Questions: Find the intercepts and asymptotes, use limits to describe the behavior at the vertical asymptotes, and analyze and draw f(x)=(x^2+2x-3)/(x^2-64) The function has no horizontal asymptote. The function has a slant asymptote.

Find the intercepts and asymptotes, use limits to describe the behavior at the vertical asymptotes, and analyze and draw

f(x)=(x^2+2x-3)/(x^2-64)

The function has no horizontal asymptote. The function has a slant asymptote.
Transcript text: Find the intercepts and asymptotes, use limits to describe the behavior at the vertical asymptotes, and analyze and draw \[ f(x)=\frac{x^{2}+2 x-3}{x^{2}-64} \] The function has no horizontal asymptote. The function has a slant asymptote, $\square$ -
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Solution

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Solution Steps

To solve the given problem, we need to find the intercepts, asymptotes, and analyze the behavior of the function f(x)=x2+2x3x264 f(x) = \frac{x^2 + 2x - 3}{x^2 - 64} .

  1. Intercepts:

    • x-intercepts: Set the numerator equal to zero and solve for x x .
    • y-intercept: Evaluate f(0) f(0) .
  2. Vertical Asymptotes:

    • Set the denominator equal to zero and solve for x x .
  3. Horizontal/Slant Asymptotes:

    • Since the degrees of the numerator and denominator are the same, there is no horizontal asymptote.
    • To find the slant asymptote, perform polynomial long division on the numerator and denominator.
  4. Behavior at Vertical Asymptotes:

    • Use limits to describe the behavior of the function as x x approaches the vertical asymptotes.
Step 1: Finding the x-Intercepts

To find the x-intercepts, we set the numerator equal to zero: x2+2x3=0 x^2 + 2x - 3 = 0 The solutions to this equation are: x=3andx=1 x = -3 \quad \text{and} \quad x = 1 Thus, the x-intercepts are (3,0) (-3, 0) and (1,0) (1, 0) .

Step 2: Finding the y-Intercept

To find the y-intercept, we evaluate the function at x=0 x = 0 : f(0)=02+2(0)30264=364=364 f(0) = \frac{0^2 + 2(0) - 3}{0^2 - 64} = \frac{-3}{-64} = \frac{3}{64} Thus, the y-intercept is (0,364) (0, \frac{3}{64}) .

Step 3: Finding the Vertical Asymptotes

Vertical asymptotes occur where the denominator is equal to zero: x264=0 x^2 - 64 = 0 Solving this gives: x=8andx=8 x = -8 \quad \text{and} \quad x = 8 Thus, the vertical asymptotes are x=8 x = -8 and x=8 x = 8 .

Step 4: Finding the Slant Asymptote

Since the degrees of the numerator and denominator are the same, we perform polynomial long division: Slant Asymptote: y=1 \text{Slant Asymptote: } y = 1

Step 5: Analyzing Behavior at Vertical Asymptotes

To analyze the behavior of the function at the vertical asymptotes, we take the limits: limx8f(x)=zooandlimx8f(x)=zoo \lim_{x \to -8} f(x) = \text{zoo} \quad \text{and} \quad \lim_{x \to 8} f(x) = \text{zoo} This indicates that the function approaches infinity as x x approaches both vertical asymptotes.

Final Answer

  • x-intercepts: (3,0) (-3, 0) and (1,0) (1, 0)
  • y-intercept: (0,364) (0, \frac{3}{64})
  • Vertical asymptotes: x=8 x = -8 and x=8 x = 8
  • Slant asymptote: y=1 y = 1
  • Behavior at vertical asymptotes: limx8f(x)=zoo,limx8f(x)=zoo \lim_{x \to -8} f(x) = \text{zoo}, \lim_{x \to 8} f(x) = \text{zoo}

Thus, the final boxed answer is: x-intercepts: (3,0),(1,0); y-intercept: (0,364); vertical asymptotes: x=8,8; slant asymptote: y=1 \boxed{\text{x-intercepts: } (-3, 0), (1, 0); \text{ y-intercept: } (0, \frac{3}{64}); \text{ vertical asymptotes: } x = -8, 8; \text{ slant asymptote: } y = 1}

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