The document provides step-by-step instructions for performing several mathematical operations and problem solving techniques, including: 1) Finding the inverse of a function by switching x and y values and manipulating the equation. 2) Finding the maximum area of a fenced rectangular field using the area and perimeter formulas. 3) Simplifying rational expressions by finding common denominators. 4) Completing the square to rewrite a quadratic equation in standard form.

1. D.E.V. PROJECT 2014
ROCIO ALMARALES

2. INVERSES!
Step #1: Switch f(x) in the
equation to x=, also switch
all x’s in the equation to
y’s.
Step #2: Multiply the
denominator (y-2) to both
sides. This way the y-2 on
the original side cancels out
the other.

3. INVERSES
CONTINUED…
Step #3: Next, Distribute the
x into the (y-2) on the left
side to get….. xy-2x=4y-9
Step #4: Next we must get all
the y’s on one side and x’s on
the other side. In order to do
this we must subtract xy from
both sides of the equation.
Then add 9 to both sides of
the equation.

4. INVERSES
CONTINUED…
Step #5: Next we must factor
out the y from 4y-xy in order
to get the y by itself,
considering that this is what
we are trying to solve for.
Step #6: Divide both sides of
the equation by 4-x to get the
y by itself.
Step #7: Change the y at the
end of the equation to f-1(x)
to identify that it is the
inverse of the function we
began with.

5. FARMER TED TAKEOVER!
Farmer Ted has 16,200ft. of fencing. He wants to fence off a
rectangular field for his magical unicorns who desperately need a
home. Please help Farmer Ted find the maximum area of his
fenced off field! He’s running out of time!

6. FARMER TED!
Step #1: We must help Farmer
Ted get his fence built quick! But
how do we begin? Well first we
must be aware of the equations
that take place when finding the
maximum area. The two
equations we use when
searching for this maximum
area is the area equation (A=xy)
and the perimeter equation
(2x=2y)

7. FARMER TED!
Step #2:Next, we must insert
the perimeter, which is
16,200ft., into the perimeter
equation.
Step #3:We must try and get y
by itself because we are going
to need it later in this
process! We are going to do
this by dividing both sides y
2!
Step #4: After dividing both
sides by 2, subtract x from
both sides to finally get y by
itself!

8. FARMER TED!
 Step #5: Now going back to the
perimeter equation, we are going
to insert what we solved for y in
the perimeter equation into the
area equation.
 Step#6: Next we distribute the x
so we can get an A and a B value.
 Step #7: Since we now do have an
A and a B value we can use the
equation –b/2a to find our x-value.
 Step #8: Insert your A value,
8,100, and your B value, -1,
solve, and you will end up with
4,050.

9. FARMER TED!
Step #9: Insert this value into
the equation we began with
when we first inserted 8,100
into the area equation.
Step #10: Calculate your
answer and you will see that
we end up with 16,402,500
square feet. Which is our
maximum area!

10. SIMPLIFYING
RATIONAL
EXPRESSIONS
 Step #1: When adding or
subtracting fractions, we must
identify that in order to do so, we
must have common
denominators.
 Step #2: To achieve common
denominators we must multiply
each of the denominators to both
sides. Also, we must realize that
if we multiply something to the
denominators, we must also
multiply those values to the
numerator to balance out the
equation.
 Step #4: When you look at the denominators of both of
the Fractions, you can realize that they are the same,
both (6b-4)(b+6). In the process of adding and
subtracting fractions, when the denominators are the
same you just leave them as they are (as (6b-4)(b+6)

11. SIMPLIFYING
RATIONAL
EXPRESSIONS
 Step #5: Next you will need to
combine like terms in the
numerator if possible. If it was
not possible then you would just
leave it as is, but in this case it is
possible.
 Step #6: After combining like
terms you will end up with the
most simplified form of this
expression.

12. COMPLETING THE
SQUARE
 Step #1: Always in a completing
the square equation one must
start off by subtracting the
beginning c-value, which in this
case is 18.
 Step #2: Now you are left with an
equation… but no C value? We
must use (b/2) squared to find
the perfect C value! After you
plug in your B value, -6, into the
equation you see that 9 is your
perfect C-value!

13. COMPLETING THE
SQUARE
 Step #3: Since it is the perfect c-value you
must also add it to the other side to keep the
equation balanced.
 Step#4: There are too ways to go about this
next step. One way you could get the value
(x-3) squared is to factor them. Factoring is
just finding the two sets of factors that if
distributed would give you the quadratic we
began with. Since the factors of this equation
are the same, they are both (x-3) you can
write them as (x-3) squared. But there is also
a shortcut, in all of these types of equations
there is a pattern, you can just divide the B-value
in the quadratic. As you can see that
would also give you -3!
 Step #5: After this long process, all that is
left to do is add nine to both sides to get
everything back on one side! And now your
done! 