Download Thursday HW Notes - Fordson High School

Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
6-1 Homework Notes
6-1 Quadratic Expressions, Rectangles, and Squares (p. 346)
________________________________ is a quadratic expression in standard form (notice no =, so can’t be an
equation)
________________________________ is a quadratic equation
________________________________ is a quadratic function (f:x-> is the same as f(x))
The product of any two linear expressions _________ and _________ is a _______________________
expression. (Remember FOIL?)
Example:
Simplify (2 x  3)(4 x  5)
When a linear expression is multiplied by itself, the result is a __________________________. Simplifying a
binomial squared as a quadratic expression is called _____________________________________________.
Example:
Expand ( x  7)
2
Complete p. 349 #5,13,15
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
6-1 In Class Notes
Real life problems can be solved using quadratic equations as well.
Example:
Suppose a rectangular swimming pool 50 m by 20 m is to be built with a walkway around it. If the walkway is
w meters wide, write the total area of the pool and walkway in standard form.
FOIL is not the only way to multiply binomials. You can also think of them as pieces of a side of a ___________
Example:
Use the Square technique to multiply the following:
(2 x  3)(4 x  5)
( x  y)2
Binomial Square theorem (the shortcut method for ( x  y ) )
2
Example:
Expand
k
(3m  )2
4
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
6-2 Homework Notes
Geometrically, the _______________________ of a number n, written ______, is the __________________
from n to ____ on the number line.
Algebraically, the ________________________ of a number can be defined piecewise as
Example 1: Solve for x: x  2  5.3
Complete p. 355 #14 – 18
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
6-2 In-Class Notes
f ( x)  x
x
is called the __________________________________ function. Graph this below.
x
You can use absolute value symbols when evaluating all solutions to ________________________.
For all real numbers x,
Example: Solve the following
x 2  81
x 2  40
Example: A square and a circle have the same area. The square has side 10. What is the radius of the circle?
Review: What is the difference between a rational number and an irrational number?
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
6-3 Homework Notes
6-3: Graph Translation Theorem
In a relation described by a sentence in x and y, replacing the ____ with ____________ translates the graph
_______________________ by _____ units. Replacing the ____ with ____________ translates the graph
_______________________ by _____ units. For the quadratic y  ax ,
2
Translates the parabola ______________________ by _____ units and _________________________ by
_____ units.
We often use ___________ to represent this translation.
Complete p. 361-632 #1,5,6,10-13
6-3 In-Class Notes
Remember from yesterday’ calculator activity, a large a value makes the graph _________________, a small a
value makes the graph _____________________, a negative a makes the graph ______________________
and a positive a makes the graph _______________________________.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
Example 1: Sketch the graph of y  7  3( x  6) by first graphing y  3 x
2
Since the vertex of the graph of
y  ax 2
2
is at _______ , then the vertex of the graph _______________________ is at
__________. Because of this, ________________________ is called the _______________________ of an
equation for a parabola. The line with the equation ___________ is called the ________________________
__________________________.
If the graph opens ________, then the vertex is the ________________________ y-value of the graph.
If the graph opens ________, then the vertex is the ________________________ y-value of the graph.
1
2
Example: Sketch the graph of y   ( x  3) and give the
equation for the axis of symmetry.
2
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
Be aware that _________ and _________ can replace the x and y in any function to translate it.
Example: Sketch the graph of y  4  x  3
6-5 Homework Notes
Finish the attached worksheets that go along with the TI-Nspire file “Completing the Square”. Use the space
below to complete any problems that you don’t have room to complete on the worksheet.
Complete p. 374 #2-5
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
