Polynomial Functions and End Behavior - AP Precalculus
Card 1 of 30
Determine the multiplicity of root $x = 2$ in $f(x) = (x-2)^3(x+1)$.
Determine the multiplicity of root $x = 2$ in $f(x) = (x-2)^3(x+1)$.
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- The exponent on factor $(x-2)$.
- The exponent on factor $(x-2)$.
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What is the y-intercept of $f(x) = 2x^4 - 3x^2 + 7$?
What is the y-intercept of $f(x) = 2x^4 - 3x^2 + 7$?
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- Evaluate $f(0)$ to find y-intercept.
- Evaluate $f(0)$ to find y-intercept.
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What is the end behavior of $f(x) = 3x^2 - 7x + 4$ as $x \to \text{infinity}$?
What is the end behavior of $f(x) = 3x^2 - 7x + 4$ as $x \to \text{infinity}$?
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$f(x) \to \text{infinity}$. Even degree with positive leading coefficient goes to $+\infty$.
$f(x) \to \text{infinity}$. Even degree with positive leading coefficient goes to $+\infty$.
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What is the degree of $f(x) = x^7 - 2x^4 + x^2$?
What is the degree of $f(x) = x^7 - 2x^4 + x^2$?
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- The highest power of x.
- The highest power of x.
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What is the end behavior of $f(x) = -4x^3 + 2x - 5$ as $x \to -\text{infinity}$?
What is the end behavior of $f(x) = -4x^3 + 2x - 5$ as $x \to -\text{infinity}$?
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$f(x) \to \text{infinity}$. Odd degree with negative leading coefficient: $x \to -\infty$ gives $f(x) \to +\infty$.
$f(x) \to \text{infinity}$. Odd degree with negative leading coefficient: $x \to -\infty$ gives $f(x) \to +\infty$.
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What is the end behavior of $f(x) = x^7 - x^5 + x$ as $x \to \text{infinity}$?
What is the end behavior of $f(x) = x^7 - x^5 + x$ as $x \to \text{infinity}$?
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$f(x) \to \text{infinity}$. Odd degree with positive leading coefficient goes to $+\infty$.
$f(x) \to \text{infinity}$. Odd degree with positive leading coefficient goes to $+\infty$.
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Find the end behavior of $f(x) = 7x^2 - 4x + 1$ as $x \to -\text{infinity}$.
Find the end behavior of $f(x) = 7x^2 - 4x + 1$ as $x \to -\text{infinity}$.
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$f(x) \to \text{infinity}$. Even degree with positive leading coefficient goes to $+\infty$.
$f(x) \to \text{infinity}$. Even degree with positive leading coefficient goes to $+\infty$.
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State the end behavior of $f(x) = -x^6 + 2x^4 - 5$ as $x \to \text{infinity}$.
State the end behavior of $f(x) = -x^6 + 2x^4 - 5$ as $x \to \text{infinity}$.
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$f(x) \to -\text{infinity}$. Even degree with negative leading coefficient goes to $-\infty$.
$f(x) \to -\text{infinity}$. Even degree with negative leading coefficient goes to $-\infty$.
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What is the degree of $f(x) = 4x^5 - 9$?
What is the degree of $f(x) = 4x^5 - 9$?
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- The highest power of x.
- The highest power of x.
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Determine the multiplicity of root $x = -1$ in $f(x) = (x+1)^2(x-2)$.
Determine the multiplicity of root $x = -1$ in $f(x) = (x+1)^2(x-2)$.
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- The exponent on factor $(x+1)$.
- The exponent on factor $(x+1)$.
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What is the axis of symmetry for $f(x) = x^2 - 4x + 3$?
What is the axis of symmetry for $f(x) = x^2 - 4x + 3$?
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$x = 2$. For quadratic $ax^2 + bx + c$, axis is $x = -\frac{b}{2a}$.
$x = 2$. For quadratic $ax^2 + bx + c$, axis is $x = -\frac{b}{2a}$.
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Identify the constant term in $f(x) = 4x^3 - x^2 + 6x - 9$.
Identify the constant term in $f(x) = 4x^3 - x^2 + 6x - 9$.
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-9. The term without any variable.
-9. The term without any variable.
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Identify the constant term in $f(x) = 4x^3 - x^2 + 6x - 9$.
Identify the constant term in $f(x) = 4x^3 - x^2 + 6x - 9$.
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-9. The term without any variable.
-9. The term without any variable.
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What is the degree of $f(x) = -x^2 + 4x - 1$?
What is the degree of $f(x) = -x^2 + 4x - 1$?
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- The highest power of x.
- The highest power of x.
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State the end behavior of $f(x) = -x^6 + 2x^4 - 5$ as $x \to \text{infinity}$.
State the end behavior of $f(x) = -x^6 + 2x^4 - 5$ as $x \to \text{infinity}$.
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$f(x) \to -\text{infinity}$. Even degree with negative leading coefficient goes to $-\infty$.
$f(x) \to -\text{infinity}$. Even degree with negative leading coefficient goes to $-\infty$.
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What is the degree of the polynomial $f(x) = 3x^4 - 5x^3 + 2x^2 - x + 7$?
What is the degree of the polynomial $f(x) = 3x^4 - 5x^3 + 2x^2 - x + 7$?
