AP Precalculus Flashcards: Polynomial Functions And End Behavior

Study Polynomial Functions And End Behavior in AP Precalculus with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Precalculus

Polynomial Functions And End Behavior

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QUESTION

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What is the end behavior of $f(x) = -4x^3 + 2x - 5$ as $x \to -\text{infinity}$ ?

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ANSWER

$f(x) \to \text{infinity}$ . Odd degree with negative leading coefficient: $x \to -\infty$ gives $f(x) \to +\infty$ .

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What this deck covers

This deck focuses on Polynomial Functions And End Behavior, 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.

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Flashcard 1: What is the end behavior of $f(x) = -4x^3 + 2x - 5$ as $x \to -\text{infinity}$ ?

Answer: $f(x) \to \text{infinity}$ . Odd degree with negative leading coefficient: $x \to -\infty$ gives $f(x) \to +\infty$ .

Flashcard 2: What is the end behavior of $f(x) = x^7 - x^5 + x$ as $x \to \text{infinity}$ ?

Answer: $f(x) \to \text{infinity}$ . Odd degree with positive leading coefficient goes to $+\infty$ .

Flashcard 3: State the polynomial type for $f(x) = x^3 - 4x + 2$ .

Answer: Cubic polynomial. Degree 3 polynomials are called cubic.

Flashcard 4: What is the end behavior of $f(x) = 3x^2 - 7x + 4$ as $x \to \text{infinity}$ ?

Answer: $f(x) \to \text{infinity}$ . Even degree with positive leading coefficient goes to $+\infty$ .

Flashcard 5: Determine the multiplicity of root $x = -1$ in $f(x) = (x+1)^2(x-2)$ .

Answer:

The exponent on factor $(x+1)$ .

Flashcard 6: What is the end behavior of $f(x) = -3x^4 + 6x^2 - 9$ as $x \to \text{infinity}$ ?

Answer: $f(x) \to -\text{infinity}$ . Even degree with negative leading coefficient goes to $-\infty$ .

Flashcard 7: What is the y-intercept of $f(x) = 2x^4 - 3x^2 + 7$ ?

Answer:

Evaluate $f(0)$ to find y-intercept.

Flashcard 8: What is the end behavior of $f(x) = 5x^2 - 3x + 1$ as $x \to \text{infinity}$ ?

Answer: $f(x) \to \text{infinity}$ . Even degree with positive leading coefficient goes to $+\infty$ .

Flashcard 9: State the end behavior of $f(x) = -x^6 + 2x^4 - 5$ as $x \to \text{infinity}$ .

Answer: $f(x) \to -\text{infinity}$ . Even degree with negative leading coefficient goes to $-\infty$ .

Flashcard 10: Identify the y-intercept in $f(x) = 6x^4 - x^2 + 2$ .

Answer:

Evaluate $f(0)$ to find y-intercept.

Flashcard 11: What is the leading coefficient of $f(x) = 10x^3 - 5x^2 + 7$ ?

Answer:

The coefficient of the highest degree term.

Flashcard 12: What is the axis of symmetry for $f(x) = x^2 - 4x + 3$ ?

Answer: $x = 2$ . For quadratic $ax^2 + bx + c$ , axis is $x = -\frac{b}{2a}$ .

Flashcard 13: What is the leading term of $f(x) = x^8 - 3x^5 + 2x - 9$ ?

Answer: x^8. The term with highest degree and its coefficient.

Flashcard 14: Identify the constant term in $f(x) = 4x^3 - x^2 + 6x - 9$ .

Answer: -9. The term without any variable.

Flashcard 15: State the end behavior of $f(x) = x^6 - 4x^3 + 5$ as $x \to -\text{infinity}$ .

Answer: $f(x) \to \text{infinity}$ . Even degree with positive leading coefficient goes to $+\infty$ .

Flashcard 16: State the y-intercept of $f(x) = -3x^3 + 4x^2 - x + 6$ .

Answer:

Evaluate $f(0)$ to find y-intercept.

Flashcard 17: What is the degree of $f(x) = 4x^5 - 9$ ?

Answer:

The highest power of x.

Flashcard 18: Determine the degree of $f(x) = 9x^6 - 7x^4 + 3x^2$ .

Answer:

The highest power of x.

Flashcard 19: Find the leading coefficient of $f(x) = 7x^5 - 4x^3 + x^2 - 8$ .

Answer:

The coefficient of the highest degree term.

Flashcard 20: What is the end behavior of $f(x) = -x^5 + 3x^2 - 7$ as $x \to -\text{infinity}$ ?

