1. How do we expand our knowledge of the distributive property in order to find the product of two binomials?
Binomial: an algebraic expression containing two terms.
This skill is one of the most important basic skill you will need when taking Algebra 1 in high school, because a huge topic in Algebra 1 is solving and understanding something called a quadratic function, and quadratic equation (we will get to a better understanding of what that is later). Unlike a linear function, quadratic functions do not have a constant rate of change, and do not graph a straight line. They are extremely useful to physicists, and engineers because they provide do a much more accurate job of modeling the motion of objects such as satellites, meteors, projectiles, planes, etc. The multiplication of two binomials is the first skill you need to solve and analyze quadratic equations.
We will learn 3 methods, or ways of thinking about, how to find the product of two binomials multiplied together.
The first method we will see is the area model…

2. AREA MODEL: don’t forget, we do LW to find the area of a rectangle, and if we know the area of different sections of a shape, we can add up the areas to find the total area of a shape. Lets multiply (x+4)(x+7) X
For this box, to find its area, multiply its side lengths together, which are x, and x.
For this box, to find its area, multiply its side lengths together, which are 7 and x. X + 4
For this box, to find its area, multiply its side lengths together, which are 4 and x.
For this box, to find its area, multiply its side lengths together, which are 7 and 4.

3. AREA MODEL: don’t forget, we do LW to find the area of a rectangle, and if we know the area of different sections of a shape, we can add up the areas to find the total area of a shape. Lets multiply (x+4)(x+7) X X2 7x X + 4 4x 28

4. Area Model continued
Now that we have the areas for each of the four sections, lets put them together in one expression, and combine any like terms…
(x+4)(x+7)= X2+ 4x +7x + 28
We see that X2+ is the only x with a power of 2, so it can’t be combined with the 4x or 7x, and the 28 is a constant so that can’t be combined with any other terms with variables.
4x and 7x are like terms because they have the same variable, raised to the same power, so x + 7x = 11x
So, (x+4)(x+7) = X2+ 4x +7x + 28 = X2+ 11x + 28 final answer.

6. F.O.I.L.
Multiply each of the following pairs of numbers: First, Outside, Inside, Last
( x + 4 )( x + 7 ) O I L F

7. Distributive Strategy: You can also think of how to multiply the binomials together like applying the distributive property TWICE.
In the left-hand binomial, we distribute the x to both the other x and the 7, AND THEN we distribute the 4 from the left-hand binomial to the other x and the 7.
* You should still end up with 4 terms before you combine like terms.
X2+ 4x +7x + 28 = X2+ 11x + 28
( x + 4 )( x + 7 )

10. Practice: Find the product
15.) (x – y)(x + y), after finding this product, and combining any like terms, compare this result to what happened in #’s 4, 10, & 13. Do you notice anything similar?
16.) (x + 1)(x2 + 4x + 5), before you attempt #16, look back at the three methods we’ve gone over for multiplying binomials, and find the one that won’t work for this. Why do the other two methods still work, but one does not?
17.) (x + 1.5)(x2 + 7x + 2.4)
18.) (x7 + x6 + x5)(x4 + x3 + x2)
19.) When multiplying any two polynomials together, how can you use the number of terms each polynomial has to know how many terms will be in your product, before you combine like terms at the end? Make a claim, and test it as it applies to #16, 17, and 18.
20.) Does your response to #19 still apply when you multiply more than one polynomial together? Test out your claim by finding the product, without combining like terms, of (x+3)(x+2)(x+1)
1.) (x+3)(x+1) 2.) (x+3)(x-1) 3.) (x-9)(x-9) 4.) (x+9)(x-9) 5.) (2x+1)(x+8) 6.) (3x+3)(8x – 7) 7.) (x4 + x3)(x2 + x1) 8.) (g8 + g9)(g2 + g10) 9.) (x + y)(w + z) 10.) (x – y)(2w – z) 11.) (0.5x )(x ) 12.) (0.5x + 0.2)(x x) 13.) (x/2 – ½) (x/4 + ½) 14.) (x2 + y)(y2 + x)

