Comparing Linear Functions

Comparing linear functions means looking at two or more linear relationships and determining which one grows faster or starts higher. These functions can be presented in different ways: as equations, tables, graphs, or word problems.

To compare them, you need to identify two key features for each function:

1. Rate of Change (Slope, $m$): This tells you how fast the function is changing. A steeper line has a greater rate of change.
2. Initial Value ($y$-intercept, $b$): This is the starting value of the function, which is the value of $y$ when $x = 0$.

Finding Key Features in Different Forms

• Equations: In the form $y = mx + b$, the slope is $m$ and the $y$-intercept is $b$.
• Tables: Find the rate of change by calculating the change in $y$ divided by the change in $x$ ($\frac{\Delta y}{\Delta x}$). Find the initial value by looking for the $y$-value when $x = 0$.
• Graphs: Find the slope by calculating the "rise over run" between two points. The $y$-intercept is exactly where the line crosses the vertical $y$-axis.

Example: Equation vs. Table

Let's compare two functions to see which has the greater rate of change and which has the greater starting value.

Function A (Equation): $y = 3x + 1$

• Rate of change (slope): $3$
• Starting value ($y$-intercept): $1$

Function B (Table):

$x$	$y$
$0$	$5$
$1$	$7$
$2$	$9$

• Rate of change: As $x$ goes up by $1$, $y$ goes up by $2$. $\text{Slope} = \frac{7 - 5}{1 - 0} = 2$
• Starting value: Look at the table where $x = 0$. The $y$-value is $5$.

The Comparison

• Rate of Change: Function A has a greater rate of change because $3 > 2$. This means Function A grows faster than Function B.
• Starting Value: Function B has a greater starting value because $5 > 1$. Function B starts higher on the $y$-axis than Function A.