Solving One Variable Equations by UNWRAPPING

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Suppose we are trying to
SOLVE: 2x + 3 = 13 Dilemma: What should be done first?
Should we divide by 2 or subtract 3?.
(One explanation is to follow "PMDAS in REVERSE!")
Subtract 3 from both sides,
then divide both sides by 2.

Here is another explanation:
Let's pretend that x is like a box.
Suppose we put 5 into the x-box.
Let's think of the expression
"2x + 3"
as a wrapped package.
We need to unwrap the package
to look inside the x-box!

"X" is wrapped first with
paper having
"MULTIPLY by 2"
as the pattern on the paper.

Then on top of the paper we have
a RIBBON with
"ADD 3"
as the pattern on the ribbon.

To unwrap "2x+3" we first
"CUT the RIBBON" ("SUBTRACT 3")

and our second task is to
"TAKE-OFF the paper" ("DIVIDE by 2").

Performing these same tasks on the
right side of the equation will produce:

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GENERAL HINTS for Praxis Test

1) No penalty for guessing. GUESS!!!
2) Use the graphing calculator as often as possible.
3) Sometimes you can just plug-in each option to see
which one is correct.
4) More difficult problems could be solved by
looking at a more simplified form.

5) If you have no idea, then choose the one

that is most likely!
OPTIONS: 1/2, -1/2, 5, -5, 4

Since 3 options are positive and 3 are whole numbers,

then pick the POSITIVE 5 since there are two 5's!

6) If the question asks for the largest, then pick the

one OPTION that has the largest ABSOLUTE VALUE!

OPTIONS: -5, -4, -2, 3, and 4 CHOOSE -5

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INVERSE FUNCTIONS

All functions have the property where each x has one y-partner.
(Graphically: It passes the "VERTICAL LINE TEST".)
If a certain function also has each y with only one x-partner
(Thus, passing a "HORIZONTAL LINE TEST"), it has an INVERSE!

Prefixes like INV or ARC are sometimes used to denote inverse.
Another form looks like f to the -1 power(Not to be confused with RECIPROCAL).



You might say that the INVERSE
undoes what the orginal function did!

To find the inverse of a function f(x)
switch x and y and THEN solve for y.


f(x) = x cubed
switch
x = y cubed
solving for y
y = cube root of x
y = x raised to 1/3 power

(We had to invent the symbols for cube root and 1/3 power.)
The graph of the inverse of f(x) we can graph f(x) and then
rotate this graph through y = x (thus, switching x and y)

Inverse Trig:

INVERSE TRIG:

How to find the partners

for 22 degrees.

TRIGONOMETRY - Temperature in Cincinnati

Take the xy-plane and
draw a circle of radius ONE.
The equation for this circle is
x^2 + y^2 = 1.
If you wish to use the graphing
calculator to draw the circle you
will need to split it into two
functions. (It is not a function itself!)
y1 = ( 1 - x^2)^(1/2)
y2 = (-1)( 1 - x^2)^(1/2)
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Draw an angle "SMILE" in a
STANDARD POSITION
(start with a side along the x-axis and then rotate
counter-clockwise a certain number of degrees).
The side along the x-axis is called the initial side
and the other side is the terminal side.
The TERMINAL SIDE will intersect the unit circle
at an ordered pair.
This ordered pair is by definition:
the COSINE and SINE of the ANGLE drawn.
Thus, a 30 degree angle will intercept
the circle at the ordered pair (.5, .8660254).

USING the TI-83
Finding roots, verifying that a point is on a curve,
locating a common point for two curves, and finding

the derivative at a certain point are just a few of

the powers hinden in a graphing calculator. In the

example above, we are looking for the zeros

of a function. After inserting the function into

the "y=" screen, we use "2nd - calc - zero"

buttons. We will need to enter a LEFT and

RIGHT boundary and then hit "ENTER"

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FUNCTIONS

We often need to pair up numbers.
(i.e. : Age with Salary,
Height in Inches with Height in Centimeters,
Number of Shirts to be printed with Total Cost,
Time in a Taxi with the Cost of the Ride, ...)

Sometimes the 1st item has more than one
2nd partner, sometimes each 1st quantity
has one 2nd partner [a FUNCTION], and there
are even cases where each 1st member of the pair
has a unique 2nd partner [This type of FUNCTION
has an INVERSE - each 2nd member has
only one first member for a partner].

A RELATION is a set of ordered pairs.
Example 1 : {(4,2) , (4,-2) , (9,3) , (9,-3)}
Example 2 : f = {(5, 13) , (7, 17) , (2,7)}

A FUNCTION is a RELATION in which
each first number "x" has
one second number "y" for a partner.

Thus, Example 2 is a FUNCTION and
Example 1 is NOT a FUNCTION.
In most cases the relationship
between "x" and "y" has some hidden rule.
Example 1's Rule has "x" as the square of "y".
x = y * y or x = y^2

Example 2's Rule
takes "x" and first multiples by 2 and then
3 is added to create the "y" value.
y = 2x +3 or f(x) = 2x + 3

OLD FAITHFULL is predictable:

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