12. Practice: Find the product
15.) (x – z)(x + z), after finding this product, and combining any like terms, compare this result to what happened in #’s 4, 10, & 13. Do you notice anything similar?
16.) (5x + 2)(6x2 + 3x + 7), before you attempt #16, look back at the three methods we’ve gone over for multiplying binomials, and find the one that won’t work for this. Why do the other two methods still work, but one does not?
17.) (x + 2.6)(2x2 + 3x + 1.4)
18.) (x8 + x7 + x6)(x5 + x4 + x3)
19.) When multiplying any two polynomials together, how can you use the number of terms each polynomial has to know how many terms will be in your product, before you combine like terms at the end? Make a claim, and test it as it applies to #16, 17, and 18.
20.) Does your response to #19 still apply when you multiply more than one polynomial together? Test out your claim by finding the product, without combining like terms, of (4x+6)(5x+5)(6x+4)
1.) (x+2)(x+1) 2.) (x+3)(x-4) 3.) (x-12)(x-12) 4.) (x+12)(x-12) 5.) (3x+1)(x+9) 6.) (4x+2)(5x – 18) 7.) (x5 + x4)(x5 + x6) 8.) (g + g9)(g2 + g10) 9.) (x + y)(x + z) 10.) (x – y)(2y – z) 11.) (0.2x )(2x ) 12.) (0.3x + 0.2)(x x) 13.) (x/2 – ¼ ) (x/3 + ½) 14.) (x2 + y)(y + x2)

13. POD (leads to graphing)
(2x – 2)(x – 1). Find the product of these two binomials.

14. POD (leading into graphing)
(2x – 2)(x – 1). Find the product of these two binomials.
2x2 – 2x – 2x + 2
2x2 – 4x + 2
The squared term makes this function non-linear. This is a new type of function which will be studied in great detail, your 1st and 3rd years of high school. Lets get an initial idea of how to understand a function like this…

15. Group A: y = x2 + 1, and y = x2 Group B: y = 2x2, and y = x2
All groups: take your function rule, and make an input/output table, with x values from -2 to 2. Find the outputs, then graph the ordered pairs on a coordinate plane. Notice the range of y values before you create your coordinate plane so you can fit all the ordered pairs.
Group A: y = x2 + 1, and y = x2
Group B: y = 2x2, and y = x2
Group C: y = (1/2)x2, and y = x2

16. Y=2x2 – 4x + 2 This type of equation, is called a quadratic function.
The curved shape that it graphs, is called a parabola.
We would say this particular parabola is pointing, facing, or opening upward
The function gets its name, quadratic, from the importance of the squared input variable. Think, “quad = 4, and a ‘square’ has 4 sides”

17. Quadratic Functions, and Parabolas
Definitions:
Quadratic function: A mathematical function having no more than 3 terms, such that 1 term contains a variable to the 2nd power, and there are no terms with an exponent greater than 2.
Parabola: A type of symmetrical “U” shaped curve, and is the graphical representation of a quadratic function on the coordinate plane.

18. Some Parabolas will face/point/open downward.

19. Some parabolas face sideways but why do they not represent quadratic functions?

20. When the parabola opens upward, the vertex is the minimum.
All quadratic functions graph a parabola which have a vertical line of symmetry. For example, if one half of the parabola reflected (folded) over its line of symmetry, it would exactly overlap the other half on the opposite side of the line of symmetry.
The point of the parabola which lies exactly on the axis of symmetry is called the vertex.
The vertex is either the highest point or the lowest point of the parabola, called the maximum, or the minimum.
When the parabola opens upward, the vertex is the minimum.
When the parabola opens downward, the vertex is the maximum.