6-5 In-Class Notes
You have now seen two forms for an equation of a parabola:
Standard Form
Vertex Form
Because they are both useful in its own way, we need to know how to convert from one to the other.
Converting Vertex Form to Standard Form will be learned in 6-4.
Converting from Standard Form to Vertex form is a bit trickier. It requires you to ______________________
___________________. In last night’s notes, you learned how to do this in order to solve for the ________
_______________________________. Today you will learn how to use this to get from standard form to
________________________ (when the equation = y).
In order to complete the square, you need to be able create a ______________________________________.
This can be done in one of two ways:
Example: Use the reverse box method to make a perfect square trinomial for x  10 x
2
Example: Use the reverse binomial square theorem to make a perfect square trinomial for
x 2  10 x .
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
Example: Now we will convert the equation
y  x 2  10 x  8 into vertex form.
Step 1:_______________________________________________________________________________
Step 2:_______________________________________________________________________________
Step 3:_______________________________________________________________________________
Step 4:_______________________________________________________________________________
Step 5:_______________________________________________________________________________
Example: Convert the equation
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
y  3x 2  8x  9 into vertex form.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
6-4 Homework Notes
6-4: Graphing
y  ax 2  bx  c
Example: Convert y  7  3( x  6) to standard form. Be sure to follow Order of Operations!
2
Graph the two equations to make sure that you get the same graph. Do this on both your TI-Nspire and your TI-84 so
that you practice graphing on both calculators.
Complete p. 367 #1-3
6-4 In-Class Notes
6-4: Graphing y  ax  bx  c
2
Example: Show that y  2( x  3)  8 and
2
y  2 x 2  12 x  10 are equivalent.
Do this first
algebraically then by graphing. Make notes if
needed so that you know how to do this both on the
Nspire as well as the TI-84.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
Theorem: The graph of the equation ___________________ is a parabola _________________ to the graph of ________
The graph of every quadratic function is a ______________________ with a y-intercept at ____.
The __________________ of all quadratic functions has a domain of _______________________________. However,
the _______________ of a quadratic function is different for every graph and depends on two things
1. Does the graph have a maximum (when ________) or does the graph have a minimum (when _________).
2. Where is that max or min (the _____________________)
The __________________ is easy to find if your quadratic is in ____________________________. In standard form, you
must use the process below to find the vertex.
a. Use the equation ___________ to find the axis of symmetry, which is also the _______________ of the
vertex.
b. Plug x into the quadratic equation to find the ______________ of the vertex.
Example: Suppose the following quadratic equation represents the vertical height h of a thrown baseball after t seconds.
h  16t 2  44t  5
a. Find the vertex of this quadratic.
b. Doe this function have a maximum or a minimum? Explain how you know.
c. Find h when t = 0, 1, 2, and 3
d. Explain what each pair (t, h) tells you about the height of the ball.
Name:
Mrs. Gorsline
Integrated Math 2
Hour:
Unit 2 Notes: Chapter 6
e. Graph this function. Find the domain and range of this function (be careful because this is a real-life problem)
In the previous problem, how high does the ball get? Approximately when does it hit the ground?
This example is a special case of a general formula for the ____________ h of an object at time t with an initial
_________________________________
v0 and initial ____________ h0
that was discovered by ______________.
That formula is
Where g is a constant measuring the _____________________________________. g is about _________ or _________.
Be careful that you understand that this does not describe the _____________ of the ball, it only describes how the
____________ of the ball changes over _____________. If you threw a ball straight up in the air, the graph would still
look like a _______________________.
