Questions: For each function, identify the end behavior asymptote. 1) f(x) = (c x^3 - 4 x) / (4 x^2 - 12 x) 2) f(x) = (x^3 - 7 x^2 + 12 x) / (4 x^2 - 4 x) 3) f(x) = (x^3 - 2 x^2 - 8 x) / (-4 x^2 + 36) 4) f(x) = 3 / (x - 2) 5) f(x) = (x + 4) / (3 x + 9) 6) f(x) = -(4 / (x^2 - x - 2)) 7) f(x) = (x^2 + 4 x) / (x^2 - 16) 8) f(x) = (x^2 + 3 x - 4) / (2 x^2 - 2 x - 4) 9) f(x) = (x^2 + 2 x - 8) / (4 x^2 - 4 x - 8) 10) f(x) = 3 / (x^2 - x - 2) 11) f(x) = (x^3 - x^2 - 12 x) / (2 x^2 - 8) 12) f(x) = (-2 x + 4) / (x - 1)

For each function, identify the end behavior asymptote. 1) f(x) = (c x^3 - 4 x) / (4 x^2 - 12 x) 2) f(x) = (x^3 - 7 x^2 + 12 x) / (4 x^2 - 4 x) 3) f(x) = (x^3 - 2 x^2 - 8 x) / (-4 x^2 + 36) 4) f(x) = 3 / (x - 2) 5) f(x) = (x + 4) / (3 x + 9) 6) f(x) = -(4 / (x^2 - x - 2)) 7) f(x) = (x^2 + 4 x) / (x^2 - 16) 8) f(x) = (x^2 + 3 x - 4) / (2 x^2 - 2 x - 4) 9) f(x) = (x^2 + 2 x - 8) / (4 x^2 - 4 x - 8) 10) f(x) = 3 / (x^2 - x - 2) 11) f(x) = (x^3 - x^2 - 12 x) / (2 x^2 - 8) 12) f(x) = (-2 x + 4) / (x - 1)
Transcript text: For each function, identify the end behavior asymptote. 1) $f(x)=\frac{c x^{3}-4 x}{4 x^{2}-12 x}$ 2) $f(x)=\frac{x^{3}-7 x^{2}+12 x}{4 x^{2}-4 x}$ 3) $f(x)=\frac{x^{3}-2 x^{2}-8 x}{-4 x^{2}+36}$ 4) $f(x)=\frac{3}{x-2}$ 5) $f(x)=\frac{x+4}{3 x+9}$ 6) $f(x)=-\frac{4}{x^{2}-x-2}$ 7) $f(x)=\frac{x^{2}+4 x}{x^{2}-16}$ 8) $f(x)=\frac{x^{2}+3 x-4}{2 x^{2}-2 x-4}$ 9) $f(x)=\frac{x^{2}+2 x-8}{4 x^{2}-4 x-8}$ 10) $f(x)=\frac{3}{x^{2}-x-2}$ 11) $f(x)=\frac{x^{3}-x^{2}-12 x}{2 x^{2}-8}$ 12) $f(x)=\frac{-2 x+4}{x-1}$
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Solution

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Solution Steps

Step 1: Identify the End Behavior Asymptote for \( f(x)=\frac{c x^{3}-4 x}{4 x^{2}-12 x} \)

To find the end behavior asymptote, we compare the degrees of the numerator and the denominator. The degree of the numerator is 3, and the degree of the denominator is 2. Since the degree of the numerator is greater than the degree of the denominator, the end behavior asymptote is a slant (oblique) asymptote.

To find the slant asymptote, perform polynomial long division of the numerator by the denominator:

  1. Divide the leading term of the numerator by the leading term of the denominator: \(\frac{c x^3}{4 x^2} = \frac{c}{4}x\).
  2. Multiply the entire denominator by \(\frac{c}{4}x\) and subtract from the numerator.
  3. Repeat the process with the new polynomial.

The slant asymptote is given by the quotient obtained from the division, which is \(\frac{c}{4}x\).

Step 2: Identify the End Behavior Asymptote for \( f(x)=\frac{x^{3}-7 x^{2}+12 x}{4 x^{2}-4 x} \)

The degree of the numerator is 3, and the degree of the denominator is 2. Since the degree of the numerator is greater than the degree of the denominator, the end behavior asymptote is a slant asymptote.

Perform polynomial long division:

  1. Divide the leading term of the numerator by the leading term of the denominator: \(\frac{x^3}{4 x^2} = \frac{1}{4}x\).
  2. Multiply the entire denominator by \(\frac{1}{4}x\) and subtract from the numerator.
  3. Repeat the process with the new polynomial.

The slant asymptote is given by the quotient obtained from the division, which is \(\frac{1}{4}x - \frac{7}{4}\).

Step 3: Identify the End Behavior Asymptote for \( f(x)=\frac{x^{3}-2 x^{2}-8 x}{-4 x^{2}+36} \)

The degree of the numerator is 3, and the degree of the denominator is 2. Since the degree of the numerator is greater than the degree of the denominator, the end behavior asymptote is a slant asymptote.

Perform polynomial long division:

  1. Divide the leading term of the numerator by the leading term of the denominator: \(\frac{x^3}{-4 x^2} = -\frac{1}{4}x\).
  2. Multiply the entire denominator by \(-\frac{1}{4}x\) and subtract from the numerator.
  3. Repeat the process with the new polynomial.

The slant asymptote is given by the quotient obtained from the division, which is \(-\frac{1}{4}x - \frac{1}{2}\).

Final Answer

  1. The end behavior asymptote for \( f(x)=\frac{c x^{3}-4 x}{4 x^{2}-12 x} \) is \(\boxed{y = \frac{c}{4}x}\).

  2. The end behavior asymptote for \( f(x)=\frac{x^{3}-7 x^{2}+12 x}{4 x^{2}-4 x} \) is \(\boxed{y = \frac{1}{4}x - \frac{7}{4}}\).

  3. The end behavior asymptote for \( f(x)=\frac{x^{3}-2 x^{2}-8 x}{-4 x^{2}+36} \) is \(\boxed{y = -\frac{1}{4}x - \frac{1}{2}}\).

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