We are asked how to compare two functions. There are two steps for comparing linear functions. Let's use the following functions as an example.
Function A:
y=x+1 [0.9em]
Function B:
y=1/2x+3
Step 1
The first step to find the rate of change of each function. When the function is given as a linear equation, the rate of change is the coefficient of x. If a variable does not have a coefficient written next to it, it is assumed that it is 1.
y=x+1 ⇓
y= 1x+1
We can now identify the rate of change of each function.
The rate of change of Function A is 1 and the rate of change of Function B is 12.
Step 2
The second step is to find the initial value of each function. When the function is given as a linear equation, the initial value is the constantterm.
The initial value of Function A is 1 and the initial value of Function B is 3.
Comparing the Functions
We can summarize our findings in the following table.
Rate of Change
Initial Value
Function A
1
1
Function B
1/2
3
Since Function B has a greater initial value, its y-intercept is greater than Function A's y-intercept. However, Function A has a greater rate of change, so its y-values will eventually be greater than those of Function B.
