Two linear functions might be described in different ways: one by an equation, one by a table, one by a graph or in words. You can still compare them.
Why it works
Every linear function has the same two features, however it is shown: a rate of change (the slope) and an initial value (the $y$-intercept). Find those two numbers for each function, and comparing them is just comparing numbers.
Key idea
To compare linear functions in different forms, find each one’s rate of change and initial value: from an equation, read $m$ and $b$; from a table or two points, compute $\frac{\text{change in } y}{\text{change in } x}$ and work back to $x = 0$; from a graph, read the $y$-intercept and count rise over run.
Worked examples
Worked example: Equation and table
Plan A costs $y = 12x + 30$ dollars for $x$ months. Plan B is shown in a table: $$\begin{array}{c|ccc} x & 0 & 2 & 4 \\ \hline y & 10 & 40 & 70 \end{array}$$ Compare the rates and the starting values.
- Plan A: rate 12 dollars per month, starting value 30.
- Plan B: rate $\frac{40 - 10}{2 - 0} = 15$ dollars per month, starting value 10.
- A starts higher; B’s cost grows faster.
Answer: A has the greater initial value; B has the greater rate
Worked example: Words and a graph
Tank C starts with 50 liters and loses 4 liters each minute. Tank D’s graph passes through $(0, 60)$ and $(5, 35)$. Which tank drains faster?
- C: rate $-4$ liters per minute.
- D: rate $\frac{35 - 60}{5 - 0} = -5$ liters per minute.
- D loses 5 per minute and C loses 4, so D drains faster.
Answer: Tank D
Common mistake
Comparing one output instead of the rates. Plan A costs more at the start, but that does not make it grow faster. Compare slopes for “faster” and $y$-intercepts for “starts higher.”