Name:Mrs. Gorsline Integrated Math 2

Hour: 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

When a linear expression is multiplied by itself, the result is a ______. Simplifying a binomial squared as a quadratic expression is called ______.

Example:

Expand

Complete p. 349 #5,13,15

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:

Binomial Square theorem (the shortcut method for)

Example:

Expand

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:

Complete p. 355 #14 – 18

6-2 In-Class Notes

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

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?

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 ,

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 ______.

Example 1: Sketch the graph of by first graphing

Since the vertex of the graph of 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.

Example: Sketch the graph of and give the equation for the axis of symmetry.

Be aware that ______and ______can replace the x and y in any function to translate it.

Example:Sketch the graph of

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

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

Example: Use the reverse binomial square theorem to make a perfect square trinomial for .

Example: Now we will convert the equation into vertex form.

Step 1:______

Step 2:______

Step 3:______

Step 4:______

Step 5:______

Example:Convert the equation into vertex form.

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

6-4 Homework Notes

6-4: Graphing

Example: Convert to standard form. Be sure to follow Order of Operations!

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

Example: Show that and 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.

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.

  1. Use the equation ______to find the axis of symmetry, which is also the ______of the vertex.
  2. 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.

  1. Find the vertex of this quadratic.
  1. Doe this function have a maximum or a minimum? Explain how you know.
  1. Find h when t = 0, 1, 2, and 3
  1. Explain what each pair (t, h) tells you about the height of the ball.
  2. 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 ______and initial ______ 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.

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 and , then

We will be completing the proof for this in class tomorrow.

Example:

Solve using the quadratic formula.

Complete p. 385 #6 – 10

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 , where

StepsJustifications

Example: Pop Fligh, the famous baseball player, hit a pitch that followed the equation . 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.

6-8 & 6-9 Homework Notes

6-8 and 6-9: Imaginary and Complex Numbers

Example: Solve the equation

What is the problem with this?

Definition:

When , the two solutions to are denoted ______and ______.

Numbers that have negatives under a square root are called ______.

Definition: ______

Example: Solve using the letter i.

Simplify the following:

a) (2i)(5i)b)c)

Complete p. 391 #3,5 and p. 397 #2,4,6

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:

When simplifying imaginary expressions, you need to be careful.

Example: Simplify the following

a) b)

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

All properties that hold true for real numbers (except inequalities) hold true for complex numbers as well.

Examples:

Add and Simplify

Simplify

Multiply and Simplify

Any number that is a ______is not allowed to be in the ______of a fraction. This includes imaginary numbers.

Example:

Write in form.

Homework (On a separate sheet)

p. 391 #7,11-18 and p. 397 #7-10, 16-18

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 coefficientthat 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:

Name:Mrs. Gorsline Integrated Math 2

Hour: Unit 2 Notes: Chapter 6

  1. 19x3 - 19x
  2. 36x3 - 24x2 + 8x
  3. -16x4 - 32x3 - 80x2
  4. x2 + 2x - 35
  5. x2 + 8x - 20
  6. x2 - 6x + 8
  7. x2 - 4x + 3
  8. x2 - 5x - 24
  9. x2 + x - 90
  10. 6x(x - 4) + 5(x - 4)
  11. (x - 10)11 + x(x - 10)
  12. (9x + 10)-10 + 7x(9x + 10)

Name:Mrs. Gorsline Integrated Math 2

Hour: 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.
  • Rewrite the original problem but replacing the middle term with the two factors you just found
  • Now you group the first two terms and the last two terms, and factor out the greatest common factor of each group.
  • After factoring you should come up with the two parentheses to be the same, which then can be factored out of both terms.

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)
  • 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.
  • 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!

Special cases:

  • If you have a binomial where each term is a ______after like terms are takenout.
  • If there is a plus sign between the two terms, such as x2 + 4, this is not factorable.
  • 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.
  • These are called ______.

Homework: Complete using whichever factoring method you are most comfortable with.

Name:Mrs. Gorsline Integrated Math 2

Hour: Unit 2 Notes: Chapter 6

  1. x3 + 3x
  2. 14x5 - 24x4
  3. x2 - 100
  4. x2 – 16
  5. (x - 3)10 + 9x(x - 3)
  6. 9x2 - 12x + 4
  7. 25x2 - 9
  8. x2 + 2x – 35
  1. 6x3 - 17x + 5x
  2. 3x2 + x - 10
  3. 8x2 - 2x - 3
  4. 16x4 - 49
  5. 4x2 + 12x + 5
  6. 6x2 - 7x + 1
  7. 6x2 - 5x + 1
  8. 3x2 - 7x - 6

Name:Mrs. Gorsline Integrated Math 2

Hour: 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 are x=

Because a and b are real numbers, the numbers ______and ______are real so only ______could possibly be nonreal. Therefore:

If is positive, then there will be ______

If is zero, then there will be ______

If is negative, then there will be ______

Complete p. 405 #3-5,7

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 . Then the equation has

(i)______if

(ii)______if

(iii)______if

(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. b. c.

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.