Transcript Algebra 1-semester exam review - Marquette University High

Algebra 1-semester exam
review
By: Ricardo Blanco
In the next slides you will review:
The properties we
learned
What are they used for
and when to recognize
them
Addition Property (of
Equality)
Multiplication Property (of
Equality)
Reflexive Property (of
Equality)
Symmetric Property (of
Equality)
Transitive Property (of
Equality)
Properties
1.Addition Property (of
Equality)
2. Multiplication Property
(of Equality)
Examples in order
1. if a= b, then a + c = b + c.
is added to both sides of an
equation, the two sides
remain equal. That is,
2.if a= b, then a + c = b + c.
. If the same number If a =
b then a·c = b·c.
Properties
3. Reflexive Property (of
Equality)
4. Symmetric Property (of
Equality)
5. Transitive Property (of
Equality)
3. a=a
4. if a=b then b=a
5. If a = b and b = c,
then a = c.
In the next slides you will review
Associative Property of Addition
Associative Property of Multiplication
Commutative Property of Addition
Commutative Property of Multiplication
Distributive Property (of Multiplication over
Addition)
Properties
6. Associative Property of
Addition
7. Associative Property of
Multiplication
6. the sum does not
change. (2 + 5) + 4 =
11 or 2 + (5 + 4) = 11
7. answer will still not
chage.(3 x 2) x 4 = 24
or 3 x (2 x 4) = 24.
Properties
8. Commutative Property
of Addition
9. Commutative Property
of Multiplication
8. As per the
commutative property
of addition, the
expression 5 + 14 =
19 can be written as
14 + 5 = 19. so, 5 +
14 = 14 + 5.
9. 4 x 2 = 2 x 4
Properties
10. Distributive Property
(of Multiplication over
Addition)
10. 3(2 + 7 5) = 3(2) + 3(7) +
(3)(-5)
3(4) = 6 + 21 - 15
12 = 12
In the next slides you will review
Prop of Opposites or Inverse Property of Addition
Prop of Reciprocals or Inverse Prop. of Multiplication
Identity Property of Addition
Identity Property of Multiplication
Properties
11. Prop of Opposites or
Inverse Property of Addition
12. Prop of Reciprocals or
Inverse Prop. of Multiplication
11. In other words, when you add a
number to its additive inverse, the
result is 0. Other terms that are
synonymous with additive inverse are
negative and opposite. a + (-a) = 0.
12. In other words, when you multiply
a number by its multiplicative inverse
the result is 1. A more common term
used to indicate a multiplicative
inverse is the reciprocal. A
multiplicative inverse or reciprocal of a
real number a (except 0) is found by
"flipping" a upside down. The
numerator of a becomes the
denominator of the reciprocal of a and
the denominator of a becomes the
numerator of the reciprocal of a.
Properties
13. Identity Property of
Addition
14. Identity Property of
Multiplication
13. Identity property of
addition states that the
sum of zero and any
number or variable is the
number or variable itself.
4+0=4
14. According to identity
property of addition, the
sum of a number and 0 is
the number itself. 4 × 1
=4
In the next slides you will review
Multiplicative Property of Zero
Closure Property of Addition
Closure Property of Multiplication
Product of Powers Property
Power of a Product Property
Power of a Power Property
Properties
15. Multiplicative Property
of Zero
16. Closure Property of
Addition
17. Closure Property of
Multiplication
15. The product of any
number and zero is zeroa×0=0
16. Closure property of
addition states that the
sum of any two real
numbers equals another
real number.
17. Closure property
of multiplication states
that the product of any
two real numbers equals
another real number.
Properties
18. Product of Powers
Property
19. Power of a Product
Property
20. Power of a Power
Property
18.when you multiply
powers having the same
amount add the
exponents.
72 × 76
(7 × 7) × (7 × 7 × 7 × 7 × 7
× 7)
19. (3t)4
(3t)4 = 34 · t4 = 81t4
20. (ab)c = abc
In the next slides you will review
Quotient of Powers Property
Power of a Quotient Property
Zero Power Property
Negative Power Property
zero product property
Properties
21. Quotient of Powers
Property
22. Power of a Quotient
Property
21. This property states
that to divide powers
having the same base,
subtract the exponents.
