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= mnx2 + 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)