21. Because we are most familiar with linear functions lets compare quadratic functions to linear functions…

22. WHY BOTHER? Applications of linear functions and quadratic functions
Suppose a car starts a NASCAR race a tenth of a mile behind the starting line, and travels at a speed of 130 miles per hour.
A linear function model of this would look like
y=130x – 0.1
We may not have realized yet at this point, but something is not right about this. Why is this description of the race, and the linear model, not sufficient to describe what REALLY happens here in real life?
A quadratic function would much more accurately model this situations’ actual change in speed.

23. Linear functions Quadratic functions
Is a function: There is 1 and only 1 output assigned to each input. Passes the vertical line test.
Graphs a straight line
No lines of symmetry
Goes infinitely in both directions.
Therefore has no max or min.
Quadratic functions
Is a function: There is 1 and only 1 output assigned to each input. Passes the vertical line test.
Graphs a parabolic curve
Parabolas are symmetrical; over an imaginary vertical “axis of symmetry.”
Either side of the axis of symmetry is an exact reflection of the other side.
Where the parabola meets the axis of symmetry, is called the vertex
Goes infinitely up, or down, but not both.
The vertex on an upward facing parabola is called the Minimum.
The vertex on a downward facing parabola is called the Maximum.

24. Linear functions Quadratic functions General Form looks like y=mx±b
Input raised to the 1st power
Constant Rate of Change (Slope is always the same)
Quadratic functions
General Form looks like y=ax2 ± bx ± C
Has an input raised to 2nd power
Rate of Change (slope) varies
Some would say there is no slope.

25. Understanding a Quadratic Function’s General Form y=ax2 ± bx ± C
Just like other functions, x represents the input, and y, the output.
It is important to understand what we mean by a, b, and C, because many properties of quadratic functions we need to study are based on the values of a, b, and C:
“a” is how we refer to the coefficient of the x squared term
a ≠ 0 because then there would be no x squared term, which is exactly what makes the function a Quadratic function.
“b” is how we refer to the x term (to the 1st power)
There is not always a “bx” in a quadratic function. This is when b = 0
“C” is referred to as the “constant” term, because there is no input variable.
There is not always a “C” term. This is when C = 0.
Ex1) In the quadratic function, y= 5x2 - 7x + 10, a=5, b= -7, C= 10
Ex2) In the quadratic function, y= -10x2 + 7x + 3, a=?, b=? , C= ?
Ex3) In the quadratic function, y= x2 + 3, a=?, b=? , C= ?
Ex4) In the quadratic function, y= x2 , a=?, b=? , C= ?

26. Graphing a Parabola from the quadratic function equation…

27. One of the main ways we will do this, is just like a way we graphed our lines, from y=mx+b… …Using a FUNCTION TABLE
Ex1) Y=2x2 – 4x + 2
Say we plugged in -2, -1, and 0, for x in our table to produce three ordered pairs we could graph. This is the table we would get, on the left, and below is some of the work that would get us there. X Y -2 18 -1 8 2
Y=2(0)2 – 4(0) + 2 Y=
Y = 2
Y=2(-1)2 – 4(-1) + 2
Y=2(1)
Y=8
Y=2(-2)2 – 4(-2) + 2
Y=2(4) Y=
Y= 18

28. Y=2x2 – 4x + 2
Unlike linear functions, quadratic functions have a minimum or maximum point. Because this display does not show us where that is, we would consider this a poorly done graph for a quadratic function…
…When we use quadratic functions to model real world phenomena, like say, the height of a projectile as it flies through the air, the maximum or minimum point illuminates important aspects of what we are modeling: like the exact point in time the projectile reaches its maximum height before gravity pulls it back down, and what the maximum height will be.
What we see here is technically the graph of our same quadratic function from earlier…
…However, this particular display of our graph is quite limiting.

