follow the steps for graphing a rational function to graph the function ( r(x)=\frac{1}{7x + 35} ).\nc. the…

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{1}{7x + 35} ).\nc. the function has no horizontal asymptote.\ndetermine the oblique asymptotes, if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice.\na. the function has one oblique asymptote.\n(type an equation. use integers or fractions for any numbers in the equation.)\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. there is 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 to complete 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. the graph of ( r ) intersects the horizontal or oblique asymptote at infinitely many points.\nc. there is no point at which the graph of ( r ) intersects the horizontal or oblique asymptote.\nd. 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 evaluating ( r ) there. select the correct choice and fill in the complete your choice.\na. the graph of ( r ) is above the ( x )-axis on the interval(s) and below the ( x )-axis on the interval(s)\n(type your answers in interval notation. use a comma to separate answers as needed.)\nb. the graph of ( r ) is above the ( x )-axis on the interval(s)\n(type your answer in interval notation. use a comma to separate answers as needed.)\nc. the graph of ( r ) is below the ( x )-axis on the interval(s)\n(type your answer in interval notation. use a comma to separate answers as needed.)

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{1}{7x + 35} ).\nc. the function has no horizontal asymptote.\ndetermine the oblique asymptotes, if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice.\na. the function has one oblique asymptote.\n(type an equation. use integers or fractions for any numbers in the equation.)\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. there is 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 to complete 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. the graph of ( r ) intersects the horizontal or oblique asymptote at infinitely many points.\nc. there is no point at which the graph of ( r ) intersects the horizontal or oblique asymptote.\nd. 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 evaluating ( r ) there. select the correct choice and fill in the complete your choice.\na. the graph of ( r ) is above the ( x )-axis on the interval(s) and below the ( x )-axis on the interval(s)\n(type your answers in interval notation. use a comma to separate answers as needed.)\nb. the graph of ( r ) is above the ( x )-axis on the interval(s)\n(type your answer in interval notation. use a comma to separate answers as needed.)\nc. the graph of ( r ) is below the ( x )-axis on the interval(s)\n(type your answer in interval notation. use a comma to separate answers as needed.)

Answer

Explanation:

Step1: Analyze the degrees of numerator and denominator

The function ( R(x)=\frac{1}{7x + 35}) has numerator degree (n = 0) (since the numerator is a constant, (y = 1)) and denominator degree (m=1) (since the denominator is (y=7x + 35)). For oblique asymptotes, we need (n=m + 1). Here (n=0) and (m = 1), so (n\neq m + 1).

Step2: Analyze intersection with asymptotes

Since there is a horizontal asymptote (y = 0) (when (n<m), the horizontal asymptote is (y = 0)). Set (R(x)=0), (\frac{1}{7x+35}=0). The equation (\frac{1}{7x + 35}=0) has no solution because the numerator (1\neq0) for all real (x).

Answer:

  • For oblique asymptotes: C. There is no oblique asymptote.
  • For intersection with horizontal/oblique asymptotes: C. There is no point at which the graph of (R) intersects the horizontal or oblique asymptote.