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When should the label of the vertical axis be $y$, and when should it be $f(x)$? As sub questions, if I'm graphing $f(x) = 2x+1$, would it be wrong to label the vertical axis as $y$, since $y$ appears nowhere in my equation? What if I wanted to compare $f(x)$ and $g(x)$? Or $f(a)$ and $g(b)$? Or $sin(\theta)$ and $cos(\theta)$? Really struggling to come up with a bp. Thanks in advance for any thoughts!

Elem-Teach-w-Bach-n-Math-Ed
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When drawing multiple curves on the same Cartesian plane because they represent functions $f(t),g(t),h(t),$ etc. that share an input/independent variable $t,$ the vertical axis can be labelled the dependent variable $y,$ and the curves separately labelled $“y=f(t)”, “y=g(t)”, “y=h(t)”,$ etc.

When there is just a single function $h(t),$ the vertical axis can alternatively be labelled $h(t)$ so that the curve doesn't need to be labelled.

In either case, the horizontal axis is labelled the input/independent variable $t.$

ryang
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It is okay to label the vertical axis as $y$ if the horizontal axis is labeled $x$ and $y = f(x)$. To compare two functions, say $f(x)$ and $g(x)$, both can be plotted in the same graph and labeling those functions through annotations or legends.

Note that $x$ and $y$ are just dummy variables. They can be replaced with anything, like $f = g(h)$, or $x = X(y)$. This means that you can compare $\sin\theta$ and $\cos\theta$ like the way you would in the previous paragraph.

However, to compare $f(a)$ and $g(b)$ would need additional information. Assuming that $f$ and $g$ are functions of $x$, comparing them would be similar to the first paragraph, just that we are tracking specific values instead of the entire function.

soupless
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  • By "dummy variables," you mean that by labeling the axes $x$ and $y$ we're simply indicating that the horizontal axis is inputs and the vertical is outputs? Therefore, comparing $sin(\theta)$ and $cos(\theta)$ on graphs labeled $x$ and $y$ wouldn't be wrong even though using $\theta$ and $y$ might be preferable? – Elem-Teach-w-Bach-n-Math-Ed Jan 05 '22 at 17:01
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    It would be wrong because the variable used in $\sin\theta$ and $\cos\theta$ is $\theta$, and the axes are $x$ and $y$, neither of which are $\theta$. To fix it, you can label the horizontal axis as $\theta$. The label for the vertical axis would be anything you desire, so long as the functions are set to it.

    For example, labeling the vertical axis as $x$, this would mean that $x = \sin\theta$. Also, $x = \cos\theta$. Note that we are not solving for $\sin\theta = \cos\theta$, just abusing notation for functions.

     – soupless Jan 05 '22 at 17:04