Gaining an intuition for how changes to the inputs of an equation affect the output I am currently taking General Chemistry 2 and Physics 2. I have been doing very well, but I am not just preparing to do well in these classes. I also want to do well on the Medical School Admissions Test or MCAT. On the MCAT, you cannot use a calculator. I have spoken with a few people who performed very well on the test, and they told me that an intuition for algebraic equations is a major factor in a good score. However, I have noticed that I do not possess this.
What I mean by an intuition for algebraic equations is the ability to look at an algebraic equation and know how the output will be affected as a result of an input changing. An example is using the Gibbs Free Energy Equation. If some inputs are negative or positive it automatically changes the output/answer.
I don't know how to go about learning this skill. Is this just a matter of going back to Algebra and learning it more deeply?
 A: If you really feel like going back to the basics, I'd advise downloading Geogebra, and plotting some functions.
For example, create 4 sliders ($a$, $b$, $c$, and $d$) by doing "$a = 1$", etc, in a given cell. Then create a function $f$ by writing "$cos(x)$" in a fifth cell. Finally, create a function $g$ by writing "$af(cx + d) + b$" in a sixth cell.
By changing the sliders, you'll realize that a change to the input of $f$ (by changing $c$ & $d$) affects things over the $x$-axis/horizontally; while changing the output of $f$ (by changing $a$ & $b$) affects things on the $y$-axis, vertically. You'll also see that additions are translations/displacements, while multiplications are scalings/dilations. Vary your $f$ function by using something else, like $exp$, or $x^3$ whatever.
This will give you a raw understanding for real functions ($\mathbb{R} \to \mathbb{R}$).
Now open Geogebra's 3D calculator. Try to invent some functions that are from $\mathbb{R}^2 \to \mathbb{R}$ by writing something along the lines of "$exp(x) + x*y$". Once you've experimented with that, try to have another function of the same kind. See what addition "$x+y$" and multiplication "$x*y$" look like. Look up things like the monkey saddle, etc.
As for functional equations: the points where $f(u) = g(u)$ correspond to the point of intersection between your surfaces. You can also plot the function "f(u) - g(u)": its points that pass by the xy-plane have output zero, and should correspond to the same inputs which cause the intersection.
For general equations, look up quadric equations: how would you draw a sphere centered around a point $p$, or a light-cone centered in $(0, 0, 0)$ in $\mathbb{R}^3$ ?
Once you've done that, study functions from $\mathbb{R} \to \mathbb{R}^2$. One example I like to give my students is $(x, 0.2 x^2, cos(x))$. By looking straight down the y axis and the z axis, you can clearly see how both 0.2x² and cos(x) combine into a common continuous curve.
Finally, understand that every solution set to a differential equation is a foliation: a partition of the Input*Output space (eg, your $\mathbb{R}^2$ plane for functions $\mathbb{R} \to \mathbb{R}$) into non-intersecting curves. The go-to basic example is the solution to the equation $f'(x) = f(x)$, which gives rise to the family of functions which are exponentials of the form $y_0 e^{x - x_0}$. Make $y_0$ and $x_0$ sliders, and try to discern the underlying foliation expressed by this differential equation. Can you see that the curves are non-intersecting ? Can you see that they cover the whole space ?
Once that's done, I think you'll have the creativity to explore the problems you are more familiar with (and that you're more interested in) using Geogebra.
Hope this helps, and best of luck !
