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Oct. 23
• A23 Due
• Friday - Test on Chapter 3
• A24: Chapter 3 Practice Test (Due Friday)
• Key is on bulletin board
• For last problem, see hint!
• Today - Start Chapter 5 on Quadratics
• A25: p. 245 #1, 5, 9, 11, 17, 29, 33, 37 (DUE TUESDAY)
Warm up – October 23 notes
Multiply and Simplify:
1) y = x(1- x) - (1- x2 )
Which of the above is a:
• Linear Function
• Quadratic Function
2)
y = (x + 3)(x -1)
5-1: Modeling Quadratic Functions
Define:
• Linear Function –
• Quadratic Function –
5-1: Modeling Quadratic Functions
Define:
• Linear Function – A function whose graph is a line
Slope-intercept form
y = mx + b
Standard form
Ax + By = C
• Quadratic Function – A function whose graph is a
parabola
Standard form
y = ax2 + bx + c
Quadratic Term
Linear Term
Constant Term
Key elements of a parabola
• Line (or axis) of Symmetry
• the line that divides a
parabola into two parts that
are mirror images
• Vertex of Parabola
• the point on the line of
symmetry.
• y-value represents the max
or min of the function.
Identify key elements of this parabola
• Axis of symmetry
P (0,6)
• Vertex
• Corresponding (mirror)
points to:
• P (0,6) => P’ (
Q(1,0)
• Q (1,0) => Q’ (
)
)
Finding Quadratic function given 3 points
Lines
• We need 2 points to find
the equation of a line
• (or 1 point and slope)
Quadratics
• We need 3 points to find
the equation of a parabola
y = mx + b
y = ax2 + bx + c
y = 2x - 4
y = -3x2 + 5x -13
To find the Quadratic Function, use the
2
y
=
ax
+ bx + c
Standard Form of Quadratics
Example:
Find the Quadratic, given the points (1, 0) , (2, -3) and (3, -10)
What do we know / not know?
y = ax + bx + c
2
• Steps:
• 1) Substitute the three points into standard form to get 3 equations
• 2) Solve the system of equations to get the values for a, b, and c
• 3) Write the standard form of the equation substituting in the values of
a, b, and c.
y = -2x + 3x -1
2
Oct 29
• Due: A25 (p. 245) and A26 (worksheet)
• Tests are graded, but 3 people still need to take it
• HW Review
• 5-2 Properties of Parabolas
• CW: Quadratic Functions Worksheet
• A27: p. 252 #1-21 EOO, & #18
5-2 Properties of Quadratics
Graphing function y = ax2+c
• What is the max or min
y value for?
y= -0.5x2 + 2
• How can we find other
points?
When no linear term, the vertex is (0,c)
• If y = ax2 + c,
• Then the vertex is (0, c) & the axis of symmetry is x = 0
• Check:
1. Sketch graph of y = 2x2 – 4
2. Sketch graph of y = 5 – 3x2
3. What are the coordinates of the vertex of the graph of a
function in the form y = ax2 ?
If y = ax2 + bx + c
• If a is positive, it opens __________
• If a is negative, it opens ___________
b
• The axis of symmetry is the line x = 2a
• The x-coordinate of the vertex is __________
• The y-coordinate of the vertex is found by____________
• The y-intercept is (
)
Steps to Graphing a quadratic
1. Find and graph the
2.
3.
4.
5.
axis of symmetry
Find and graph the
vertex
Find and graph the yintercept and its
reflection
Pick another value for
x, evaluate for y and
graph that point and its
reflection
Sketch the curve
y = -x2 + 4x + 2
y= - 4x2 + 8x - 5
Oct 31
• Due: A26 (worksheet) and A27
• Warm up
• HW Review
• 5-2 Properties of Parabolas / 5-3 Transforming Parabolas
• CW: Quadratic Functions Worksheet
• A28: p. 252 #26-32, 37-39, 55; p. 259 #1-37 EOO
Warm up Quiz – in your Notes
• 1) Find a Quadratic function to model the points below
(-2, -18)
(0, -4)
(2, -14)
• 2) Sketch the Graph of the following function. Label the
vertex and axis of symmetry.
y = -2x - 4x + 2
2
HW Questions?
5-2 Using Vertex to find Min/Max
• You can find the min or max of a quadratic function by
finding the vertex.
• A collage frame must be a rectangle with a perimeter of
25 cm. What dimensions give the maximum area?
5-3 Vertex form of Parabolas
• We have been studying
• Standard Form:
y = ax + bx + c
2
• Today you will be learning vertex form
y = a(x - h) + k
2
Complete empty columns
Standard Form
y = ax2 + bx + c
b
x=2a
Vertex form
y = a(x - h)2 + k
y = x - 4x + 4
y = (x - 2)2
y = x + 6x + 8
y = (x + 3)2 -1
y = -3x2 -12x - 8
y = -3(x + 2)2 + 4
2
2
a
Vertex is (h, k)
a is same as in Standard Form
h
Complete empty columns
Standard Form
y = ax2 + bx + c
b
x=2a
Vertex form
y = a(x - h)2 + k
y = x - 4x + 4
y = (x - 2)2
y = x + 6x + 8
y = (x + 3)2 -1
y = -3x2 -12x - 8
y = -3(x + 2)2 + 4
2
2
a
h
Attributes of
•a
•h
•k
y = a(x - h)2 + k
Using Vertex form to graph
1. Graph the vertex &
y = -3(x + 2)2 + 4
label the axis of
symmetry
2. Find another point
and its reflection.
