Introduction to Equations and Graphs in College Algebra
College Algebra
Let’s get something clear. Algebra is not just solving for x. It’s not just shifting numbers around. That’s arithmetic. Real algebra is the foundational language of all higher mathematics. It gives you the mental tools to tackle calculus, physics, …, engineering problems—everything. If you don’t know how graphs work, if you can’t recognize inverse trig functions, if you don’t know synthetic division, long division, partial fractions, … you’re setting yourself up for failure.
Here’s what you need to know if you are serious:
- Algebra is required for everything that comes after. Every advanced math or science course depends on your ability to handle complex algebraic structures and reasoning. You can’t “skip” this and expect to succeed in calculus, physics, or engineering.
- Graph recognition is essential. Know what every function looks like, how to interpret graphs, and how they connect to equations. It’s not optional.
- You must be able to do the hardest problems. Go to the back of each section in your algebra textbook. If you can solve those questions, you’ll be ready for calculus. If not, you’ve got work to do.
- Study habits matter more than GPA. Nobody cares that you got an A if you can’t perform. Passing a course without reading the book isn’t a badge of honor—it’s a red flag. A 4.0 GPA in STEM is the result of a cheat.
- Your degree doesn’t make you smart. You’re in college because you’re not educated yet. Embrace that. Don’t try to impress people. Simply learn.
- Use your time wisely. Practice during breaks, revisit algebra throughout your education, and avoid the trap of thinking you’ve already learned it.
- Face your ego. If you’re doing a science degree to look smart, stop now. Science is about finding truth, not embellishing one’s ego.
This isn’t just advice, it’s a wake-up call. If you treat college algebra like a throwaway course, you’ll regret it. For many students, it’s the first and last warning before they fall behind in every math-heavy course that follows.
If you’re not here to be a scientist, engineer, or researcher—if you’re just passing through—do what you need to do and move on. But if you’re serious, take this to heart: mastering algebra is mastering your future. If you can’t, you’re in the wrong major, or you’re not ready yet. And that’s okay—just be honest about it.
Prepare – Practice – Understand the material—not just the procedure. Open the book. Solve the problems. Own your education. “No one can be taught; they can only choose to learn.”
Summary Checklist for Mastery:
- Learn all function graphs and their behaviors.
- Master all types of factoring and equation solving.
- Understand synthetic division, long division, partial fraction decomposition, … .
- Practice every hard question at the end of textbook sections.
- Study consistently—don’t cram.
- Stop chasing ego, start chasing understanding and acceptance.
NOTE If you can’t do the hardest algebra problems, you’re not ready for calculus. No exceptions.
PLEM vs. STEM
Understanding STEM, CS, and PLEM Before Starting a Degree
Suppose you’re thinking about pursuing a STEM degree. In that case, you need to understand the difference between STEM, CS, and what I call PLEM—Physics, Language, Engineering, and Mathematics. These categories aren’t interchangeable. They represent different levels of depth, seriousness, and expectations. If you don’t get this early, you’ll waste years heading down the wrong path.
Let’s start with STEM. It stands for Science, Technology, Engineering, and Mathematics, but the word(s) are too broad. Every college major touches science or math at some point, but not every significant field demands deep technical thinking. STEM is thrown around like a badge of honor, but most students using the term don’t even understand what it really implies. Choosing a STEM major doesn’t automatically make you smart or job ready. It’s what you study and how seriously you take it that matters.
Now let’s talk about CS. Computer Science gets grouped under STEM, but it often doesn’t require advanced math unless you choose to go further. You can get a CS degree and avoid real mathematics entirely. But if you add linear algebra, differential equations, statistics, and a few upper-level math electives, you’ll become highly employable. Most CS majors don’t do this, which is why so many of them graduate and realize they aren’t helpful in the work force.
Then there’s PLEM. This is the real deal. PLEM majors go deep—physics, math, real engineering, and the one everyone overlooks: language. If you can’t communicate with clarity, you won’t make it. Math is a language. Science is a language. And if you don’t know what the words you’re using actually mean, you’re faking it. You say, “insanity is doing the same thing repeatedly,” but you don’t even know where that comes from. Look it up. Learn to define your terms. Suppose you can’t master the language you speak every day. How will you master the most challenging language humans have ever created—mathematics?
Here’s what you need to hear: if you rely on tutors, YouTube videos, or Khan Academy to get through your degree, you’re not cut out for it. These things may help you pass your homework, but they won’t help you become someone worth hiring. I’ve been there. I made thousands of videos like that a decade ago. They were trash. They helped people pass tests but not graduate with an understanding. I deleted them all because I realized I was assisting people with failure.
You need to learn how to think. That means opening a textbook, defining your words, understanding the laws, properties, and rules that govern your subject. You don’t have time for shortcuts if you’re doing physics, math, or engineering. The further you get in your degree with cheating platforms like Khan Academy, the more likely you will drop out junior year. By the time you reach vector calculus and differential equations, only the real ones are left: the mathematicians, the engineers, and the physicists that will change the world.
Here’s the breakdown:
STEM is a vague umbrella—don't rely on the label.
CS is helpful only if you push yourself beyond the bare minimum.
PLEM is where the serious thinkers live. It's intense, it's language-based, and it's not forgiving.
Shortcut learning doesn't work—it gives you false confidence and no fundamental skills.
The job market doesn't care if you got through college—it cares whether you can actually think and work hard.This isn’t meant to scare you. It’s to wake you up. If you’re serious about learning, you’ll either get on track, or you will change your major. Hopefully, you do this early enough to avoid failing out of college all together. That’s the whole point of my crash course. Not to baby you, but to make you face reality before it’s too late.
If you are a PLEM or CS major or a STEM major that requires calculus in the sense of doing research such as a chemist, do yourself the biggest favor, and delete social media platforms and never visit a .com or watch the news. You will be shocked to find that nothing you do on a daily basis has and truth to it.
Any source that comes from the internet that is not a .edu or .gov, is not a trustworthy or legitimate source. If you cannot cite a legit source or quote (at least verbatim) the definitions of the words you speak, then you don’t know what the F*CK you are talking about. You are like an infant lion attacking a flower pretending it is prey. It is just cute and sad.
A highly learned STEM major knows the definitions of the words they say. They know the sources of the information they speak of. This means, a true scientist doesn’t argue because it is pointless as they no for a fact what the studies actually say. A child says, “I read somewhere” “I’ve got a guy.” “They say.” … You are not scientists if you are arguing with people!
