follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x^{2}+9 x + 18}{x + 6}…

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x^{2}+9 x + 18}{x + 6} ) \n\nb. the function has two oblique asymptotes. the oblique asymptote with a negative slope is, and the oblique asymptote with a positive slope is \n(type equations. use integers or fractions for any numbers in the equations ) \nc. the function has no oblique asymptote \ndetermine the points, if any, at which the graph of ( r ) intersects the horizontal or oblique asymptote. if one exists. select the correct choice and, if necessary, fill in the answer box within your choice \na. the graph of ( r ) intersects the horizontal or oblique asymptote at \n(simplify your answer. type an ordered pair. use a comma to separate answers as needed ) \nb. there is no point at which the graph of ( r ) intersects the horizontal or oblique asymptote \nc. there is no horizontal or oblique asymptote \nuse the real zeros of the numerator and denominator of ( r ) to divide the ( x )-axis into intervals. determine where the graph of ( r ) is above or below the ( x )-axis by choosing a number in each interval and evaluatin complete your choice \n(type your answer in interval notation. use a comma to separate answers as needed ) \na. the graph of ( r ) is below the ( x )-axis on the interval(s) \nb. the graph of ( r ) is above the ( x )-axis on the interval(s) \nc. the graph of ( r ) is above the ( x )-axis on the interval(s) and below the ( x )-axis on the interval(s) \nuse the results from the previous steps to graph ( r ). choose the correct graph \na. \nb. \nc.

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x^{2}+9 x + 18}{x + 6} ) \n\nb. the function has two oblique asymptotes. the oblique asymptote with a negative slope is, and the oblique asymptote with a positive slope is \n(type equations. use integers or fractions for any numbers in the equations ) \nc. the function has no oblique asymptote \ndetermine the points, if any, at which the graph of ( r ) intersects the horizontal or oblique asymptote. if one exists. select the correct choice and, if necessary, fill in the answer box within your choice \na. the graph of ( r ) intersects the horizontal or oblique asymptote at \n(simplify your answer. type an ordered pair. use a comma to separate answers as needed ) \nb. there is no point at which the graph of ( r ) intersects the horizontal or oblique asymptote \nc. there is no horizontal or oblique asymptote \nuse the real zeros of the numerator and denominator of ( r ) to divide the ( x )-axis into intervals. determine where the graph of ( r ) is above or below the ( x )-axis by choosing a number in each interval and evaluatin complete your choice \n(type your answer in interval notation. use a comma to separate answers as needed ) \na. the graph of ( r ) is below the ( x )-axis on the interval(s) \nb. the graph of ( r ) is above the ( x )-axis on the interval(s) \nc. the graph of ( r ) is above the ( x )-axis on the interval(s) and below the ( x )-axis on the interval(s) \nuse the results from the previous steps to graph ( r ). choose the correct graph \na. \nb. \nc.

Answer

Explanation:

Step1: Simplify the rational function

First, factor the numerator (x^{2}+9x + 18=(x + 3)(x+6)). Then (R(x)=\frac{x^{2}+9x + 18}{x + 6}=\frac{(x + 3)(x + 6)}{x+6}=x + 3) for (x\neq-6).

Step2: Analyze asymptotes

Since (R(x)=x + 3) (a linear function) after simplification (with a hole at (x=-6)), there is no oblique asymptote (because the simplified function is linear, not a non - linear polynomial divided by a linear polynomial in the form that would create an oblique asymptote).

Step3: Check intersection with asymptote

Since there is no horizontal or oblique asymptote (the function (y=x + 3) is not an asymptote in the traditional sense for a rational function here, as the original rational function simplifies to a linear function with a removable discontinuity), we consider the nature of the function. The (x) - intercept is found by setting (y = 0): (0=x+3\Rightarrow x=-3), so the point is ((-3,0)). The (y) - intercept is found by setting (x = 0): (y=0 + 3=3), so the point is ((0,3)). The function (y=x + 3) is a straight line. The domain is all real numbers except (x=-6).

Answer:

C. The function has no oblique asymptote. C. There is no horizontal or oblique asymptote. Since (R(x)=x + 3) (a straight line with a hole at (x=-6)), it is above the (x) - axis when (x+3>0\Rightarrow x>-3) and below the (x) - axis when (x + 3<0\Rightarrow x<-3) (but (x\neq-6)). So the graph is above the (x) - axis on the interval ((-3,\infty)) and below the (x) - axis on the interval ((-\infty,-6)\cup(-6,-3))