Complete p. 367 #2,3,5-14 on a separate sheet.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
6-7 Homework Notes
6-7: The quadratic formula
Any quadratic equation equaling zero can be solved using _________________________________. However, there are
times when the arithmetic for this is quite complicated. When this happens, you can use the
_______________________________.
Quadratic Formula Theorem
If
ax 2  bx  c  0 and a  0 , then
We will be completing the proof for this in class tomorrow.
Example:
Solve
3x 2  11x  4  0 using the quadratic formula.
Complete p. 385 #6 – 10
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
6-7 In-Class Notes
FYI: You will have a 5-point quiz on Monday that deals with the quadratic formula with no calculator.
Proof of the Quadratic Formula:
Given: the equation
Steps
ax 2  bx  c  0 , where a  0
Justifications
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
Example: Pop Fligh, the famous baseball player, hit a pitch that followed the equation
h( x)  .005 x 2  2 x  3.5 .
Find out when the ball was exactly 8 feet high.
Remember that the quadratic formula can only be applied when an equation is in the ________________________ of a
quadratic equation that is equal to _____.
Example: The 3-4-5 right triangle has sides with consecutive integers. Are there any others with the same property?
Complete p. 385-386, #4,5,17-20 on a separate sheet.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
6-8 & 6-9 Homework Notes
6-8 and 6-9: Imaginary and Complex Numbers
Example: Solve the equation t  400
2
What is the problem with this?
Definition:
When k  0 , the two solutions to x  k are denoted _______________ and ___________.
2
Numbers that have negatives under a square root are called ______________________.
Definition: __________
Example: Solve
t 2  400 using the letter i.
Simplify the following:
a) (2i)(5i)
b)
9  25
Complete p. 391 #3,5 and p. 397 #2,4,6
c)
27  3
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
6-8 & 6-9 In-Class Notes
6-8 and 6-9: Imaginary and Complex Numbers
____________ is called the _____________________ number. When you take ___ to different powers, a pattern
emerges:
i
i2 
i3 
i4 
i5 
i6 
i100 
i50 
When simplifying imaginary expressions, you need to be careful.
Example: Simplify the following
a)
16  25
b)
9
81
When multiplying square roots, you can combine and multiply or divide the numbers under the square root together
unless both numbers are ______________________. To be safe, always pull the ____ out before taking the square root.
When you take the sum of an imaginary number with a real number, you get a _____________________ number.
Definition: A _______________________________ is a number of the form ______________, where a and b are real
numbers and ________________; a is called the ___________ part and b is called the _______________________ part.
Example: Name the real and imaginary parts of
3  4i
All properties that hold true for real numbers (except inequalities) hold true for complex numbers as well.
Name:
Hour:
Examples:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
Add and Simplify
Simplify
(3  4i)  (7  8i)
2i(8  5i)
Multiply and Simplify
(8  3i)(4  5i)
Any number that is a _________________ is not allowed to be in the ______________________ of a fraction. This
includes imaginary numbers.
Example:
Write
3  4i
2  5i
in
a  bi form.
Homework (On a separate sheet)
p. 391 #7,11-18 and p. 397 #7-10, 16-18
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
Factoring Homework Notes
There are many types of factoring problems. These are tips that I use when I factor.
Problems:
1. 6a4b2 + 9a3b
This problem is a binomial so look at the two terms and see if there is a common factor in the two coefficients.
If so, write that factor down. Now look at the variables and see if any of them repeat. If so, take the smallest
exponent of that variable out. In this problem we would take out ____________. Now we divide this factor
into both factors and write the result in parentheses or we can look at it as what times the common factor to get
the original problem. Our answer is _________________. Always check your final answer to be sure it can’t
be factored any more.
2. Problem 1 process can be applied to x(a + 3) – 4(a + 3) to get___________________.
3. To factor trinomials that don’t have a leading coefficient, follow the following steps. Ex. x4 – 7x3 + 10x2




