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.

\underline{Function A :} y = x + 1 \underline{Function B :} y = \frac{1}{2} x + 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 = 1 x + 1

We can now identify the rate of change of each function.

$the functions y=1x+1 and y = \dfrac{1}{2}x+3 with a flag pointing to the number that describes their rate of change$

The rate of change of Function A is $1$ and the rate of change of Function B is $\frac{1}{2} .$

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 constant term.

$the functions y=1x+1 and y = \dfrac{1}{2}x+3 with a flag pointing to the number that describes their initial value$

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	$\frac{1}{2}$	$3$

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.

graph of the linear functions y=x+1 and y=(1/2)*x+3 plotted on the first quadrant using legends to show their function rules with matching colors

Step 1

Step 2

Comparing the Functions

Step $1$

Step $2$

Exercise