math

functions

So, a function is like that super loyal friend who never double-texts different things when you ask the same question. In other words, each input (x-value) gets exactly ONE output (y-value).

Why is the second list not a function?

Look at (3, lock) and (3, index). Same x, different y’s?! That’s like your bestie telling one person they love pineapple on pizza and telling someone else they hate it—total contradiction! That’s why the second list isn’t a function. Messy, right?

The Players in This Drama:

  • Domain (x-values, independent) → The people asking the questions.

  • Range (y-values, dependent) → The answers they get.

  • Relation → Any situation where inputs and outputs are connected.

But a function? That’s when there’s NO cheating—each input commits to exactly one output.

Types of Functions:

  • Discrete: Like your group chat, messages (points) are separate, not connected.

  • Continuous: Like scrolling on TikTok, smooth and never-ending.

  • Linear Function: A straight-up, no-nonsense friend—forms a straight line.

Slopes

  • Slope (m) → The energy of the drama (positive means hype, negative means shady).

  • Intercepts → Where the drama crosses the x-axis (x-intercept) or y-axis (y-intercept).

Tea on Transformations:

  • Reflections → Like flipping a rumor—over the x-axis makes everything opposite in attitude, over the y-axis flips the whole perspective.

  • Dilations → Making things louder (steeper) or quieter (flatter).

  • Inverse Relations → Switching roles—flipping (x, y) to (y, x), like when someone suddenly becomes the one getting ghosted instead of doing the ghosting.

The Ultimate Scandal: A Circle Isn’t a Function?!

That’s right! A circle fails the "loyalty test"—one x-value can have multiple y-values. It’s like a two-timing cheater, so we kick it out of the function club.

System Solving

  • Consistent Independent: Just one perfect solution—rare but ideal.

  • Consistent Dependent: They match completely—same slope, same intercept—total twin flames.

  • Inconsistent: Parallel lines, never meet—aka "they were never meant to be."

So, the moral of the story? Functions are all about commitment—if an x-value tries to juggle multiple y-values, that’s function fraud, and we don’t accept that in math or relationships. 🔥💅

correlations

correlations

describe the relationship between two variables, indicating whether they move together (positive correlation) or in opposite directions (negative correlation), much like how partners must align their goals and values for a healthy relationship.

Positive Correlation: The Power Couple 💕

This is when both variables rise together—like two besties who always hype each other up.
📈 Example: The more someone posts on social media, the more likes they get.

  • They’re thriving together!

  • Their energy matches—as one goes up, so does the other.

It’s giving "celebrity and their PR team" vibes—always working together to stay relevant.

Negative Correlation: The Messy Breakup 💔

Here, one variable goes up while the other goes down—like a toxic situationship.
📉 Example: The more time you spend on TikTok, the lower your sleep hours.

  • One is winning, the other is struggling.

  • Just like when one ex moves on and glows up while the other spirals.

A total “when you level up, they start acting different” situation.

No Correlation: The Irrelevant Ex 🙅‍♀

No connection at all—one thing changes, and the other just… doesn’t care.
⚠ Example: The number of iced coffees you drink vs. the number of texts from your crush.

  • They have nothing to do with each other (sadly).

  • No drama, no connection, just two completely separate storylines.

It’s like when two celebs are randomly rumored to be dating, but there’s literally zero evidence. 🚫

In Conclusion:

  • Positive correlation = Ride or die, glowing up together. ✨

  • Negative correlation = One’s up, one’s down, total heartbreak energy. 💔

No correlation = No chemistry, no drama, just vibes. 😴

inverse relations

Inverse Relations

An inverse relation happens when you flip the roles of x and y—like when a celebrity assistant suddenly becomes the star, and the star gets ghosted.

  • If you start with (a, b) and take the inverse, it becomes (b, a).

  • Basically, whoever was in the driver’s seat is now the passenger.

Examples of This Plot Twist in Action:

📌 Before Fame: (Kylie Jenner, "Kim Kardashian’s little sister")
📌 After the Inverse Relation: ("Billionaire CEO," Kylie Jenner)

Oop—power shift detected. 🚨

Math Example (Still Spicy, I Promise! 💅)

Let’s say you have the point (3, -5). If the roles reverse, it becomes (-5, 3).

  • It’s like when someone was totally ignored in high school (3, -5) but then glowed up and now everyone wants to be them (-5, 3).

  • A true revenge arc.

Inverse Functions: The Ultimate Uno Reverse Card 🔄

If a function and its inverse are both functions, they’re like two people who respect the rules—they switch but still keep things in order.

  • Example: y = 2x becomes x = 2y, which we rearrange to y = x/2.

  • They’re still stable, just swapped.

BUT, sometimes, when you take the inverse, it’s no longer a function (scandalous!).

  • Example: The equation of a circle—if you swap x and y, suddenly, it fails the function test.