(am)n = amn
22. This property states that
the power of a quotient can
be obtained by finding the
powers of numerator and
denominator and dividing
them.
Properties
23. Zero Power Property
24. Negative Power
Property
23. If a variable has an
exponent of zero, then it must
equal one 30=1
24. When a fraction or a
number has negative
exponents, you must
change it to its reciprocal
in order to turn the
negative exponent into a
positive exponent
Properties
25. zero product property
25. when your
variables are equal to
zero then one or the
other must be zero.
In the next slides you will review
Product of Roots Property
Quotient of Roots Property
Root of a Power Property
Power of a Root Property
Properties
26. Product of Roots
Property
26. The product is the
same as the product of
square roots
a
X
b
=
AB
Properties
27. Quotient of Roots
Property
27. the quotient is the
same as the quotient
of the square roots
Properties
28. Root of a Power
Property
29. Power of a Root
Property
28.
29.
Property quiz
Problems in which you
determine the property.
You will fill in the answer
on the power point
when finished go back
through the properties to
make sure you have the
correct answers.
1.
3.
4.
5.
A. if a= b, then a + c = b
+c.
B. a=a
C. If a = b and b = c,
then a = c.
D. answer will still not
chage.(3 x 2) x 4 = 24
or 3 x (2 x 4) = 24.
E. 4 x 2 = 2 x 4
Solving1st power equations
In the next slides you will see how toA. with only one inequality sign
B. conjunction
C. disjunction
Solving1st power equationswith only one inequality sign
This will only be true if
x is equal to four
The answer will be
x>4
Which on a number
line is
6x = 24
6x > 24
x>4
Solving1st power equationsconjunction
A conjunction is true
only if both the
statements in it are
true
A conjunction is a
mathematical operator
that returns an output
of true if and only
if all of
its operands are true.
-2 < x <= 4
Solving1st power equationsdisjunction
A disjunction is statement
which connects two other
statements using the word
or.
To solve a disjunctions of
two open sentences, you
find the variables for which
at least one of the
sentences is true. The
graph consists of all points
that are in the graph
Ex. -3<x or x<4
Line where the lines
Linear equations in two variables
Standard form
Next determine
whether or not the
equations is linear or
not.
Next subtract 5x from
both sides
Ax + By = C
y = 5x - 3
5x + y = -3
This would be -5x + y
= -3 it would become
a straight line
Linear equations in two variables
cont.
A graphed linear
equation
Linear systems
A. substitution
B. addition/subtraction
C. check for
understanding of terms1.dependent
2. inconsistent
3. consistent
Solving equations in two
variables
Graphing points
Standard/General Form
Slope- Intercept Form
Point-Slope Form
Slopes
Linear systems-substitution
1.looks like it would be
easy to solve for x, so we
take it and isolate x:
2. Now that we have y,
we still need to substitute
back in to get x. We could
substitute back into any
of the previous equations,
but notice that equation 3
is already conveniently
solved for x:
3. answer is 1
1.2y + x = 3
2. 2y + x = 3
3.x=3-2y
x=3-2(1)
x=3-2
x=1
Linear systems-add/sub
(elimination)
1. Note that, if I add
down, the y's will cancel
out. So I'll draw an
"equals" bar under the
system, and add down:
2. Now I can divide
through to solve for x = 5,
and then back-solve,
using either of the original
equations, to find the
value of y. The first
equation has smaller
numbers, so I'll backsolve in that one:
1. 2x + y = 9
3x – y = 16
2. 2x + y = 9
3x – y = 16
5x
=25
3. 2(5) + y = 9
10 + y = 9
y = –1
Linear systems-understanding
terms
1. inconsistent
2. consistent
3. dependent
A system is inconsistent
if it has no solutions
A system is consistent if
there is at least one
solution
A system is dependent if
it has many solutions
Factoring-methods and techniques
A. Factoring GCF
B. Difference of squares
C. Sum and difference of
cubes
D. Reverse of foil
E. PST
F. Factoring by grouping
In the next slides you
would learn each.
