AP Precalculus Flashcards: Polynomial Functions And Rates Of Change

What this deck covers

This deck focuses on Polynomial Functions And Rates Of Change, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: State the derivative of $5x^3 - 2x^2 + x - 8$.

Answer: $15x^2 - 4x + 1$. Apply power rule to each term separately.

Flashcard 2: Find the roots of $x^2 - 4x + 4 = 0$.

Answer: $x = 2$ (multiplicity 2). Perfect square: $(x-2)^2 = 0$ gives repeated root.

Flashcard 3: What is the end behavior of $f(x) = -3x^2 + 5x$?

Answer: As $x \to \pm \infty$, $f(x) \to -\infty$. Even degree with negative leading coefficient.

Flashcard 4: Determine the polynomial degree of $7x^5 - 4x^3 + x - 9$.

Answer:

1. The highest power of $x$ is 5.

Flashcard 5: What is the leading coefficient of $x^4 - 3x^3 + 5x - 2$?

Answer:

1. Coefficient of highest degree term when no coefficient shown is 1.

Flashcard 6: What is the degree of the polynomial $3x^4 + 5x^2 - 7$?

Answer:

1. The highest power of $x$ determines the degree.

Flashcard 7: Which term is the cubic term in $7x^3 + 2x^2 - x + 5$?

Answer: $7x^3$. The term containing $x$ to the third power.

Flashcard 8: Identify the leading coefficient of $6x^3 - 4x + 2$.

Answer:

1. The coefficient of the highest degree term.

Flashcard 9: Find the derivative of $f(x) = x^3 + 2x^2 - x + 1$.

Answer: $3x^2 + 4x - 1$. Apply power rule to each term.

Flashcard 10: What is the rate of change of $f(x) = x^2 + 3$ at $x = 2$?

Answer:

1. Evaluate $f'(2) = 2(2) = 4$ where $f'(x) = 2x$.

Flashcard 11: Identify the quadratic term in $x^2 - 4x + 4$.

Answer: $x^2$. The term containing $x$ to the second power.

Flashcard 12: Which polynomial has a degree of 0?

Answer: Constant polynomials. Non-zero constants have degree 0.

Flashcard 13: What does it mean for a polynomial to be monic?

Answer: Leading coefficient is 1. A monic polynomial has leading coefficient equal to 1.

Flashcard 14: What is the derivative of $f(x) = 7x^4 - 2$?

Answer: $28x^3$. Apply power rule: $4 \cdot 7x^3 = 28x^3$.

Flashcard 15: What is the sum of the roots of $x^2 - 5x + 6 = 0$?

Answer:

1. For $ax^2 + bx + c = 0$, sum of roots is $-\frac{b}{a}$.

Flashcard 16: Which term in $5x^3 + 2x^2 - x + 7$ is the constant term?

Answer:

1. The term with no variable (degree 0).

Flashcard 17: What is the standard form of a polynomial?

Answer: $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$. Terms arranged by decreasing powers of $x$.

Flashcard 18: What is the leading term of $-5x^4 + 3x^2 - x + 8$?

Answer: $-5x^4$. The term with highest power and its coefficient.

Flashcard 19: What is the root multiplicity of $f(x) = (x-2)^3$ at $x=2$?

Answer:

1. The exponent indicates how many times the root repeats.

Flashcard 20: What is the constant term of $4x^3 + 9x - 7$?

Answer: -7. The term with no variable attached.

Flashcard 21: Determine the degree of $f(x) = (x+1)(x-1)(x+3)$.

Answer:

1. Product of three linear factors gives degree 3.

Flashcard 22: What is the coefficient of $x^2$ in $3x^2 + 4x + 5$?

Answer:

1. The numerical factor multiplying $x^2$.

Flashcard 23: What is the vertex form of the quadratic $x^2 + 6x + 9$?

Answer: $(x + 3)^2$. Complete the square or recognize perfect square trinomial.

Flashcard 24: Find the derivative of $f(x) = 3x^2 - 4x + 6$ at $x = 1$.

Answer:

1. Evaluate $f'(1) = 6(1) - 4 = 2$ where $f'(x) = 6x - 4$.

Flashcard 25: Find the sum of coefficients in $f(x) = 2x^3 - 4x + 6$.

Answer:

1. Evaluate $f(1) = 2(1)^3 - 4(1) + 6 = 4$.

Flashcard 26: Determine the y-intercept of $f(x) = x^3 - 4x + 7$.

Answer:

1. Evaluate $f(0) = 0^3 - 4(0) + 7 = 7$.

Flashcard 27: Find the $x$-intercepts of $f(x) = x^2 - 4$.

Answer: $x = 2, x = -2$. Set $f(x) = 0$ and solve for $x$.

Flashcard 28: State the formula for the derivative of $ax^n$.

Answer: $nax^{n-1}$. Power rule: bring down exponent, reduce power by 1.

Flashcard 29: What is the end behavior of $f(x) = 2x^3 - 5x + 1$?

Answer: As $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to -\infty$. Odd degree with positive leading coefficient.

Flashcard 30: Find the product of the roots of $x^2 - 5x + 6 = 0$.

Answer:

1. For $ax^2 + bx + c = 0$, product of roots is $\frac{c}{a}$.

Flashcard 31: Convert $f(x) = (x-1)(x+2)$ to standard form.

Answer: $x^2 + x - 2$. Use FOIL to expand the product.

Flashcard 32: What is the constant term of $(x+2)(x-3)$?

Answer: -6. Expand and identify the constant term.

Flashcard 33: Determine the y-intercept of $f(x) = 7x^2 - 3x + 5$.

Answer:

1. Set $x = 0$ and evaluate the function.

Flashcard 34: Determine the leading term of $2x^4 + 3x^3 - x + 5$.

Answer: $2x^4$. The term with the highest power of $x$.

Flashcard 35: Identify the polynomial with zero leading coefficient.

Answer: Not possible; $a_n \neq 0$ for leading term. A polynomial must have a non-zero leading coefficient.