All formulas you need for the Precalc Review Test. Each formula is shown in textbook style and how you would write it on paper.
Circles
Standard Form of a Circle
Write: (x - h)² + (y - k)² = r²
Center at (h, k) with radius r
Distance Formula (Radius)
Write: r = sqrt((x₂ - x₁)² + (y₂ - y₁)²)
Distance between two points = radius if one is the center
Completing the Square
Write: x² + bx → (x + b/2)² - (b/2)²
Take half of b, square it. Add and subtract to convert general form to standard form.
Functions
Function Composition
Write: (f ∘ g)(x) = f(g(x))
Replace every x in f with the entire g(x) expression
Difference Quotient
Write: [f(x+h) - f(x)] / h
Measures average rate of change. Foundation for derivatives.
Even Function
Write: f(-x) = f(x)
Symmetric about the y-axis. All exponents are even.
Odd Function
Write: f(-x) = -f(x)
Symmetric about the origin. All exponents are odd.
Inverse Function
Steps: 1) Replace f(x) with y 2) Swap x and y 3) Solve for y
Domain of f⁻¹ = Range of f, and Range of f⁻¹ = Domain of f
Quadratics & Parabolas
Quadratic Formula
Write: x = (-b ± sqrt(b² - 4ac)) / (2a)
Solves ax² + bx + c = 0. Works for any quadratic.
Vertex Formula
Write: x = -b / (2a)
x-coordinate of the vertex for f(x) = ax² + bx + c. Plug back in for y.
Vertex Form
Write: f(x) = a(x - h)² + k
Vertex at (h, k). a > 0 opens up, a < 0 opens down.
Discriminant
Write: Δ = b² - 4ac
Δ > 0: two real roots | Δ = 0: one repeated root | Δ < 0: two complex roots
Polynomials
Rational Zeros Theorem
Write: ±(factors of constant term) / (factors of leading coefficient)
p = factors of constant term, q = factors of leading coefficient. List all ±p/q combinations.
Remainder Theorem
Write: f(c) = remainder of f(x) ÷ (x - c)
To evaluate f at c, just do synthetic division by c. The remainder IS f(c).
Complex Conjugate Theorem
Write: If a+bi is a root → a-bi is also a root
For polynomials with real coefficients. Complex roots always come in conjugate pairs.
Factor Theorem
Write: (x - c) is a factor ↔ f(c) = 0
If plugging c into f gives 0, then (x - c) divides f evenly.
Equations & Inequalities
Linear Inequality Rule
Key rule: Multiply/divide by negative → flip < to > (or ≤ to ≥)
Solve like a normal equation, just watch the direction when dividing by negatives.
Quadratic Inequality
Steps: 1) Find roots r₁, r₂ 2) Plot on number line 3) Test a point in each interval
The sign of the expression can only change at the roots.
Graph Transformations
Vertical Shift
Write: y = f(x) + k
+k shifts UP, -k shifts DOWN
Horizontal Shift
Write: y = f(x - h)
Positive h shifts RIGHT, negative h shifts LEFT (opposite of what you'd expect!)
Vertical Stretch / Compression
Write: y = a · f(x)
|a| > 1: stretch (taller/narrower) | |a| < 1: compress (shorter/wider)
Reflections
Write: -f(x) = reflect over x-axis | f(-x) = reflect over y-axis
Negative outside flips up/down. Negative inside flips left/right.
Full Transformation Template
Write: y = a · f(x - h) + k → a = stretch/reflect, h = horizontal, k = vertical
Read transformations in order: horizontal shift, stretch/reflect, vertical shift.
Synthetic Division
How It Works: (2x⁴ - 4x + 5) ÷ (x - 3)
Step 1: Write coefficients. Include 0 for missing terms (x³ and x² are missing).
Step 2: Bring down the first coefficient.
Step 3: Multiply by c (=3), write below next column, add. Repeat.
Step 4: Last number = remainder. Others = quotient coefficients.
| 3 | 2 | 0 | 0 | -4 | 5 |
| 6 | 18 | 54 | 150 | ||
| 2 | 6 | 18 | 50 | 155 |
Result: Quotient = 2x³ + 6x² + 18x + 50, Remainder = 155