Answer: $f(x) \to \text{infinity}$ . Odd degree with negative leading coefficient: $x \to -\infty$ gives $f(x) \to +\infty$ .

Flashcard 21: What is the degree of $f(x) = -x^2 + 4x - 1$ ?

Answer:

The highest power of x.

Flashcard 22: What is the degree of the polynomial $f(x) = 3x^4 - 5x^3 + 2x^2 - x + 7$ ?

Answer:

The highest power of x determines the degree.

Flashcard 23: State the number of real roots for $f(x) = x^4 - 1$ .

Answer:

Factor as $(x^2-1) = (x-1)(x+1)$ to find roots $\pm 1$ .

Flashcard 24: What is the degree of $f(x) = x^7 - 2x^4 + x^2$ ?

Answer:

The highest power of x.

Flashcard 25: State the end behavior of $f(x) = x^9 - x^5$ as $x \to -\text{infinity}$ .

Answer: $f(x) \to -\text{infinity}$ . Odd degree with positive leading coefficient: $x \to -\infty$ gives $f(x) \to -\infty$ .

Flashcard 26: Identify the constant term in $f(x) = 5x^3 - 2x + 7$ .

Answer:

The term without any variable.

Flashcard 27: State the polynomial function given roots $x = 1$ , $x = -2$ , and $x = 3$ .

Answer: $(x-1)(x+2)(x-3)$ . Use $(x-r)$ for each root r.

Flashcard 28: Identify the degree of polynomial $f(x) = -5x^7 + 2x^4 - x$ .

Answer: $7$ . The highest power of x.

Flashcard 29: State the leading coefficient of $f(x) = -8x^4 + 2x^3 - x + 1$ .

Answer: -8. The coefficient of the highest degree term.

Flashcard 30: State the constant term in $f(x) = 3x^5 - 5x^3 + 2$ .

Answer:

The term without any variable.

Flashcard 31: Find the end behavior of $f(x) = 7x^2 - 4x + 1$ as $x \to -\text{infinity}$ .

Answer: $f(x) \to \text{infinity}$ . Even degree with positive leading coefficient goes to $+\infty$ .

Flashcard 32: Identify the leading term in $f(x) = 6x^7 - 3x^5 + 2x - 8$ .

Answer: 6x^7. The term with highest degree and its coefficient.

Flashcard 33: Determine the multiplicity of root $x = 2$ in $f(x) = (x-2)^3(x+1)$ .

Answer:

The exponent on factor $(x-2)$ .

Flashcard 34: State the leading coefficient of $f(x) = -2x^3 + 4x^2 - x + 5$ .

Answer: -2. The coefficient of the highest degree term.

Flashcard 35: What is the end behavior of $f(x) = 2x^3 - 4x + 1$ as $x \to \text{infinity}$ ?

Answer: $f(x) \to \text{infinity}$ . Odd degree with positive leading coefficient goes to $+\infty$ .

AP Precalculus Flashcards: Polynomial Functions And End Behavior

Polynomial Functions And End Behavior

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: What is the end behavior of f(x)=−4x3+2x−5f(x) = -4x^3 + 2x - 5f(x)=−4x3+2x−5 as x→−infinityx \to -\text{infinity}x→−infinity?

Flashcard 2: What is the end behavior of f(x)=x7−x5+xf(x) = x^7 - x^5 + xf(x)=x7−x5+x as x→infinityx \to \text{infinity}x→infinity?

Flashcard 3: State the polynomial type for f(x)=x3−4x+2f(x) = x^3 - 4x + 2f(x)=x3−4x+2.

Flashcard 4: What is the end behavior of f(x)=3x2−7x+4f(x) = 3x^2 - 7x + 4f(x)=3x2−7x+4 as x→infinityx \to \text{infinity}x→infinity?

Flashcard 5: Determine the multiplicity of root x=−1x = -1x=−1 in f(x)=(x+1)2(x−2)f(x) = (x+1)^2(x-2)f(x)=(x+1)2(x−2).

Flashcard 6: What is the end behavior of f(x)=−3x4+6x2−9f(x) = -3x^4 + 6x^2 - 9f(x)=−3x4+6x2−9 as x→infinityx \to \text{infinity}x→infinity?

Flashcard 7: What is the y-intercept of f(x)=2x4−3x2+7f(x) = 2x^4 - 3x^2 + 7f(x)=2x4−3x2+7?

Flashcard 8: What is the end behavior of f(x)=5x2−3x+1f(x) = 5x^2 - 3x + 1f(x)=5x2−3x+1 as x→infinityx \to \text{infinity}x→infinity?