  • That’s like a couple breaking up and realizing they were never meant to be. 🚫💔

Final Tea on Inverses:

  • An inverse relation flips x and y, changing who’s in charge.

  • If it stays a function, it’s a peaceful power transfer.

  • If it stops being a function, it’s a total messy breakup.

point slope form

Point-Slope Form

Point-slope form is basically when you have tea on one specific point and the slope, and you use it to expose the entire story (aka the equation of the line).
The formula is:
y - y₁ = m(x - x₁)

Translation:
➡ You know where the drama starts (the point (x1,y1)(x₁, y₁)(x1​,y1​))
➡ You know how the drama escalates (the slope, m)
➡ You use that tea to write the whole plot.

Why is it DRAMATIC?

Imagine you know:

  • Where the fight went down (the point)

  • How fast it got messy (the slope)

With just that info, you can write the WHOLE storyline. That’s what point-slope form does—it gives you the receipts. 🧾

Example (Real Housewives Edition):

Let’s say the point is (6, -4) and the slope is 1/9. The equation is: y - (-4) = 1/9(x - 6)
or
y + 4 = 1/9(x - 6)

It’s like saying: "Listen, the drama started at (6, -4), and every time something happened, it escalated by 1/9. Here’s the whole thread!"

Why Do We Use It?

✅ When you have one point and the slope—the point-slope form is your go-to.
✅ It’s a quick way to get the receipts without needing both intercepts.
✅ You can easily turn it into slope-intercept form later, to get the full storyline (y = mx + b) if you want to clean it up.

Tea Summary:

📝 Formula → y - y₁ = m(x - x₁)
📍 You need → One point and the slope
🔥 Purpose → So you can spill the full story with minimal info

converting standard into y intercept

Standard Form

Standard form is like showing up in last season’s outfit—technically fine, but it’s not serving slope and y-intercept realness.
The equation looks like this:
💀 Ax + By = C

  • It’s stiff.

  • It’s clunky.

  • It doesn’t immediately tell us the drama (slope, y-intercept).
    So what do we do? We give it a makeover. 💅

Step 1: Isolate “y” Like You’d Isolate a Fake Friend

The goal is to get it in y = mx + b form, so we need to kick “Ax” to the other side.

Example:

Let’s say we have:
✨ 4x - 3y = 7
➡ That 4x is blocking y from shining, so we subtract it from both sides:
💃 -3y = -4x + 7
Now “y” is still stuck with that -3 … so time to break up.

Step 2: Give “y” Its Solo Era

Right now, “y” is still tied down by -3.
We divide everything by -3 to set it free:
💖 y = (4/3)x - 7/3

🔥 BOOM! Glow-up complete. 🔥
Now we can clearly see:
✅ The slope (m) = 4/3 (How steep the drama escalates)
✅ The y-intercept (b) = -7/3 (Where it all started)

Why Do We Do This?

Because slope-intercept form tells us everything we need to know about the line!

  • We can spill the tea on where it starts (y-intercept)

  • We can track the drama escalation (slope)

  • It’s just cleaner, smoother, and more stylish

Tea Summary ☕🔥

✨ Standard form (Ax + By = C) is stiff and outdated
✨ We move x over so y can shine
✨ We divide to isolate y, completing the glow-up to y = mx + b
✨ Now we can clearly see who’s messy (slope) and where the drama starts (y-intercept)

converting point slope into standard

Point-Slope Form into standard

So we start with the drama-filled equation:
💃 y - y₁ = m(x - x₁)

This form is like someone just dropped the tea but hasn’t cleaned it up yet—it’s all messy and personal (with a specific point in it), and we need to make it look polished for public consumption (aka standard form).

Step 1: Expand the Drama (Distribute the Slope)

Let’s say we start with:
y - 3 = 2(x - 7)

We gotta spill everything out by distributing that 2 across the parentheses:
y - 3 = 2x - 14

Now, we’ve got a clearer storyline, but it’s still not "mathematically classy."

Step 2: Move “x” and “y” to One Side Like a Celeb Couple Breakup

Standard form is Ax + By = C, which means:
✅ x and y on the same side
✅ No fractions, no mess—just clean numbers

So, we subtract y from both sides to move it over:
✨ -3 = 2x - y - 14

Now let’s kick that lonely -14 to the other side (we add 14 to both sides):
✨ 11 = 2x - y

Now we just flip it for aesthetic reasons (so x comes first):
✨ 2x - y = 11

💅 BOOM. The drama has been cleaned up into a classy press statement (Standard Form). 💅

Tea Summary:

1⃣ Point-Slope Form starts as a personal, messy situation.
2⃣ Expand the drama by distributing the slope.
3⃣ Move x and y to the same side—no one should be left out!
4⃣ Make sure everything looks neat and polished.

