Math  /  Algebra

QuestionPart 12 of 13 Score: 6.67%,0.936.67 \%, 0.93 of 14 points Save
Follow the steps for graphing a rational function to graph the function R(x)=x+8x(x+12)R(x)=\frac{x+8}{x(x+12)} A. The graph of RR intersects the horizontal or oblique asymptote at (8,0)(-8,0). (Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.) B. The graph of R-intersects the horizontal or oblique asymptote at infinitely many points. C. There is no point at which the graph of R intersects the horizontal or oblique asymptote. D. There is no horizontal or oblique asymptote.
Use the real zeros of the numerator and denominator of RR to divide the xx-axis into intervals. Determine where the graph of RR is above or below the xx-axis by choosing a number ii each interval and evaluating R there. Select the correct choice and fill in the answer box(es) to complete your choice. A. The graph of RR is above the x -axis on the interval(s) \square . (Type your answer in interval notation. Use a comma to separate answers as needed.) B. The graph of RR is below the xx-axis on the interval (s)(\mathrm{s}) \square (Type your answer in interval notation. Use a comma to separate answers as needed.) C. The graph of RR is above the xx-axis on the interval(s) \square and below the xx-axis on the interval(s) \square (Type your answers in interval notation. Use a comma to separate answers as needed.)

Studdy Solution

STEP 1

1. The function R(x)=x+8x(x+12) R(x) = \frac{x+8}{x(x+12)} is a rational function.
2. We need to analyze the asymptotic behavior and the sign of the function over different intervals.

STEP 2

1. Determine the horizontal or oblique asymptote.
2. Determine the zeros of the numerator and denominator.
3. Analyze the intervals based on the zeros.
4. Determine where the graph is above or below the x-axis.

STEP 3

To find the horizontal or oblique asymptote, compare the degrees of the numerator and denominator. The numerator x+8 x+8 is degree 1, and the denominator x(x+12) x(x+12) is degree 2. Since the degree of the denominator is greater, the horizontal asymptote is y=0 y = 0 .

STEP 4

Find the zeros of the numerator and denominator: - The numerator x+8=0 x+8 = 0 gives x=8 x = -8 . - The denominator x(x+12)=0 x(x+12) = 0 gives x=0 x = 0 and x=12 x = -12 .

STEP 5

Divide the x-axis into intervals based on the zeros: - (,12) (-\infty, -12) - (12,8) (-12, -8) - (8,0) (-8, 0) - (0,) (0, \infty)

STEP 6

Evaluate R(x) R(x) at test points in each interval to determine the sign of R(x) R(x) :
- For x(,12) x \in (-\infty, -12) , choose x=13 x = -13 : $ R(-13) = \frac{-13 + 8}{-13(-13 + 12)} = \frac{-5}{13} < 0 \] So, \( R(x) \) is below the x-axis.
- For x(12,8) x \in (-12, -8) , choose x=10 x = -10 : $ R(-10) = \frac{-10 + 8}{-10(-10 + 12)} = \frac{-2}{20} > 0 \] So, \( R(x) \) is above the x-axis.
- For x(8,0) x \in (-8, 0) , choose x=1 x = -1 : $ R(-1) = \frac{-1 + 8}{-1(-1 + 12)} = \frac{7}{11} > 0 \] So, \( R(x) \) is above the x-axis.
- For x(0,) x \in (0, \infty) , choose x=1 x = 1 : $ R(1) = \frac{1 + 8}{1(1 + 12)} = \frac{9}{13} > 0 \] So, \( R(x) \) is above the x-axis.
The graph of R R is: - Below the x-axis on the interval (,12) (-\infty, -12) . - Above the x-axis on the intervals (12,8) (-12, -8) , (8,0) (-8, 0) , and (0,) (0, \infty) .
The correct choice is C: The graph of R R is above the x-axis on the interval(s) (12,8)(8,0)(0,) (-12, -8) \cup (-8, 0) \cup (0, \infty) and below the x-axis on the interval(s) (,12) (-\infty, -12) .

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