29. So how do we ensure that our graph clearly displays the vertex (maximum or minimum), and axis of symmetry?
1.) We use the axis of symmetry formula, which tells us the x coordinate location of the axis of symmetry: For Y=2x2 – 4x + 2, a=2, and b= -4 so x = - (-4) therefore x = 1 2(2) and so we then fill in the x coordinate in the middle row of our table. Leave 3 rows above and 3 rows below your middle row. X Y 1
and x = 4 4

30. X Y -2 18 -1 8 2 1 3 4

33. Y=2x2 – 4x + 2 This type of equation, is called a quadratic function.
The curved shape that it graphs, is called a parabola.
This particular parabola is said to be pointing or facing upward

34. Quadratics in Athletics
In athletic events that involve throwing things, quadratic equations are highly useful. Say, for example, you want to throw a ball into the air and have your friend catch it, but you want to give her the precise time it will take the ball to arrive. To do this, you would use the velocity equation, which calculates the height of the ball based on a parabolic (quadratic) equation. So, say you begin by throwing the ball at 3 meters, where your hands are. Also assume that you can throw the ball upward at 14 meters per second, and that the earth's gravity is reducing the ball's speed at a rate of 5 meters per second squared. This means that we can calculate the height, using the variable t for time, in the form of h = t - 5t^2. If your friend's hands are also at 3 meters in height, how many seconds will it take the ball to reach her? To answer this, set the equation equal to 3 = h, and solve for t. The answer is approximately 2.8 seconds (adopted from Reference 2).

35. POD (leading into factoring)
Find the product:
(x – 5)(x + 6)
(x + 5)(x – 6)

36. POD (leading into factoring)
Find the product:
(x – 5)(x + 6)

37. Solving a quadratic equation by Factoring.
When we multiply two

38. Quadratic Functions
As discussed earlier, one important application of quadratic functions by engineers and physicists, is for modeling and predicting the motion of projectiles. We will see soon, the shape of a quadratic function when graphed, looks just like the path a ball takes when thrown through the air.

39. Quadratic Functions: Background
Before we learn exactly how to graph a quadratic function, we will need a little background information…
Quadratic function definition: A function where the x is raised to the 2nd power and there are no powers greater than 2.

40. General Form: y = mx + b ----- linear function general form
y = ax2 + bx + c (where a ≠ 0) quadratic function general form
Why are general forms important for our understanding?
a is coefficient of x2 term
b is coefficient of x term
c is a constant, and also the y intercept.

41. Quadratic Functions:
The simplest quadratic function is also called an exponential function:
y = x where a=1, b=0, c=0 (why can a never be zero?)
the graph of a quadratic functions forms what is called a PARABOLA.

42. Factoring a Quadratic Expression
Many quadratic expressions (if a=1) can be factored in the following way…
(Factor: to divide one or more terms into two or more factors)
… in the form, ax2 + bx + c, you will have two binomial factors, in the form (x ± P)(x ± Q), such that P and Q…
are factors of “c”, and
Have a sum equal to “b”
y = x2 + 5x + 6

43. Some practice: y = x2 + 13x + 12 y = x2 + 8x + 12 y = x2 + 9x + 20

44. Graph of Quadratic Function: Parabolas
Graph of y = x2
What we need to know about our parabola…
has a “u” shape or and upside-down “U” shape
Has a vertical line of symmetry called the AXIS OF SYMMETRY
The axis of symmetry is the line, over which exactly half of the parabola can be folded, and will overlap exactly on top of the other half.
Notice that for every point on the parabola there is another point equidistant from the axis of symmetry and shares the same y coordinate (but different x coordinate.)

45. Graph of Quadratic Function: Parabolas
Graph of y = x2
…parabola characteristics continued…
the point where the parabola intersects with the axis of symmetry is called the VERTEX
The vertex is either the maximum y value, or it is the minimum y value.

46. When the parabola opens upward, the vertex is the____________________
When the parabola opens downward, the vertex is the_____________________

47. Quick Check:
When the vertex is a maximum, the parabola opens_________________.
When the vertex is a minimum, the parabola opens_________________.

48. Homework: #17: Find the products below: Ex1: (x + 3)(x + 12)