Polynomial Functions

Polynomials are the most common functions you'll encounter in Classic Mode. They are formed by adding integer powers of $x$.

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$

In Function Guessr, determining the degree ($n$) and the leading coefficient ($a_n$) is half the battle.

Common Types

1. Linear Functions ($n=1$)

$f(x) = mx + b$

• Shape: Straight line.

• Strategy: Two points are enough to define it.
  • Find the y-intercept ($b$) by checking $x=0$.
  • Find the slope ($m$) by calculating "rise over run".

2. Quadratic Functions ($n=2$)

$f(x) = ax^2 + bx + c$

• Shape: Parabola (U-shape).

• Strategy:
  • Concavity: If it opens up, $a > 0$. If down, $a < 0$.
  • Vertex: The turning point is at $x = -b/2a$.
  • Roots: Where does it cross the x-axis? If it touches at one point, it's a perfect square like $(x-k)^2$.

3. Cubic Functions ($n=3$)

$f(x) = ax^3 + bx^2 + cx + d$

• Shape: "S" shape. Starts low and ends high (if $a>0$), or vice versa.

• Strategy: Look for up to 3 roots. The inflection point is key.

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🧠 Advanced Guessing Strategies

Strategy 1: The "Limit Test" for Degree & Leading Coefficient

How do you know if it's $x^2$, $x^3$, or $x^4$? Look at the End Behavior.

1. Zoom Out: Use the graph tool to look at large $x$ values (e.g., $x=10, x=100$).
2. Divide by $x^n$: If you suspect the degree is $n$, try evaluating $f(x) / x^n$ for a large $x$.
   • If the result converges to a non-zero constant, that constant is your leading coefficient ($a_n$).
   • If it goes to $0$, the degree is lower than $n$.
   • If it goes to $\infty$, the degree is higher than $n$.

Example: You see a U-shape. You guess it's quadratic ($x^2$). Check $f(100)$. If $f(100) \approx 20000$, then $f(100) / 100^2 = 20000 / 10000 = 2$. The function likely starts with $2x^2$.

Strategy 2: Root Finding ($f(x)=0$)

In Function Guessr, coefficients are often simple integers. This makes the roots (x-intercepts) extremely powerful clues.

• If the graph crosses the x-axis at $x = 2$, then $(x-2)$ is likely a factor.
• If it crosses at $x = -3$ and $x = 2$, try multiplying $(x+3)(x-2)$.
• combine this with the leading coefficient found in Strategy 1.

$f(x) = a \cdot (x - \text{root}_1) (x - \text{root}_2) \dots$

Strategy 3: Parity (Symmetry)

• Even Function ($f(-x) = f(x)$): Symmetric across the Y-axis. Contains only even powers (e.g., $x^2, x^4, \text{constants}$).
• Odd Function ($f(-x) = -f(x)$): Point symmetric about the origin $(0,0)$. Contains only odd powers (e.g., $x, x^3$).

Use this to eliminate half the possible terms instantly!