**I want to explore the behavior
of three selected linear equations f(x), g(x), and h(x). I want
to observe the behavior of these equations when they meet the
follow criteria:**

h(x) = f(x) + g(x)

**For my explorations, I want to
set f(x) = x+9 and g(x) = 2x - 1. First I want to look at the
equations of f(x), g(x), and h(x) =f(x) + g(x). Here are the graphs
of f(x) = x + 9, g(x) = 2x - 1, and h(x) = (x+ 9 ) + (2x - 1).**

** It
looks like h (x) can be rewritten as 3x-8. If this is true, then
h (x) is also a linear function. Let's look at a general case.**

**Using the commutative property,
this can be rewritten as**

**Therefpre h(x) = (x + 9) + (
2x - 1) = 3x + 8 is true. Also true is that h(x) is always linear
when you add to linear equations. The slope of h(x) is equal to
the sum of the slope of f(x) plus slope of g(x). The y-intercept
of h(x) is also equal to the sum of the y-intercept of f(x) plus
the y-intercept of g(x). That is**

**Next, I want to look at the equations
of f(x), g(x), and h(x) = f(x) * g(x). Here are the graphs of
f(x) = x + 9, g(x) = 2x-1, and h(x) = (x+9) * (2x -1).**

**It looks like h(x) can be rewritten
as 2x^2 +17x -9. If this is true then h(x) is a quadratic equation.
So let's look at the general case.**

**Using double distributive property
or the FOIL method, h(x) can be rewritten as**

**and using the associative property**

**since integer addition and multiplication
are closed,**

**and this is the general form
od the quadratic equation.**

**Now I want to look at the equations
of f(x), g(x), and h(x) = f(x)/g(x).**

**Here are the graphs of f(x),
g(x), and h(x) = (x+9)/(2x-1).**

**Looking at the graph, h(x) has
an asymptote. Where does this asymptote occur? From general knowledge
of functions, I know that an asymptote occurs when the equation
is discontinous. I know that the equation is discontinous when
the denominator is zero becuase division by zero is undefined.**

**Therefore in this case, h(x)
is discontinous when 2x-1 =0. So when does 2x-1 =0?**

**Let's look at the general case.**

**Discontinuity occurs when cx+d
=0. There is an asymptote in the graph of h(x) at cx+d = 0**

**Therefore, if h(x) = (ax + b)/
(cx + d), then h(x) will always have a asymptoe at the point where
x = -d/c.**

**Finally I want to look at the
equations of f(x), g(x), and h(x) = f(g(x)).**

**Here are the graphs of f(x),
g(x), and h(x) = (2x - 1) + 9.**

**We can simplify h(x), and then
we see that h(x) = 2x - 1 + 9 = 2x + 8. This is also a linear
function. Does this always happen? Let's look at the general case.**

**Multiplication and addition of
integers are closed. Then**

**Then we get h(x) = sx + u, which
is the general form of a linear function. Therefore, the compostion
of two linears functions is a linear function.**