Now, instead of "he said, she said" energy (point-slope form), we’ve got a well-crafted, PR-approved statement (standard form). ✨

It’s giving "cleaning up receipts before the group chat sees them." 📲🔥

4o

set builder notation

Set-Builder Notation

Instead of writing out every single number, set-builder notation is like saying,
"Only the people who meet my criteria can enter this club."

💃 The format:
x∣condition about x{x | \text{condition about } x}x∣condition about x
(Translation: "The set of all x such that x follows these rules.")

Example 1: The VIP List (x < 3)

x∣x<3{x | x < 3}x∣x<3
📌 Translation: "X can be anything, as long as it’s LESS than 3."
📌 It’s like saying, "Only people younger than 3 hours in the drama can speak on it."
📌 If x = 4, security says NOPE. ❌
📌 If x = 2, welcome to the party! ✅

Example 2: The Strict Guest List (2 ≤ x ≤ 6)

x∣2≤x≤6{x | 2 \leq x \leq 6}x∣2≤x≤6
📌 Translation: "X can be between 2 and 6, including both."
📌 This is like saying, "Only people between 2 and 6 seasons deep in the show can discuss spoilers."
📌 If x = 7, too new, sweetie. No entry. 🚫
📌 If x = 3, you're in! 🎉

Example 3: The Exclusive Inner Circle (x ≠ 5)

x∣x≠5{x | x \neq 5}x∣x=5
📌 Translation: "X can be ANY number except 5."
📌 It’s like saying, "Everyone’s invited, EXCEPT Sarah—she started drama at the last party."
📌 If x = 5, blocked and deleted. ✋
📌 If x = 6, come on in! 😌

Final Tea on Set-Builder Notation:

  • It’s all about setting boundaries—who can and can’t join.

  • It’s the VIP list of math—only certain numbers get in.

  • The rules can be strict or loose, depending on the drama level.

So, next time you see set-builder notation, just think of it as the guest list for the hottest event in town—math edition! 💅✨

system solving

System of Equations

A system of equations is like multiple storylines happening at the same time, and we’re trying to see if:
1⃣ They intersect (one solution, one juicy moment) 💕
2⃣ They’re the same line (infinite solutions, inseparable BFFs) 👯‍♀
3⃣ They never meet (no solution, total drama, parallel lives) 🚫

Type 1: One Solution (The Shocking Plot Twist 😱💖)

👉 The equations meet at one point, meaning there's ONE perfect answer—like when two characters who’ve been flirting all season finally get together!
💅 Math version: The lines intersect at a single point (x, y).
📌 Example:

y=2x+3y = 2x + 3y=2x+3y=−x+7y = -x + 7y=−x+7

📌 The tea? When we solve, we get one solution (x, y). That’s the exact moment the drama unfolds!

🔮 Interpretation: This is like two exes running into each other at a party—their lives only cross once, and it’s messy but important.

Type 2: Infinite Solutions (The Ride-or-Die Duo 👯‍♀💖)

👉 The equations are literally the same, just written differently. These lines overlap forever—like besties who are always on the same page or that one couple that’s been together since high school.
💅 Math version: The equations simplify to something like 3x + y = 5 and 6x + 2y = 10 (which is really the same thing).
📌 The tea? These equations are saying the exact same thing, just in different words—so there are infinite solutions!

🔮 Interpretation: This is like a best friend duo that’s always inseparable, finishing each other’s sentences. No matter what, their paths always align.

Type 3: No Solution (The Total Fallout 🚫💔)

👉 The equations never, EVER intersect—meaning they’re parallel lines and will never be in the same place at the same time.
💅 Math version: The slopes are the same, but the y-intercepts are different.
📌 Example:

y=2x+3y = 2x + 3y=2x+3y=2x−5y = 2x - 5y=2x−5

📌 The tea? They have the same slope but different starting points, so they never meet—they’re running in parallel but separate storylines.

🔮 Interpretation: This is like two former best friends who had a falling out and now just avoid each other completely. No chance of a reunion, ever. 🚷

How Do We Solve These Drama Situations?

1⃣ Substitution (Secret Spilling Method) – Replace one equation into another like sneaky gossip to uncover the truth.
2⃣ Elimination (Drama Cleanup Method) – Add or subtract equations to eliminate a variable and get to the truth faster.
3⃣ Graphing (The Overanalyzing Everything Method) – Literally plot them out to see where the drama unfolds (or doesn’t).

Final Gossip Summary:

  • One solution = A shocking intersection where drama unfolds. (Exes meeting up, unexpected plot twist!)

  • Infinite solutions = Bestie energy, always together. (They finish each other’s sentences!)

  • No solution = Total fallout, parallel lives. (Ghosted forever.)

Math is just one big reality show, and systems of equations are the tangled relationships we live for. 💅✨