Graph these points
3. Sketch the curve
Free Plain Graph Paper from http://incompetech.com/graphpaper/plain/
Graph the function
y = 2(x +1) - 4
2
Free Plain Graph Paper from http://incompetech.com/graphpaper/plain/
Writing the equation of a parabola
• Vertex (3, 4)
• Point (5, -4)
1. Write vertex form
2. Substitute the vertex
for h, k
3. Plug in 2nd point and
solve for a.
4. Write equation using
vertex form
You Try!
• Vertex (-1, 0)
• Point (-2, 2)
Converting to Vertex form
1. Find the x-
coordinate of the
vertex
2. Find the y-
coordinate by
plugging in x
3. a is given
4. Substitute vertex for
(h, k) and a from
initial equation
y = x2 - 4x + 4
You try!
y = -3x +12x + 5
2
Review
• Page 262, #94
Nov 4
• Due: A28: p. 252 #26-32, 37-39, 55; p. 259 #1-37 EOO
• Warm up
• 5-4 Factoring Quadratics
• Quiz
• A29: p. 267 #1-49 EOO
Warm up Quiz – in your Notes
• 1) Rewrite the equation in vertex form. Identify the vertex
and the axis of symmetry. Sketch the graph of the
function.
y = -3x - 6x - 8
2
Multiply using Generic Rectangles
(aka Area Models)
(x -1)(3x +12)
5-4: Factoring Quadratics
• If possible, always factor out GCF first!
• GCF: Greatest Common Factor
4x + 20x -12
2
9n - 24n
2
9x + 3x -18
2
4w + 2w
2
When a = 1
• When the coefficient of the x2 term is 1, use diamonds
When a is not 1: Rectangles / Diamonds
Try:
2x +11x +12
2
3x +12x -15
2
Quiz
• Individual. You can use your notes.
• After quiz, work on HW
• A29: p. 267 #1-49 EOO
Nov 6
• Due: A29: p. 267 #1-49 EOO
• Finish 5-4 Factoring Quadratics
• Start 5-5 Quadratic Equations
• Finish Quiz
• A30: p. 270 Odds, (Due Friday)
• A31: p. 274 #1-19 all (TBD – might change on Friday!)
Warm up – In your Notes
• Factor:
2x +10x - 28
2
4x -1
2
Other methods for factoring exist
• Your text has a method similar to box/diamond, (see
examples on pages 264-265)
• There is also the AC method, which works, but I’m not a
huge fan
• http://www.regentsprep.org/regents/math/algtrig/ATV1/Ltri3.htm
• And many others, e.g. Guess and Check
• What works best is…… what works best for you!
Two Special Expressions that are good to
know:
• Perfect Squares
a2 + 2ab + b2 = (a + b)2
a - 2ab + b = (a - b)
2
2
2
If you can find the square root
of the first and last terms,
AND the middle term is twice
the product of the roots
• Difference of two Squares
a - b = (a + b)(a - b)
2
2
If you can find the square root
of the first and last terms,
AND there is no middle term
AND the sign in between is -
Examples
4x + 20x + 25
2
4x - 20x + 25
2
16x - 25
2
16x + 25
2
Nov 8
• Due today: A30: p. 270 Odds
• Warm up
• Start 5-5 Quadratic Equations
• A31: p. 274 #1-19 odd, 37-53 EOO
Warm up – in your notes!
• Factor
16x + 40x + 25
2
36x - 24x + 4
2
49x -1
2
16x + 64
2
5-5 Quadratic Equations
• What values of x would make this equation true?
( x + 3)( x - 7) = 0
• What point(s) would that be?
• Where would that be on a graph?
Ways to solve quadratics
• Zero Product Property
• Square Roots
• Graphing Calculators (Graph/Table)
Zero-Product Property
Used when we have a quadratic equation that can be
factored.
if ab = 0, then a = 0 or b = 0
If two things are being multiplied together and the result is zero,
then one of those has to equal Zero.
Example
Steps
1. Solve for zero. (Write
equation on standard form)
2. Factor
3. Set each factor = 0 and
solve.
4. Check!
2x -11x = -15
2
You Try!
x + 7x = 18
2
2x + 4x = 6
2
16x = 8x
2
Solving by square roots
5x -180 = 0
2
• This is useful when we have no linear term (no bx term)
Steps
1. ax2 term on left, number right
2. Solve for x2 (divide by a)
3. Find square roots
You Try!
