Precalculus Flashcards
Free precalculus flashcards for functions and transformations, polynomials, exponentials and logarithms, conic sections, sequences and series, and complex numbers. Flip each card, mark what you know and drill what you miss.
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Functions and transformations
| Question | Answer |
|---|---|
| Vertical shift of y = f(x) + k? | Up k units if k > 0, down if k < 0. |
| Horizontal shift of y = f(x − h)? | Right h units if h > 0. |
| Reflection over the x-axis? | y = −f(x) |
| Reflection over the y-axis? | y = f(−x) |
| Vertical stretch by factor a? | y = a f(x) |
| Composition of functions? | (f ∘ g)(x) = f(g(x)) |
| How do you find an inverse function? | Swap x and y, then solve for y. |
| Horizontal line test? | A function has an inverse function if no horizontal line crosses its graph more than once. |
| Even function? | f(−x) = f(x); symmetric about the y-axis. |
| Odd function? | f(−x) = −f(x); symmetric about the origin. |
| Average rate of change on [a, b]? | (f(b) − f(a)) / (b − a) |
Polynomials and rational functions
| Question | Answer |
|---|---|
| Quadratic formula? | x = [−b ± √(b² − 4ac)] / 2a |
| Discriminant? | b² − 4ac. Positive: two real roots. Zero: one. Negative: two complex roots. |
| Vertex of y = ax² + bx + c? | x = −b / 2a |
| Remainder theorem? | The remainder of f(x) ÷ (x − c) is f(c). |
| Factor theorem? | (x − c) is a factor of f(x) if and only if f(c) = 0. |
| Fundamental theorem of algebra? | A degree-n polynomial has exactly n complex roots, counting multiplicity. |
| How do you find a vertical asymptote of a rational function? | Set the denominator to zero (after cancelling common factors). |
| Horizontal asymptote when degree of top < degree of bottom? | y = 0 |
Exponentials and logarithms
| Question | Answer |
|---|---|
| Definition of logₐ x = y? | aʸ = x |
| logₐ(xy) = ? | logₐ x + logₐ y |
| logₐ(x/y) = ? | logₐ x − logₐ y |
| logₐ(xⁿ) = ? | n logₐ x |
| Change of base formula? | logₐ x = ln x / ln a |
| logₐ 1 = ? | 0 |
| logₐ a = ? | 1 |
| Compound interest formula? | A = P(1 + r/n)ⁿᵗ |
| Continuous compounding? | A = Peʳᵗ |
| Exponential growth model? | y = a bˣ with b > 1 |
| What is e, approximately? | 2.71828 |
Conic sections
| Question | Answer |
|---|---|
| Equation of a circle? | (x − h)² + (y − k)² = r² |
| Equation of an ellipse centered at the origin? | x²/a² + y²/b² = 1 |
| Equation of a hyperbola centered at the origin (opens left/right)? | x²/a² − y²/b² = 1 |
| Parabola with vertex (h, k) opening up or down? | y = a(x − h)² + k |
| For an ellipse, how are a, b and c related? | c² = a² − b² (a is the larger semi-axis) |
| For a hyperbola, how are a, b and c related? | c² = a² + b² |
Sequences and series
| Question | Answer |
|---|---|
| nth term of an arithmetic sequence? | aₙ = a₁ + (n − 1)d |
| Sum of the first n terms of an arithmetic series? | Sₙ = n(a₁ + aₙ) / 2 |
| nth term of a geometric sequence? | aₙ = a₁ rⁿ⁻¹ |
| Sum of the first n terms of a geometric series? | Sₙ = a₁(1 − rⁿ) / (1 − r) |
| Sum of an infinite geometric series? | S = a₁ / (1 − r) when |r| < 1 |
| Sigma notation Σ? | The sum of the terms of an expression over a range of values. |
| Binomial theorem? | (a + b)ⁿ = Σ nCₖ aⁿ⁻ᵏ bᵏ |
Complex numbers
| Question | Answer |
|---|---|
| What is i? | The imaginary unit, with i² = −1. |
| Standard form of a complex number? | a + bi |
| Modulus of a + bi? | √(a² + b²) |
| Complex conjugate of a + bi? | a − bi |
| Product of a complex number and its conjugate? | a² + b² |
| Polar form of a complex number? | r(cos θ + i sin θ) |
| De Moivre’s theorem? | [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ) |
| i³ = ? and i⁴ = ? | −i and 1 |
Frequently asked questions
What topics are in precalculus?
Precalculus typically covers functions, polynomial and rational functions, exponentials and logarithms, trigonometry, conic sections, sequences and series, and sometimes vectors, matrices and limits.
Is precalculus needed before calculus?
Most calculus courses expect strong algebra, function and trigonometry skills, which precalculus reviews and extends.
Sources
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