Which graphs should I be able to recognize before starting calculus?

Question

Which graphs should I be able to recognize by the time I start calculus?

Here is the list I've compiled so far:

• quadratic
• cubic
• log
• exponent
• $\frac {1}{x^2}$
• sin
• cos
• tan
• normal distribution
• CDF
• Inverses of these where they exist

Any others?

Answer 1

I can say definitively that knowing sine and cosine apart visually has helped me a lot, but you want to link their values at $0$ to the geometric definition of the two. It's worth noting that audio programs like Audacity will usually have sine generators and not cosine generators, even if you can generate a cosine wave through it. Starting audio at a non-zero amplitude will make it play a popping noise, because this is a rapid change - a high-frequency, or high-pitched sound.

Besides what you have there, there's the general pattern of $x^n$ and $\frac{1}{x^n}$ as $n$ alternates between even and odd numbers, which you should be familiar with and be able to justify off the top of your head which shape belongs to the odd $n$ and which to the even. The behavior of $\frac{1}{x}$ on $(0,1]$ and $[1,\infty)$ is very important in itself.

Deeper into calculus, frequency content (sine, cosine, square wave, triangle wave, sawtooth wave) and growth ($x$, $e^x$, $ln(x)$, polynomial, constant) come into play. Most likely you'll be talking about the basics of continuity first, and square waves and $\frac{1}{x}$ are good to keep in mind just because their singularities are found differently, and I noticed some people who don't know about square waves will leap to various assumptions about behavior at singularities.

The biggest trouble is if you aren't up on the singularities or regions of definition of secant, cosecant, tangent, and cotangent. You also want to know what arcsine and arccos look like, where they're defined, and what the logic is for that.

In 2D, $x^2+y^2=1$ should be familiar. It's good to have seen $sin(\frac{1}{x})$ before, because even though the professor or teacher will probably try to draw it, it's not necessarily going to look right. $e^{-x}$ is a good perspective to have. $|x|$ (absolute value) and $\lfloor x \rfloor$ (floor) are good to know visually.