4x - 25 = 0
2
3x = 24
2
1
x - =0
4
2
Solving with a Graphing Calculator
• Enter the following function into Y=
y = x - 5x + 2
2
Classwork – in pairs
• Put both names on one page (OR turn in 2 pages, your
choice)
• Solve by Graphing/Table: p. 274 #23, 27, 31
• Solve any way: p. 275 #35
• A31: p. 274 #1-19 odd, 37-53 EOO
Nov 13
• Due today: A31: p. 274 #1-19 odd, 37-53 EOO
• Complex Numbers
• Quadratic Equation Activity
• A32: p. 282 #3-43 EOO, 49, 52a, 55
•
p. 285 #1-9 odd
• A35: Ch 5 Review (DUE NEXT THURS 11/21)
• CHAPTER 5 TEST NEXT THURS `11/21
Nov 13
• Due today: A31: p. 274 #1-19 odd, 37-53 EOO
• Complex Numbers
• Quadratic Equation Activity
• A32: p. 282 #3-43 EOO, 49, 52a, 55
•
p. 285 #1-9 odd
• A35: Ch 5 Review (DUE NEXT THURS 11/21)
• CHAPTER 5 TEST NEXT THURS `11/21
Nov 15
• Due today: A32: p. 282 #3-43 EOO, 49, 52a, 55
•
p. 285 #1-9 odd
• Complex Numbers (Division and solving)
• Completing the square
• A33: p. 289 #1-33 EOO (do #25, 29 in class)
• A35: Ch 5 Review (DUE NEXT THURS 11/21)
• CHAPTER 5 TEST NEXT THURS 11/21
Warm up:
• Sketch the graph of this function
• Label key information
Quadratics…what have you learned?
• 5-1: vertex and axis of symmetry; how to find standard
•
•
•
•
•
•
•
•
form
5-2: standard form and how to find vertex, a.o.s; which
way will the parabola open, min/max, y-intercept?
5-3: vertex form; how to find vertex
5-4: factoring; what does that mean in the graph?
<simplifying square roots>
5-5: solving by factoring; solving by square roots
(table/graph too)
5-6: complex numbers
5-7: solving by completing the square; vertex form
5-8: solving using quadratic formula; discriminant
Complex Numbers
• Division
Completing the square
• OBJECTIVE: You’ll use completing the square to solve
quadratic equations.
• The process of completing the square is a really useful
method that can help solve quadratic equations.
• The best way to show how useful completing the square
can be is with an example…
EXAMPLE: Solve x2 – 10x + 21 = 0 by
completing the square.
• Step 1: Move the constant (c) to the right by subtracting
from both sides
• Step 2: Figure out what you need to complete the square
on the left side (to make a perfect square)
• Step 3: ADD THIS NUMBER TO BOTH SIDES to
preserve the equality
• Step 4: Factor the left and simplify the right
• Step 5: Now solve by taking the square root of both sides
You Try!
x2 – 6x + 5 = 0
x2 + 4x – 5 = 0
What if there is an (a) term?
• Step 0: Divide
everything by a
• Step 1: Move the
constant (c) to the right
by subtracting from both
sides
• Step 2: Take half of b
and square it
• Step 3: ADD TO BOTH
SIDES
• Step 4: Factor the left
and simplify the right
• Step 5: Solve
• Example:
4x2 – 9x + 2 = 0
You Try!
3a2 + 12a – 15 = 0
y = x2 + 6x + 2
Vertex form using CTS
• Before you take
square root to solve,
just move number
back!
Nov 19
Due today:
A33: p. 289 #1-33 EOO (do #25, 29 in class)
• Quadratic Formula / Discriminant
• A34: p. 297 #1-49 EOO, 57-59
• A35: Ch 5 Review (due Monday after Thanksgiving)
Warm up
• Rewrite the following equation in Vertex Form.
• Then sketch the graph.
y = 2x - 8x +1
2
Deriving the Quadratic Formula
-b ± b2 - 4ac
x=
2a
ax2 + bx + c = 0
The quadratic formula gives roots to our
equations
• Use quadratic formula to solve
3x2 - 5x = 2
We can get complex solutions
2x = -6x - 7
2
You Try~!
3x - x = 4
2
-2x2 = 4x + 3
How can we tell the quadratic has real
roots or not?
-b ± b2 - 4ac
x=
2a
The Discriminant
• If
b - 4ac > 0
2
b - 4ac = 0
2
b - 4ac < 0
2
Then:
Determine the type and # of solutions
x2 + 6x + 8 = 0
x + 6x + 9 = 0
2
x2 + 6x +10 = 0
Determining how to solve quadratics
If the
Discriminant
is….
a Positive
Square
Number
Factor
Table/
Graph
Quadratic
Formula
Complete the
Square
✔
✔
✔
✔
(approximate)
✔
✔
✔
✔
✔
✔
✔
a Positive
non-square
number
Zero
a Negative
✔
• https://www.youtube.com/watch?v=z6hCu0EPs-o
• https://www.youtube.com/watch?v=2lbABbfU6Zc
Nov 21
Objective:
Today you will
assess your
readiness for
Chapter 5 test!
Method: You will
take a partner
quiz – Great
chance to
LEARN, debate,
figure out what
you need to
practice!
Due today:
• A34: p. 297 #1-49 EOO, 5759
• PARTNER QUIZ
• A35: Ch 5 Review (due
Monday after Thanksgiving)
• Ch 5 TEST on Wednesday
after Break!