**Assignment #1**

**Explorations
of Linear Functions**

**by**

**Kim Burrell**

**The study of functions is
taught across the world. Mathematical operations can be applied
to functions. A linear function is often represented by f(x). The
graph of a linear function, f(x) = x, is a straight line (figure
below).**

**The following is a visual
investigation of the linear functions and four mathematical
operations on them. We will now take a look at graphs when the
coefficients of the linear functions are greater than 2.
Explorations will include when the coefficients are both
positive, both negative, and one of each sign (one positive &
one negative).**

**I will use the following
notation:**

**h(x) = f(x) + g(x) ****represents addition,**

**h(x) = f(x)g(x) ****represents multiplication,**

**h(x) = f(x)/g(x)**** represents division, and**

**h(x) = f(g(x))**** represents composition.**

**When the
coefficients are both positive**

**Let's let f(x) = 2x + 4 and
g(x) = 4x + 6. These graphs are shown below. Compare these graphs
with the linear function of f(x) = x. What is different? What is
similar? What type of funtion is formed? If a linear function, is
the slope positive or negative?**

**When the
coefficients are both negative**

**Let's let f(x) = -2x - 4and
g(x) = -4x - 6. These graphs are shown below. Compare these
graphs with the linear function of f(x) = x. What is different?
What is similar? What type of funtion is formed? If a linear
function, is the slope positive or negative?**

**When the
coefficients have different signs**

**Let's let f(x) = -2x + 4and
g(x) = 4x - 6. These graphs are shown below. Compare these graphs
with the linear function of f(x) = x. What is different? What is
similar? What type of funtion is formed? If a linear function, is
the slope positive or negative?**

**Make a conjecture about the functions
f(x) and g(x) with the above mentioned operations performed on
them.**

**How do you think the graphs of****
****f(x) = 2x - 4 and g(x)
= -4x + 6 will look like when the above operations are performed
on them? **

**Summary**

**From the above observations,
you can see that the graphs of functions are different for
different signs of the coefficients.**

**When the coefficients are
both positive****, the
graph by addition is still a linear function with positive slope.
The graph by multiplication is a quadratic function. The graph by
division is a hyperbola. The graph by composition is a linear
function with positive slope, but not the same slope as the graph
by addition.**

**When the coefficients are
both negative****, the
graph by addition is still a linear function with negative slope.
The graph by multiplication is a quadratic function. The graph by
division is a hyperbola. The graph by composition is a linear
function with positive slope.**

**When the coefficients are
different signs****, the
graph by addition is still a linear function with positive slope.
The graph by multiplication is a quadratic function. The graph by
division is a hyperbola. The graph by composition is a linear
function with negative slope.**