Question

follow the steps for graphing a rational function to graph the function ( g(x)=\frac{4 x}{x^{2}-36} ).
(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)
it approaches ( infty ) on one side of the asymptote(s) at ( x=-6,6 ) and ( -infty ) on the other.
(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)
d. there is no vertical asymptote.
determine the horizontal asymptote(s). if any exist. select the correct choice and, if necessary, fill in the answer box(ex) to complete your choice.
a. the function has one horizontal asymptote. ( y = 0 ).
(type an equation. use integers or fractions for any numbers in the equation.)
b. the function has two horizontal asymptotes. the top asymptote is and the bottom asymptote is
(type equations. use integers or fractions for any numbers in the equations.)
c. there is no horizontal asymptote.
determine the oblique asymptote(s), if any exist. select the correct choice and, if necessary, fill in the answer box(ex) to complete your choice.
a. the function has one oblique asymptote.
(type an equation. use integers or fractions for any numbers in the equation.)
b. the function has two oblique asymptotes. the oblique asymptote with negative slope is and the oblique asymptote with positive slope is
(type equations. use integers or fractions for any numbers in the equations.)
c. there is no oblique asymptote.
determine the points, if any, at which the graph of ( g ) 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.
a. the graph of ( g ) intersects the horizontal or oblique asymptote at ( y = 4 ).
(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)
b. the graph of ( g ) intersects the horizontal or oblique asymptote at infinitely many points.
c. there is no point at which the graph of ( g ) intersects the horizontal or oblique asymptote.
d. there is no horizontal or oblique asymptote.

Explanation:

Step1: Find vertical asymptotes

Set the denominator \(x^{2}-36 = 0\). Using the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\), where \(a=x\) and \(b = 6\), we have \((x + 6)(x - 6)=0\). Solving for \(x\), we get \(x=-6\) and \(x = 6\).

Step2: Find horizontal asymptotes

For a rational function \(y=\frac{f(x)}{g(x)}=\frac{a_{n}x^{n}+\cdots+a_{0}}{b_{m}x^{m}+\cdots+b_{0}}\), if \(n

Step3: Check for oblique asymptotes

A rational function \(y=\frac{f(x)}{g(x)}\) has an oblique asymptote if \(n=m + 1\). Since \(n = 1\) and \(m = 2\) (\(n
eq m + 1\)), there is no oblique asymptote.

Step4: Find intersection with horizontal asymptote

Set \(y = 0\) (the horizontal asymptote). So, \(\frac{4x}{x^{2}-36}=0\). Multiply both sides by \(x^{2}-36\) (assuming \(x
eq\pm6\)), we get \(4x=0\), which gives \(x = 0\). When \(x = 0\), \(y=\frac{4\times0}{0^{2}-36}=0\). The intersection point is \((0,0)\).

Answer:

• Vertical asymptotes: \(x=-6,x = 6\)
• Horizontal asymptote: \(y = 0\)
• No oblique asymptote
• Intersection with horizontal asymptote: \((0,0)\)