Flashcard 9: State the end behavior of f(x)=−x6+2x4−5f(x) = -x^6 + 2x^4 - 5f(x)=−x6+2x4−5 as x→infinityx \to \text{infinity}x→infinity.

Flashcard 10: Identify the y-intercept in f(x)=6x4−x2+2f(x) = 6x^4 - x^2 + 2f(x)=6x4−x2+2.

Flashcard 11: What is the leading coefficient of f(x)=10x3−5x2+7f(x) = 10x^3 - 5x^2 + 7f(x)=10x3−5x2+7?

Flashcard 12: What is the axis of symmetry for f(x)=x2−4x+3f(x) = x^2 - 4x + 3f(x)=x2−4x+3?

Flashcard 13: What is the leading term of f(x)=x8−3x5+2x−9f(x) = x^8 - 3x^5 + 2x - 9f(x)=x8−3x5+2x−9?

Flashcard 14: Identify the constant term in f(x)=4x3−x2+6x−9f(x) = 4x^3 - x^2 + 6x - 9f(x)=4x3−x2+6x−9.

Flashcard 15: State the end behavior of f(x)=x6−4x3+5f(x) = x^6 - 4x^3 + 5f(x)=x6−4x3+5 as x→−infinityx \to -\text{infinity}x→−infinity.

Flashcard 16: State the y-intercept of f(x)=−3x3+4x2−x+6f(x) = -3x^3 + 4x^2 - x + 6f(x)=−3x3+4x2−x+6.

Flashcard 17: What is the degree of f(x)=4x5−9f(x) = 4x^5 - 9f(x)=4x5−9?

Flashcard 18: Determine the degree of f(x)=9x6−7x4+3x2f(x) = 9x^6 - 7x^4 + 3x^2f(x)=9x6−7x4+3x2.

Flashcard 19: Find the leading coefficient of f(x)=7x5−4x3+x2−8f(x) = 7x^5 - 4x^3 + x^2 - 8f(x)=7x5−4x3+x2−8.

Flashcard 20: What is the end behavior of f(x)=−x5+3x2−7f(x) = -x^5 + 3x^2 - 7f(x)=−x5+3x2−7 as x→−infinityx \to -\text{infinity}x→−infinity?

Flashcard 21: What is the degree of f(x)=−x2+4x−1f(x) = -x^2 + 4x - 1f(x)=−x2+4x−1?

Flashcard 22: What is the degree of the polynomial f(x)=3x4−5x3+2x2−x+7f(x) = 3x^4 - 5x^3 + 2x^2 - x + 7f(x)=3x4−5x3+2x2−x+7?

Flashcard 23: State the number of real roots for f(x)=x4−1f(x) = x^4 - 1f(x)=x4−1.

Flashcard 24: What is the degree of f(x)=x7−2x4+x2f(x) = x^7 - 2x^4 + x^2f(x)=x7−2x4+x2?

Flashcard 25: State the end behavior of f(x)=x9−x5f(x) = x^9 - x^5f(x)=x9−x5 as x→−infinityx \to -\text{infinity}x→−infinity.

Flashcard 26: Identify the constant term in f(x)=5x3−2x+7f(x) = 5x^3 - 2x + 7f(x)=5x3−2x+7.

Flashcard 27: State the polynomial function given roots x=1x = 1x=1, x=−2x = -2x=−2, and x=3x = 3x=3.

Flashcard 28: Identify the degree of polynomial f(x)=−5x7+2x4−xf(x) = -5x^7 + 2x^4 - xf(x)=−5x7+2x4−x.

Flashcard 29: State the leading coefficient of f(x)=−8x4+2x3−x+1f(x) = -8x^4 + 2x^3 - x + 1f(x)=−8x4+2x3−x+1.

Flashcard 30: State the constant term in f(x)=3x5−5x3+2f(x) = 3x^5 - 5x^3 + 2f(x)=3x5−5x3+2.

Flashcard 31: Find the end behavior of f(x)=7x2−4x+1f(x) = 7x^2 - 4x + 1f(x)=7x2−4x+1 as x→−infinityx \to -\text{infinity}x→−infinity.

Flashcard 32: Identify the leading term in f(x)=6x7−3x5+2x−8f(x) = 6x^7 - 3x^5 + 2x - 8f(x)=6x7−3x5+2x−8.

Flashcard 33: Determine the multiplicity of root x=2x = 2x=2 in f(x)=(x−2)3(x+1)f(x) = (x-2)^3(x+1)f(x)=(x−2)3(x+1).

