Follow the seven step strategy to graph the following rational function.

Question

f left parenthesis x right parenthesis equals StartFraction x plus 2 Over x squared plus x minus 20 EndFraction
Question content area bottom
Part 1
To graph the​ function, first determine the symmetry of the graph of f. Choose the correct answer below.
neither​ y-axis symmetry nor origin symmetry
origin symmetry
​y-axis symmetry

What is the​ y-intercept and x-intercept? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The​ y-intercept is
  
enter your response here.
​(Type an integer or a simplified​ fraction.)
B.
There is no​ y-intercept.

Find the vertical​ and horizontal asymptote(s). Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The​ equation(s) of the vertical​ asymptote(s) is/are
  
enter your response here.
​(Type an equation. Use a comma to separate answers as​ needed.)
B.
There is no vertical asymptote.

Answer 1

To graph the rational function \( f(x) = \frac{x + 2}{x^2 + x - 20} \), we will follow the seven step strategy as outlined.

Part 1: Determine Symmetry

1. Symmetry:
   • y-axis symmetry: \( f(-x) = f(x) \)
   • origin symmetry: \( f(-x) = -f(x) \)
   • For our function: \[ f(-x) = \frac{-x + 2}{(-x)^2 + (-x) - 20} = \frac{-x + 2}{x^2 - x - 20} \] This is not equal to \( f(x) \) nor is it equal to \( -f(x) \). Therefore, this function has neither y-axis symmetry nor origin symmetry.

Conclusion:

Neither y-axis symmetry nor origin symmetry

Part 2: Find the y-intercept and x-intercept

2. Y-intercept:

   • To find the y-intercept, set \( x = 0 \): \[ f(0) = \frac{0 + 2}{0^2 + 0 - 20} = \frac{2}{-20} = -\frac{1}{10} \]
   • Thus, the y-intercept is \( -\frac{1}{10} \).

3. X-intercept:

   • To find the x-intercept, set \( f(x) = 0 \):
   • \( \frac{x + 2}{x^2 + x - 20} = 0 \) implies \( x + 2 = 0 \) or \( x = -2 \).
   • Thus, the x-intercept is \( -2 \).

Conclusion:

A. The y-intercept is \( -\frac{1}{10} \).
A. The x-intercept is \( -2 \).

Part 3: Find the vertical and horizontal asymptotes

4. Vertical Asymptotes:

   • Vertical asymptotes occur where the denominator is zero, provided that these points do not cancel with the numerator.
   • Factor the denominator: \[ x^2 + x - 20 = (x - 4)(x + 5) \]
   • Set the denominator to zero: \[ (x - 4)(x + 5) = 0 \Rightarrow x - 4 = 0 \rightarrow x = 4 \] \[ x + 5 = 0 \Rightarrow x = -5 \]
   • Thus, the vertical asymptotes are \( x = 4 \) and \( x = -5 \).

5. Horizontal Asymptotes:

   • For rational functions where the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
   • The degrees are:
     • Degree of numerator: 1
     • Degree of denominator: 2
   • Thus, the horizontal asymptote is \( y = 0 \).

Conclusion:

A. The equations of the vertical asymptotes are \(x = 4, x = -5\).
A. The horizontal asymptote is \(y = 0\).

With this information, you can proceed to graph the rational function.