**Mathematics
Education**

**EMAT 6680,
Professor Wilson**

**
Exploration 1, Combining Functions by Ursula Kirk**

Make up
a linear function f(x) and g(x). Explore with different pairs of f(x) and g(x)
the graph for:

·

·

·

·

Summarize, explain and illustrate

** Graph Number 1**

For and, we will calculate

;

Now,
we can graph our new function

Here we can observe
that when we add two linear functions, the new function is also a linear
function. Our new function is a straight line with a positive slope.

** Graph Number 2**

For and, we will calculate

);

Now,
we can graph our new function

Here we can observe that when we multiply two linear functions,
the result is a quadratic function. Our new function is a concave up parabola.

** Graph
Number 3**

For and, we will calculate

);

Now, we can graph our new function

Here we can observe that when we
divide two linear functions the result is a rational function. Our new function
has an asymptote at

** Graph
Number 4**

For and, we will calculate

;

Now, we can graph our new function

Here we can observe that when we
find the composite of two linear functions, the result is another linear
function. Our new function is a straight line with a positive slope.

**Exploration
Number 2**

Make up
a linear function f(x) and g(x). Explore with different pairs of f(x) and g(x)
the graph for:

·

·

·

·

Summarize, explain and illustrate

** Graph Number 1**

** **Next, we will repeat the
exploration with a new set of functions

For and, we will calculate

;

Now,
we can graph our new function

Here we can observe
that when we add two linear functions, the new function is also a linear
function. Our new function is a straight line with a negative slope.

** Graph Number 2**

For and, we will calculate

);

Now,
we can graph our new function

Here we can observe that when we multiply two linear functions,
the result is a quadratic function. Our new function is a concave down
parabola.

** Graph
Number 3**

For and, we will calculate

);

Now, we can graph our new function

Here we can observe that when we
divide two linear functions the result is a rational function. Our new function
has an asymptote at .

** Graph
Number 4**

For and, we will calculate

;

Now, we can graph our new function

Here we can observe that when we
find the composite of two linear functions, the result is another linear
function. Our new function is a straight line with a negative slope.

**Exploration
Number 3**

Make up
a linear function f(x) and g(x). Explore with different pairs of f(x) and g(x)
the graph for:

·

·

·

·

Summarize, explain and illustrate

** Graph Number 1**

For and, we will calculate

;

Now,
we can graph our new function

Here we can observe
that when we add two linear functions, the new function is also a linear
function. Our new function is a straight
line with a slope of zero.

** Graph Number 2**

For and, we will calculate

);

Now,
we can graph our new function

Here we can observe that when we multiply two linear functions,
the result is a quadratic function. Our new function is a concave down
parabola.

** Graph
Number 3**

and, we will calculate

);

Now, we can graph our new function

Here we can observe that when we
divide two linear functions the result is a rational function. Our new function
has an asymptote at .

** Graph
Number 4**

For and, we will calculate

;

Now, we can graph our new function

Here we can observe that when we
find the composite of two linear functions, the result is another linear
function. Our new function is a straight line with a negative slope.

**Conclusion**

After completing the three
explorations, we can conclude that:

·
When we add two linear functions, the result
is another linear function.

·
When we multiply two linear functions,
the result is a quadratic function.

·
When we divide two linear functions, the
result is a rational function.

·
When we find the composite of two linear
functions, the result is a linear function.

1. When
we add two linear functions, the result is another linear function.

Given two functions f(x) = mx + b and g(x) = nx + d, we can add them
together so:

h(x) = f(x) + g(x)= (mx + b) + (nx + d)= (mx +
nx) + (b + d)= **(m + n)x + (b + d)**

As we can see, the addition results in a new linear function, having a
slope of m + n and a y-intercept of b + d.

2.
When we multiply two linear functions, the result is a quadratic function.

Given two
functions f(x) = mx + b and g(x) = nx + d, we
can multiply them together so:

f(x)g(x) = (mx + b)(nx
+ d)= mxnx + mxd + bnx + bd= **mnx ^{2} + mdx + bd**

The
coefficients are: a = mn, b = md and c = bd

3.
When we divide two linear functions, the result is a rational function.

Given
two functions f(x) = mx + b and g(x) = nx + d, we can divide them together so:

h(x)
= f(x)/g(x) = **(mx + b)/(nx + d)**

Vertical
Asymptote

nx
+ d = 0

x
= -d/n

Horizontal
Asymptote

yn
- m = 0

y
= m/n

4.
When we find the composite of two linear functions, the result is a linear
function.

Given two functions f(x) = mx + b and g(x) = nx + d, we can find their composition so:

h(x) = f(g(x))= m(nx + d) + b=
mnx + md + b= **(mn)x + (md + b)**

k(x) = g(f(x))= n(mx + b)
+ d= nmx + nb + d= **(nm)x + (nb + d)**