Question

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{3x + 3}{7x+35}

the graph will cross the ( x )-axis at ( x=-1 ) (type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once.)

determine the vertical asymptote(s). if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice

a. the function has two vertical asymptotes. the leftmost asymptote is, and the rightmost asymptote is (type equations. use integers or fractions for any numbers in the equations )
b. the function has one vertical asymptote, ( x = -5 ) (type an equation. use integers or fractions for any numbers in the equation )
c. the function has three vertical asymptotes. the leftmost asymptote is, the middle asymptote is, and the rightmost asymptote is (type equations. use integers or fractions for any numbers in the equations.)
d. the function has no vertical asymptote

determine the hole, if it exists. select the correct choice and, if necessary, fill in the answer box to complete your choice

a. there is a hole in the graph at the point (type an ordered pair using integers or fractions )
b. there are no holes in the graph

determine the behavior of the graph on either side of any vertical asymptotes, if one exists. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice

a. it approaches ( infty ) on one side of the asymptote(s) at ( x= ) and ( -infty ) on the other. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x= ) (type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)
b. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x= ) (type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once )
c. it approaches ( infty ) on one side of the asymptote(s) at ( x=-5,-1 ) 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

Explanation:

Step1: Find the x - intercept

Set \(R(x)=0\), so \(\frac{3x + 3}{7x+35}=0\). A rational function \(y=\frac{f(x)}{g(x)}\) is zero when \(f(x)=0\) (and \(g(x)
eq0\)).
Solve \(3x + 3=0\), \(3x=-3\), \(x=-1\). And when \(x = - 1\), \(g(-1)=7\times(-1)+35=28
eq0\).

Step2: Find the vertical asymptote

Set the denominator \(g(x)=7x + 35=0\).
Solve \(7x+35 = 0\), \(7x=-35\), \(x=-5\).

Step3: Check for holes

Factor the numerator and denominator. \(R(x)=\frac{3(x + 1)}{7(x + 5)}\). There is no common factor (other than 1) in the numerator and denominator, so there are no holes.

Step4: Determine the behavior near the vertical asymptote \(x=-5\)

Take a test point to the left of \(x = - 5\), say \(x=-6\). \(R(-6)=\frac{3\times(-6 + 1)}{7\times(-6+35)}=\frac{3\times(-5)}{7\times19}=-\frac{15}{133}\).
Take a test point to the right of \(x=-5\), say \(x=-4\). \(R(-4)=\frac{3\times(-4 + 1)}{7\times(-4 + 35)}=\frac{3\times(-3)}{7\times31}=-\frac{9}{217}\).
As \(x\to-5^{-}\) (approaching from the left), \(R(x)\to-\infty\); as \(x\to-5^{+}\) (approaching from the right), \(R(x)\to+\infty\)

Answer:

The graph will cross the \(x -\)axis at \(x=-1\). The function has one vertical asymptote \(x = - 5\). There are no holes in the graph. It approaches \(-\infty\) on one side of the asymptote at \(x=-5\) and \(+\infty\) on the other.