class 051007
What are the basic points of the basic linear, quadratic and power functions?
The first graph below represents a linear function, the second represents a quadratic function and the third represent a p = -1 power function.
What are the basic points for y=f(x)?
Difference Equations
f(0) = 2
f(1) = 5
f(2) = 8
f(3) = 11
etc.
The pattern here goes up by 3 and starts at 2. In more precise language we say
f(0) = 2
f(n+1) = f(n) + 3.
To use this, we first let n = 0.
Substituting 0 for n into f(n+1) = f(n) + 3 we get
f(0+1) = f(0) + 3
f(1) = 2 + 3
f(1) = 5.
Next, we let n = 1. Substituting 1 for n into f(n+1) = f(n) + 3 we get
f(1 + 1) = f(1) + 3
f(2) = f(1) + 3. Since f(1) = 5 from the previous step:
f(2) = 5 + 3
f(2) = 8.
Next, we let n = 2. Substituting 2 for n into f(n+1) = f(n) + 3 we get
f(2 + 1) = f(2) + 3
f(3) = f(2) + 3. Since f(2) = 8 from the previous step:
f(3) = 11.
It's clear that this process just keeps adding 3 to the previous value to get the next value.
What linear function f(x) would give us the same values?
A graph of f(n) vs. n goes up 3 for every unit of run so its slope is 3.
The graph goes through (0, 2) so the y intercept is 2.
The function is therefore y = m x + b with m = 3 and b = 2.
The function is y = 3x + 2.
The linear function y = f(x) = m x + b can be modeled by the equations
f(0) = b
f(n+1) = f(n) + m.