SYSTEMS OF EQUATIONS AND INEQUALITIES: The Function Machine
NOTES:
Find the Expression: A finite difference game for... ax + b
2x + 10
The method of finding an expression of the form ax + b is called the method of... FINITE DIFFERENCE.
When you choose any two consecutive whole numbers for x, the difference is a.
Think about how to find b: An easy way to find a and b in an expression of the form:
ax + b
is to use x=0 and x=1.
The function machine will pick an expression of the form
ax + b
When you fill in values of x, the function machine will give you values of the expression.
You will then guess the expression.
***
Here is a practice game...
The function machine has a secret expression. It has the form:
ax + b
You are to guess the expression replacing a and b with numbers.
WORKSPACE:
ANS:
4 , 5 , -9 , -7 , a , a , a , 8 , 20 , 7 , 19
*END*
TEST: Systems of Equations II
This lesson will evaluate your skills in solving and graphing systems of equations and inequalities.
NOTES:
WORKSPACE:
x + y = 2
2x - 3y = -6
x = 2 - y
2(2 - y) - 3y = -6
4 - 2y - 3y = -6
-5y = -10
y = 2
x + 2 = 2
x = 0
***
3y = 2x - 10
y = -2x + 2
3(-2x + 2) = 2x - 10
-6x + 6 = 2x - 10
-8x = -16
x = 2
y = -2(2) + 2
y = -4 + 2
y = -2
***
ANS:
0,2 , D , C , 2,-2 , B , D , A , B , A , NOT:A,C
*END*
INTRO: Polynomial Concepts I
This lesson will evaluate your knowledge about multiplying, factoring, and graphing polynomials.
NOTES:
WORKSPACE:
(m + 2n)(m - 2n)
m2 - 2mn + 2mn - 4n2
***
(3b + n3)(5b - 3n3)
15b2 - 9bn3 + 5bn3 - 3n6
***
(3x2 - y)(2x2 - 7y)
6x4 - 21x2y - 2x2y + 7y2
***
(x + 2)(x2 - 5x - 3)
x3 - 5x2 - 3x + 2x2 - 10x - 6
x3 - 3x2 - 13x - 6
***
ANS:
A , B , A , D , C , C , NOT:A , NOT:A , NOT:C , B
MULTIPLYING POLYNOMIALS: Multiplying
- Multiply polynomials
- Relate factoring to multiplying
- Relate factoring to graphing
NOTES:
Figuring out missing factors:
Let's look at the same problem written in horizontal form.
5x2 + 7x - 6 = (? - 3)(x + ?)
Now:
5x2 + 7x - 6 = (5x - 3)(x + 2)
Before pressing ENTER, desribe in your own words how multiplying binomials and factoring the results are related.
Factoring is the reverse of multiplying.
***
Multiply (3x + 5)(x - 4).
y = 3x2 + x - 14
Compare the x-intercepts on the graph with the given binomial factors. (SEE PAINT CLIPPING)
*HAVING SEVERE TROUBLE WITH THIS PART*
Before pressing ENTER, desribe in your own words what factoring tells you about the graph of a trinomial.
The factors indicate where the curve intersects the x-axis.
*FLAG, MAY BE SOME EXPLANATION HERE*
Factoring a trinomial into two binomials tells us where the graph crosses the x-axis.
Now let's multiply a binomial times a trinomial.
2
x - 2x - 3
x + 4
-----------
2
4x - 8x - 12
3 2
x - 2x - 3x
-----------------
3 2
x + 2x - 11x - 12
*OR MAYBE NOT... :-| *
...
*OH, NO! NOT AGAIN*
The product of 2x - 1 and 3x2 - 11x - 4 is 6x3 - 25x2 + 3x + 4.
Here's another graph of:
3 2
y = 6x - 25x + 3x + 4
(SEE PAINT CLIPPING)
2
Factor 3x - 11x - 4 using the graph to help you. (3x + 1)(x - 4)
The x-intercepts are -1/3, 1/2, and 4.
***
Before pressing ENTER, desribe in your own words how to multiply polynomials.
Multiply each term of the first polynomial by each term of the second polynomial and add to the results.
***
WORKSPACE:
(7x - 4)(2x - 1)
14x2 - 7x - 8x + 4
14x2 - 15x + 4
***
(3x + 5)(x - 4)
3x2 - 12x + 5x - 20
3x2 - 7x - 20
***
(3x + 7)(x - 2)
3x2 - 6x + 7x - 14
3x2 + x - 14
***
(4x - 1)(2x + 11)
8x2 + 44x - 2x - 11
8x2 + 42x - 11
***
(5x + 6)(3x - 10)
15x2 - 50x + 18x - 60
15x2 - 32x - 60
***
ANS:
14x , 1x , 24x , 4x2+23x-6 , 1 , 5x , 5x , 2 , 7x , 1 , 14x2 - 15x + 4 , 7 , 2 , 3x2 + x - 14 , x , 3x , 6x2 + 17x + 5 , 9 , 5x , 20x2 - 57x + 27 , 3x2 - 7x - 20 , 3x2 + x - 14 , 15x2 - 32x - 60 , 6x2 + 13x - 5 , 20x2 - 19x - 6 , 5x2 + 7x - 6 , 14x2 - 15x + 4 , 3x2 - 7x - 20 , 6x , 37x2 , 4 , 2x , 6x3 - 22x2 - 8x , 2x3 + 5x2 - 4x - 3 , x3 + 2x2 - 8x , 2x3 + 17x2 + 42x + 27 , x2 , 18x , 2x4 , 5x2 , 6x , 2x3 , 9x2 , 3x3 , 40 , 13x
MULTIPLYING POLYNOMIALS: Factoring Polynomials
- Factor and graph polynomials
NOTES:
We now know how to multiply polynomials and have seen how graphing and factoring are related.
In this activity we'll combine some of the methods for factoring polynomials.
4 2
3x - 48 = 3(x - 2)(x + 2)(x + 4)
Let's factor this binomial:
2
2x - 18
First factor out a constant:
2( ? )
2
2(x - 9)
Then factor the difference of squares:
2(x + 3)(x - 3)
We've graphed differences of squares and compared the graphs witht the factored form.
2
y = x - 9
y = (x + 3)(x - 3)
Now let's include a constant factor.
y = 2(x + 3)(x - 3)
***
WORKSPACE:
3
3t - 48t
2
3t(t - 16)
3t(t + 4)(t - 4)
***
2
6n - 54
2
6(n - 54)
6(n + 3)(n - 3)
***
3
4p - 16p
2
4p(p - 4)
4p(p + 2)(p - 2)
***
4 2
5z - 5z
...
2
5z ()
2
5z (z + 1)(z - 1)
ANS:
25 , x-5 , 3t(t + 4)(t - 4) , 6(n + 3)(n - 3) , 4p(p + 2)(p - 2) , 5z2(z + 1)(z - 1) , (x + 1)(x - 1) , 4 , (x + 5)(x - 5) , 2 , (x + 4)(x - 4) , 3