Exploring Linear Functions

Write-up #1

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)

h(x) = f(x) * g(x)

h(x) = f(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.

f(x) = ax + b

g(x) = cx + d

for any real numbers a, b, c,and d.

and h(x) = f (x) + g (x), then h(x) = (ax + b) + (cx + d).

Using the commutative property, this can be rewritten as

h(x) = (ax + cx) + (b + d)

h(x) = (a+c)x + (b + d).

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

(a + c) = s

(b + d) = t

for some real numbers s and t and

h(x) = sx + t which is the general form for linear equations.

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.

f(x) = ax + b

g(x) = cx + d

for any real numbers a, b, c, and d, and

h(x) = f(x) * g(x) then

h(x) =(ax + b) * (cx + d)

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

h(x) = acx^2 + adx + bcx +bd

and using the associative property

h(x) = (ac) x^2 + (ad + bc)x + bd

since integer addition and multiplication are closed,

a * c = e

b * d = i

ac + bd = e + i = k

for some real numbers e, i , and k

h (x) = ex^2 + kx + i

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?

2x-1 =0

2x=1

x=1/2=.5

Let's look at the general case.

f(x) = ax + b

g(x) = cx + d

h(x) = (ax+b)/(cx+d)

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

cx= -d

x = -d/c

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.

f(x) = ax + b

g(x) = cx + d

h(x) = a(cx + d) + b

= acx + ad + b

= acx +(ad + b) for some a, b, c, d that are integers.

Multiplication and addition of integers are closed. Then

ac = s