As in problem 1, factor out any like terms. x2 (x2 –7x + 10)
Put two sets of parentheses. x2( )( )
Put the variable inside each parentheses as the first term, (either first or last) x2 (x )(x ).
Find numbers that multiply to give you the last coefficient that also add up to get the middle term.
In this case -5 times -2 is 10 and -5 + -2 is -7, so place the factors in the appropriate place.
x2 (x – 2)(x – 5). Check your work by using FOIL and distributing.
Factor the following for Preview Homework:
1. 19x3 - 19x
7. x2 - 4x + 3
2. 36x3 - 24x2 + 8x
8. x2 - 5x - 24
3. -16x4 - 32x3 - 80x2
9. x2 + x - 90
4. x2 + 2x - 35
10. 6x(x - 4) + 5(x - 4)
5. x2 + 8x - 20
11. (x - 10)11 + x(x - 10)
6. x2 - 6x + 8
12. (9x + 10)-10 + 7x(9x + 10)
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
Factoring In-Class Notes
To factor a trinomial with the leading coefficient is not one, there are multiple ways to do it. I am going to
teach you 3 different ways. You do it the way you like best. We will use 2x2 – x – 6 for our examples.
Method 1: Factor Sum Method





Multiply the coefficients of the first and last term. 2 times -6 = ______.
Find the factors of -12 that will add up to give us the coefficient of the second term -1.
o
Rewrite the original problem but replacing the middle term with the two factors you just found
o
Now you group the first two terms and the last two terms, and factor out the greatest common factor of
each group.
o
After factoring you should come up with the two parentheses to be the same, which then can be factored
out of both terms.
o
Method 2: The reverse box method


Begin just like the last problem where you multiply the coefficients of the first and last term, and then
find the factors that will add up to the coefficient of the second term (we already know they are 3 and -4)
Now draw your 2x2 box and write in the 4 factors that you have


Now you need to factor out the greatest common factors across both rows
and down both columns.
You have magically found the factors!
Method 3: Modified Factor Sum Method




Again, begin the same way by multiplying the coefficients of the first and last term, and then find the
factors that will add up to the coefficient of the second term (again, we know they are 3 and -4)
Take those two factors, and divide them both by the leading coefficient (keep as fractions if they don’t
divide evenly)
o
Now, draw your two empty parentheses, placing x as the first term in both sets, and the numbers you just
found as the second term.
o
If you have any fractions, take the denominator of that fraction and bring it up as the coefficient of the x
in that group. You have found your factors!
o
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes: Chapter 6
Special cases:
 If you have a binomial where each term is a ________________________ after like terms are taken out.
o If there is a plus sign between the two terms, such as x2 + 4, this is not factorable.
o If there is a minus sign between the two terms, such as x2 – 4, then put two parentheses with
opposite signs and fill in with the square roots of each term.

o These are called _____________________.
Homework: Complete using whichever factoring method you are most comfortable with.
1. x3 + 3x
9. 6x3 - 17x + 5x
2. 14x5 - 24x4
10. 3x2 + x - 10
3. x2 - 100
11. 8x2 - 2x - 3
4. x2 – 16
12. 16x4 - 49
5.
13. 4x2 + 12x + 5
(x - 3)10 + 9x(x - 3)
6. 9x2 - 12x + 4
14. 6x2 - 7x + 1
7. 25x2 - 9
15. 6x2 - 5x + 1
8. x2 + 2x – 35
16. 3x2 - 7x - 6
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes (Chapter 6)
6-10 Homework Notes
6-10: Analyzing Solutions to Quadratic Equations (p. 402)
How many real solutions does a Quadratic Equation Have?
Quadratic equation ax  bx  c  0 are x=
2
Because a and b are real numbers, the numbers _________ and _______ are real so only ______________________
could possibly be nonreal. Therefore:
If
b2  4ac is positive, then there will be _______________________________________________
If
b2  4ac is zero, then there will be _______________________________________________
If
b2  4ac is negative, then there will be _______________________________________________
Complete p. 405 #3-5,7
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 2 Notes (Chapter 6)
6-10 In-Class Notes
From last night’s notes, the part of the quadratic formula that determines whether the solution is going to be _________
or ___________________ is the part under the ________________________________, which is __________________.
Because this is so important, it is given a special name, the _________________________.
______________________ Theorem
Suppose a, b, and c are real numbers with
a  0.
Then the equation
ax 2  bx  c  0 has
(i)
______________________________________ if
b2  4ac  0
(ii)
______________________________________ if
b2  4ac  0
(iii)
_________________________________________________________ if
b2  4ac  0
(remember that _________________________________ mean that they have an ________________ part)
Solutions to quadratic (and other) equations that equal ________ are often called _____________. The number ____
allows square roots of _________________ numbers to be considered as __________________ solutions.
Example: Determine the nature of the roots of the following equations. Then solve.
a.
4 x2  12 x  9  0
b.
2 x 2  3x  4  0
Example: Does Pop Fligh’s Ball ever reach a height of 40 feet?
Complete p. 405 #9,10,11,14,15 on a separate sheet.
c.
2 x 2  3x  9  0