Question

1. g(x) = \frac{x + 2}{x - 5} graph (with a grid graph below)

Explanation:

Step1: Find Vertical Asymptote

For a rational function \( g(x)=\frac{f(x)}{h(x)} \), vertical asymptotes occur where \( h(x) = 0 \) (and \( f(x)
eq0 \) at those points). Here, \( h(x)=x - 5 \), so set \( x-5 = 0 \), we get \( x = 5 \). So vertical asymptote is \( x = 5 \).

Step2: Find Horizontal Asymptote

For rational functions where the degrees of numerator and denominator are equal (both degree 1 here), the horizontal asymptote is the ratio of leading coefficients. Leading coefficient of numerator is 1, denominator is 1, so horizontal asymptote is \( y=\frac{1}{1}=1 \).

Step3: Find x - intercept

Set \( g(x)=0 \), so \( \frac{x + 2}{x - 5}=0 \). A fraction is zero when numerator is zero (denominator non - zero). So \( x+2 = 0\Rightarrow x=-2 \). So x - intercept is \( (-2,0) \).

Step4: Find y - intercept

Set \( x = 0 \) in \( g(x) \), \( g(0)=\frac{0 + 2}{0-5}=-\frac{2}{5}=-0.4 \). So y - intercept is \( (0,-\frac{2}{5}) \).

Now, when graphing:

• The vertical asymptote \( x = 5 \) is a vertical line.
• The horizontal asymptote \( y = 1 \) is a horizontal line.
• The x - intercept at \( (-2,0) \) and y - intercept at \( (0,-\frac{2}{5}) \) help in plotting the curve. For \( x>5 \), as \( x \) increases, \( g(x)=\frac{x + 2}{x - 5}=\frac{x-5+7}{x - 5}=1+\frac{7}{x - 5} \), so as \( x

ightarrow\infty \), \( g(x)
ightarrow1 \), and for \( x \) just greater than 5, \( g(x)
ightarrow\infty \). For \( x<5 \), as \( x
ightarrow-\infty \), \( g(x)
ightarrow1 \), and for \( x \) just less than 5, \( g(x)
ightarrow-\infty \). The curve passes through \( (-2,0) \) and \( (0,-\frac{2}{5}) \) and approaches the asymptotes.

Answer:

To graph \( g(x)=\frac{x + 2}{x - 5} \), we use the vertical asymptote \( x = 5 \), horizontal asymptote \( y = 1 \), x - intercept \( (-2,0) \), y - intercept \( (0,-\frac{2}{5}) \) and the behavior of the function around the asymptotes. The graph should have two branches: one to the left of \( x = 5 \) (passing through \( (-2,0) \) and \( (0,-\frac{2}{5}) \), approaching \( y = 1 \) as \( x
ightarrow-\infty \) and \( x = 5 \) from the left) and one to the right of \( x = 5 \) (approaching \( y = 1 \) as \( x
ightarrow\infty \) and \( x = 5 \) from the right).