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- The highest power of x determines the degree.
- The highest power of x determines the degree.
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State the leading coefficient of $f(x) = -2x^3 + 4x^2 - x + 5$.
State the leading coefficient of $f(x) = -2x^3 + 4x^2 - x + 5$.
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-2. The coefficient of the highest degree term.
-2. The coefficient of the highest degree term.
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What is the end behavior of $f(x) = 2x^3 - 4x + 1$ as $x \to \text{infinity}$?
What is the end behavior of $f(x) = 2x^3 - 4x + 1$ as $x \to \text{infinity}$?
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$f(x) \to \text{infinity}$. Odd degree with positive leading coefficient goes to $+\infty$.
$f(x) \to \text{infinity}$. Odd degree with positive leading coefficient goes to $+\infty$.
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What is the end behavior of $f(x) = -x^5 + 3x^2 - 7$ as $x \to -\text{infinity}$?
What is the end behavior of $f(x) = -x^5 + 3x^2 - 7$ as $x \to -\text{infinity}$?
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$f(x) \to \text{infinity}$. Odd degree with negative leading coefficient: $x \to -\infty$ gives $f(x) \to +\infty$.
$f(x) \to \text{infinity}$. Odd degree with negative leading coefficient: $x \to -\infty$ gives $f(x) \to +\infty$.
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State the polynomial function given roots $x = 1$, $x = -2$, and $x = 3$.
State the polynomial function given roots $x = 1$, $x = -2$, and $x = 3$.
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$(x-1)(x+2)(x-3)$. Use $(x-r)$ for each root r.
$(x-1)(x+2)(x-3)$. Use $(x-r)$ for each root r.
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What is the end behavior of $f(x) = -3x^4 + 6x^2 - 9$ as $x \to \text{infinity}$?
What is the end behavior of $f(x) = -3x^4 + 6x^2 - 9$ as $x \to \text{infinity}$?
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$f(x) \to -\text{infinity}$. Even degree with negative leading coefficient goes to $-\infty$.
$f(x) \to -\text{infinity}$. Even degree with negative leading coefficient goes to $-\infty$.
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State the end behavior of $f(x) = x^6 - 4x^3 + 5$ as $x \to -\text{infinity}$.
State the end behavior of $f(x) = x^6 - 4x^3 + 5$ as $x \to -\text{infinity}$.
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$f(x) \to \text{infinity}$. Even degree with positive leading coefficient goes to $+\infty$.
$f(x) \to \text{infinity}$. Even degree with positive leading coefficient goes to $+\infty$.
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Find the leading coefficient of $f(x) = 7x^5 - 4x^3 + x^2 - 8$.
Find the leading coefficient of $f(x) = 7x^5 - 4x^3 + x^2 - 8$.
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- The coefficient of the highest degree term.
- The coefficient of the highest degree term.
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State the polynomial type for $f(x) = x^3 - 4x + 2$.
State the polynomial type for $f(x) = x^3 - 4x + 2$.
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Cubic polynomial. Degree 3 polynomials are called cubic.
Cubic polynomial. Degree 3 polynomials are called cubic.
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Identify the degree of polynomial $f(x) = -5x^7 + 2x^4 - x$.
Identify the degree of polynomial $f(x) = -5x^7 + 2x^4 - x$.
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$7$. The highest power of x.
$7$. The highest power of x.
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What is the end behavior of $f(x) = 5x^2 - 3x + 1$ as $x \to \text{infinity}$?
What is the end behavior of $f(x) = 5x^2 - 3x + 1$ as $x \to \text{infinity}$?
Tap to reveal answer
$f(x) \to \text{infinity}$. Even degree with positive leading coefficient goes to $+\infty$.
$f(x) \to \text{infinity}$. Even degree with positive leading coefficient goes to $+\infty$.
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Identify the leading term in $f(x) = 6x^7 - 3x^5 + 2x - 8$.
Identify the leading term in $f(x) = 6x^7 - 3x^5 + 2x - 8$.
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6x^7. The term with highest degree and its coefficient.
6x^7. The term with highest degree and its coefficient.
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State the constant term in $f(x) = 3x^5 - 5x^3 + 2$.
State the constant term in $f(x) = 3x^5 - 5x^3 + 2$.
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- The term without any variable.
- The term without any variable.
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What is the end behavior of $f(x) = -4x^3 + 2x - 5$ as $x \to -\text{infinity}$?
What is the end behavior of $f(x) = -4x^3 + 2x - 5$ as $x \to -\text{infinity}$?
Tap to reveal answer
$f(x) \to \text{infinity}$. Odd degree with negative leading coefficient: $x \to -\infty$ gives $f(x) \to +\infty$.
$f(x) \to \text{infinity}$. Odd degree with negative leading coefficient: $x \to -\infty$ gives $f(x) \to +\infty$.
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What is the axis of symmetry for $f(x) = x^2 - 4x + 3$?
What is the axis of symmetry for $f(x) = x^2 - 4x + 3$?
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$x = 2$. For quadratic $ax^2 + bx + c$, axis is $x = -\frac{b}{2a}$.
$x = 2$. For quadratic $ax^2 + bx + c$, axis is $x = -\frac{b}{2a}$.
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