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Both function types contain a straight line on a plane, so if a student asks for the difference between them what would be a standard or recommended way to explain it?

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Mathematically: $y = mx +b$, with $m = 0$ (constant function), or $m \neq 0$ (linear).

Intuitively: discuss something that is responsive or not. Perhaps adding weight to a scale, before (or after) it hits the highest level on the indicator.

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  • $\begingroup$ I find the example about adding a weight to a scale genius --- from constant to linear (by change, not by diagonality). $\endgroup$ Oct 19 '21 at 9:33
  • $\begingroup$ @chichorozov, I see you've accepted this answer. Just a note that you can also give it an upvote. $\endgroup$
    – J W
    Oct 19 '21 at 10:18
  • $\begingroup$ @JW I cannot because I am not registered and will not register. If you like this answer, please up vote it yourself. $\endgroup$ Oct 19 '21 at 12:40
  • $\begingroup$ @chichorozov, ahh, I see. Thanks for clarifying that. $\endgroup$
    – J W
    Oct 19 '21 at 13:23
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A constant function does not vary with the independent variable: its value remains the same no matter what $x$ is (assuming $x$ is the independent variable).

As a formula, a line can be described by $y = ax + b$. The special case of a constant function has $a = 0$, so the formula can be written succinctly as $y = b$.

If you want to bring in the idea of slope, note that a constant function has slope zero and that the graph is a horizontal line.

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