Factoring GCF
grouping is important
pulling out the GCF
will take one or two
times
EXAMPLE
these are the steps you'll
need to go through.
1.3x2 + 6x - 4x - 8
2. (3x2 + 6x) - (4x + 8)
3 3x (x + 2) - 4 (x + 2)
4.(3x - 4) (x + 2)
Difference of squares-binomials
you must find out
what is a common
factor
then make into
binomials
You must watch
squares in case
answer might be
prime
EXAMPLE
1.a2-b2
2.(a+b)(a-b)=a2-b2
Prime example
EXAMPLE
1.a2+b2
Sum and difference of cubesbinomials
find difference
opposite product in
the middle
Use parenthesis very
important.
EXAMPLE
1. x3 -8
2.x3 – 23
3. (x-2)(x2+2x+22)
4.(x-2)(x2+2x+4)
Reverse of foil-trinomials
Just do foil in reverse
Trial and error it may
take you a couple of
tries to find the
correct answer.
EXAMPLE
1.3x2 - 6x + x - 2
2.(3x+1)(x-2)
PST-perfect square trinomial
The first term and the last
term will be perfect
squares.
The coefficient of the
middle term will be
double the square root of
the last term multiplied by
the square root of the
coefficient of the first
term.
There will be many
different problems that
will be PST
EXAMPLE
1.x2 + 6x + 9 = 0
2.x2 + 2(3)x + 32= 0
3.(x + 3)2 = 0
4. x+3=0
5.x=-3
EXAMPLE
(ax)2 + 2abx + b2
Factoring by grouping-four or more
items
remember it is a binomial
and make sure you set
problem up for globs
the key is to find a
common factor and keep
factoring out the problem
EXAMPLE
1. x3-4x2+3x-12
2.x3-4x2+3x-12=x2(x4)+3(x-4)
3.(x-4)(x2+3)
Functions
A Function is a
correspondence between
two sets, the domain and
the range, that assigns to
each member of the
domain exactly one
member of the range.
Each member of the
range must be assigned
to at least one member of
the domain.
example of
equation h(k)=
x2 - 2x -2
Simplifying expressions with
exponents
You would use
properties when doing
this.
The x6 means six copies of x
multiplied together and the x5
means five copies of x multiplied
together. So if I multiply those two
expressions together, I will get
eleven copies of x multiplied
together.
x6 × x5
x6 × x5 = (x6)(x5)
=
(xxxxxx)(xxxxx) (6
times, and then 5 times)
=
xxxxxxxxxxx
(11
times)
= x11
Simplifying expressions with
exponents cont.
The exponent rules tell me to subtract the exponents. But let's
suppose that I've forgotten the rules again. The " 68 " means I have
eight copies of 6 on top; the " 65 " means I have five copies of 6
underneath.
Then you would cancel out the top and bottom then you would have
your simplified expression.
Word problems
In three more years,
Jack's grandmother
will be six times as
old as Jack was last
year. If Jack's present
age is added to his
grandmother's
present age, the total
is 68. How old is each
one now?
Let 'g' be Jack's grandmother's current
age
Let 'j' be Jack's grandmother's current
age
If Jack's present age is added to his
grandmother's present age, the total is
68
j + g = 68
In six more years, Jack's grandmother
will be six times as old as Jack was
last year
(g+3) = 6 (j-1)
If Jack's present age is added to his
grandmother's present age, the total is
68
j+g=68
Solving both equations we get Jack's
age (j) as 11 and Jack's grandmother's
age (g) as 57
Lines best fit or regression
A Regression line is a line
draw through and scatterplot of two variables. The
line is chosen so that it
comes as close to the
points as possible.
When asked to draw a
linear regression line or
best-fit line, you have to
to draw a line through
data point on a scatter
plot. In order to solve
these problems a
calculator will be needed
Lines best fit or
regression
Conclusion
These slides should have gave you
information on what we worked on during
semester two and what you will have to
know for the test.