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Concept Review
Purpose: The purpose of the following set of slides is to review the major
concepts for the math course this year. Please look at each problem and how to
solve them. Then attempt to solve the second problem on your own. If you
have difficulty it would be good to come see me and ask questions.
Exponents
• With exponents it is important to remember that the
variables can represent any number. We cannot do
anything with a variable that we could not do with a
number, because variables are any number.
• A good strategy when working with exponents is to put
a number in for the variable. If it works with the
number than this would be how you would solve the
problem with exponents.
• *** Refer back to the exponent lesson under the
trimester two heading for a more in depth explanation.
Exponents continued
• Simplify
1)
X
A)
X
SOLUTION:
*X = 1
X
X=0
X
= (XXXXXXXX)
X
1
(XXX)
B)
C
(xxx)
1
**Multiply by reciprocal
(XXXXXXXX) (XXX) = (XXXXXXXXXX) or X
Practice
• Simplify
1
X
X
X
X
Exponents Continued
Simplify:
X F X F X
Solution
Rewrite it so that all of the X’s are together
and all of the F’s are together then add the
exponents.
X
F
Exponents Continued
Solve:
4
Pattern
• Another way to understand exponents and x to
the negative n is to look at the pattern
5
5
5
5
5
= 125
= 25
=
5
=
1
= 1/5
Each time we divide by 5. Therefore
1 5 =
1 x 1/5 when we
invert and multiply we get 1/5.
Concept 2
Solve
4 * 7
We use the same method as with
exponents.
Solution: 1
X 49
4
1
1
256
X 49
1
= 49
256
Practice
Solve
1) 5
7
* 2
6
Defining y=mx
using similar triangles
and Linear Equations
Y = mx
Defining y=mx
using similar triangles
and Linear Equations
y = mx
The equation y = mx is the algebraic way of representing what is drawn on the graph.
The m is the y or rise over the run. Looking at the coordinate plane above we can then
x
begin to graph the line. Caution : Remember that points are expressed as (x,y) Don’t
Reverse the m and place the x or y in the incorrect position. This will flip the line your
Intending to express.
Note that the line on the graph forms triangles in a linear fashion. You can use similar
Triangles to determine if a line is linear. The slopes should be the same when simplified.
Example Problem
y = mx
Given an m value of 1 please graph 3 (x,y) points and use similar triangles to show
2
that the line is linear.
Solution:
Note: the y or rise value is 1 and the x or
run value is 2. That means for every 1 I
rise I move over two. This pattern continues.
There are three triangles found at each point
That all simplify to a 1:2 ratio.
Triangle 1 (2,1)
Triangle 2 (4,2)
Triangle 3 (6,3)
Practice Problem
y = mx
Given an m value of 3 please graph 3 (x,y) points and use similar triangles to show
4
that the line is linear.
Solution:
Practice Problem
y = mx
Given the equation y = -4 x please graph.
2
Note: The negative refers to the rise
Not the run so we will be in quadrant 4
The next x, y would be either (-8,4) or 0,0
Solution:
Practice Problem
y = mx
Given the equation y = 4 x please graph.
-2
Note: The negative refers to the run
Solution:
Practice Problems
Given the equation y = 3 x please graph.
2
Given the equation y = -3 x please graph.
-2
Given the equation y = 3 x please graph.
-2
Given the equation y = -3 x please graph.
2
y = mx + b
Thus far I have only presented you
with lines that pass through the point
of origin or (0,0). By adding b to the
equation y = mx +b we can shift the
line up or down on the y axis
The original lines
depending on whether the value for
Slope of ½ has not
changed it simply has
b is positive or negative.
been shifted from (0,0)
on the y axis to (3,0)
making the equation
y=1 x+3
2
Practice Problems
with y = mx + b
Given the equation y = 3 x + 3please graph.
2
Given the equation y = -3 x + -3 please graph.
-2
Given the equation y = -3 x + 5 please graph.
Given the equation y = 3 x +3 please graph.
-2
2
Caution: When we did not have a
value for b the first point matched
the slope values. This will not be the
case with a b value. Follow the slope
as a like a set of directions form the
b value or new origin.