Graph:
(-1, -8), (2, 1), (3, 8)
End of Day 9
Finding a Quadratic Model
1) Turn on plot:
Press [2nd] [Y=], [ENTER], Highlight “On”,
Press [ENTER]
2) Turn on diagnostic:
Press [2nd] [0] (for catalog),
Scroll down to find DiagonsticOn.
Press [ENTER] to select.
Press [ENTER] again to activate.
Finding a Quadratic Model
3) Enter data values:
Press [STAT], [ENTER] (for EDIT),
Enter x-values (independent) in L1
Enter y-values (dependent) in L2
Clear Lists (if needed):
Press [STAT], [ENTER] (for EDIT),
Highlight L1 or L2 (at top)
Press [CLEAR], [ENTER].
Finding a Quadratic Model
4) Graph scatter plot:
Press [ZOOM], 9 (zoomstat)
5) Find quadratic equation to fit data:
Press [STAT], over to CALC,
For Quadratic Model - Press 5: QuadReg
Press [ENTER] 4 times, then Calculate.
Write quadratic equation using the values of a, b,
and c rounded to the nearest thousandths if needed.
Write down the R2 value!
Find a quadratic equation to model the values
in the table.
X
-1
2
3
Y
-8
1
8
𝑹𝟐
• is a measure of the “goodness-of-fit” of a
regression model.
• the value of R2 is between 0 and 1 (0 ≤ R2 ≤ 1)
• R2 = 1 means all the data points “fit” the model
(lie exactly on the graph with no scatter) –
“knowing x lets you predict y perfectly!”
• R2 = 0 means none of the data points “fit” the
model – “knowing x does not help predict y!”
• An R2 value closer to 1 means the better the
regression model “fits” the data.
Find a quadratic equation to model the values
in the table.
X
2
3
4
Y
3
13
29
Find a quadratic equation to model the values
in the table.
X
-5
0
2
Y
-18
-4
-14
Find a quadratic equation to model the values
in the table.
X
-2
1
5
7
Y
27
10
-10
12
Apply!
The table shows data about
Wavelength Wave Speed
the wavelength (in meters)
(m)
(m/s)
and the wave speed (in meters
per second) of the deep water
3
6
ocean waves. Model the data
5
16
with a quadratic function then
use the model to estimate:
7
31
a) the wave speed of a deep
8
40
water wave that has a
wavelength of 6 meters.
b) the wavelength of a deep
water wave with a speed
of 50 meters per second.
Apply!
The table at the right shows the
height of a column of water as it
drains from its container. Model the
data with a quadratic function then
use the model to estimate:
a)
b)
c)
d)
the water level at 35 seconds.
the waver level at 80 seconds.
the water level at 3 minutes.
the elapsed time for the water
level to reach 20 mm.
Honors Math 2 Assignment
In the Algebra 2 textbook,
pp. 237-238
#16-22, 30, 31, 38
Write down the R² value for each
equation!
End of Day 10
Unit 5 Test Review: “Quadratics”
 Quadratic Function
 Standard form: 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
 Vertex form: 𝒚 = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌
 Change Forms!
 Direction - parabola opens up or down?
 Vertex (𝒙 =
−𝒃
𝟐𝒂
, substitute x to find y) or (h, k )
 Vertex – is a Maximum or Minimum?
 Axis of Symmetry 𝒙 =
−𝒃
𝟐𝒂
or
x=h
 y-intercept (0, c) or (0, substitute 0 to find y)
 Graph (at least 5 points – vertex and 2 points on
each side of axis of symmetry)
Unit 5 Test Review: “Quadratics”
 Solve Quadratic Equations by:
 Factoring – Zero Product Property
 Square Root – Don’t forget ±
 Quadratic Formula 𝒙 =
−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄
𝟐𝒂
 Use Discriminant for Number & Types of Solutions
 Graphing – Find Intersection on Calculator
 Solve System with Quadratic by Graphing
 Quadratic Model for a Set of Data
 Quadratic Regression Model: 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
 Find R² value and what it means
 Predict Values (x or y) using Quadratic Model
Math 2 Assignment
In the Algebra 2 textbook,
pp. 293-295
#2-11, 12ce, 13-38, 70-72
Math 2 homework
In the Algebra 2 textbook,
p. 296
#6, 14, 24, 25, 27
28, 30, 33, 38, 39