Flashcard 34: State the leading coefficient of f(x)=−2x3+4x2−x+5f(x) = -2x^3 + 4x^2 - x + 5f(x)=−2x3+4x2−x+5.

Flashcard 35: What is the end behavior of f(x)=2x3−4x+1f(x) = 2x^3 - 4x + 1f(x)=2x3−4x+1 as x→infinityx \to \text{infinity}x→infinity?

Flashcard 1: What is the end behavior of $f(x) = -4x^3 + 2x - 5$ as $x \to -\text{infinity}$ ?

Flashcard 2: What is the end behavior of $f(x) = x^7 - x^5 + x$ as $x \to \text{infinity}$ ?

Flashcard 3: State the polynomial type for $f(x) = x^3 - 4x + 2$ .

Flashcard 4: What is the end behavior of $f(x) = 3x^2 - 7x + 4$ as $x \to \text{infinity}$ ?

Flashcard 5: Determine the multiplicity of root $x = -1$ in $f(x) = (x+1)^2(x-2)$ .

Flashcard 6: What is the end behavior of $f(x) = -3x^4 + 6x^2 - 9$ as $x \to \text{infinity}$ ?

Flashcard 7: What is the y-intercept of $f(x) = 2x^4 - 3x^2 + 7$ ?

Flashcard 8: What is the end behavior of $f(x) = 5x^2 - 3x + 1$ as $x \to \text{infinity}$ ?

Flashcard 9: State the end behavior of $f(x) = -x^6 + 2x^4 - 5$ as $x \to \text{infinity}$ .

Flashcard 10: Identify the y-intercept in $f(x) = 6x^4 - x^2 + 2$ .

Flashcard 11: What is the leading coefficient of $f(x) = 10x^3 - 5x^2 + 7$ ?

Flashcard 12: What is the axis of symmetry for $f(x) = x^2 - 4x + 3$ ?

Flashcard 13: What is the leading term of $f(x) = x^8 - 3x^5 + 2x - 9$ ?

Flashcard 14: Identify the constant term in $f(x) = 4x^3 - x^2 + 6x - 9$ .

Flashcard 15: State the end behavior of $f(x) = x^6 - 4x^3 + 5$ as $x \to -\text{infinity}$ .

Flashcard 16: State the y-intercept of $f(x) = -3x^3 + 4x^2 - x + 6$ .

Flashcard 17: What is the degree of $f(x) = 4x^5 - 9$ ?

Flashcard 18: Determine the degree of $f(x) = 9x^6 - 7x^4 + 3x^2$ .

Flashcard 19: Find the leading coefficient of $f(x) = 7x^5 - 4x^3 + x^2 - 8$ .

Flashcard 20: What is the end behavior of $f(x) = -x^5 + 3x^2 - 7$ as $x \to -\text{infinity}$ ?

Flashcard 21: What is the degree of $f(x) = -x^2 + 4x - 1$ ?

Flashcard 22: What is the degree of the polynomial $f(x) = 3x^4 - 5x^3 + 2x^2 - x + 7$ ?

Flashcard 23: State the number of real roots for $f(x) = x^4 - 1$ .

Flashcard 24: What is the degree of $f(x) = x^7 - 2x^4 + x^2$ ?

Flashcard 25: State the end behavior of $f(x) = x^9 - x^5$ as $x \to -\text{infinity}$ .

Flashcard 26: Identify the constant term in $f(x) = 5x^3 - 2x + 7$ .

Flashcard 27: State the polynomial function given roots $x = 1$ , $x = -2$ , and $x = 3$ .

Flashcard 28: Identify the degree of polynomial $f(x) = -5x^7 + 2x^4 - x$ .

Flashcard 29: State the leading coefficient of $f(x) = -8x^4 + 2x^3 - x + 1$ .

Flashcard 30: State the constant term in $f(x) = 3x^5 - 5x^3 + 2$ .

Flashcard 31: Find the end behavior of $f(x) = 7x^2 - 4x + 1$ as $x \to -\text{infinity}$ .

Flashcard 32: Identify the leading term in $f(x) = 6x^7 - 3x^5 + 2x - 8$ .

Flashcard 33: Determine the multiplicity of root $x = 2$ in $f(x) = (x-2)^3(x+1)$ .

Flashcard 34: State the leading coefficient of $f(x) = -2x^3 + 4x^2 - x + 5$ .

Flashcard 35: What is the end behavior of $f(x) = 2x^3 - 4x + 1$ as $x \to \text{infinity}$ ?