Question

determine each feature of the graph of the given function.
$f(x) = \frac{3x - 7}{4x + 14}$
answer attempt 1 out of 2
horizontal asymptote: $y = \square$ $\boxed{\text{no horizontal asymptote}}$
vertical asymptote: $x = \square$ $\boxed{\text{no vertical asymptote}}$
x-intercept: $(\square, 0)$ $\boxed{\text{no } x\text{-intercept}}$
y-intercept: $(0, \square)$ $\boxed{\text{no } y\text{-intercept}}$
hole: $(\square, \square)$ $\boxed{\text{no hole}}$

Explanation:

Step1: Find Horizontal Asymptote

For a rational function \( f(x)=\frac{ax + b}{cx + d} \) (where \( a,c
eq0 \)), the horizontal asymptote is \( y=\frac{a}{c} \). Here, \( a = 3 \), \( c = 4 \), so \( y=\frac{3}{4} \).

Step2: Find Vertical Asymptote

Set the denominator equal to zero: \( 4x + 14 = 0 \). Solve for \( x \): \( 4x=-14 \), \( x = -\frac{14}{4}=-\frac{7}{2} \).

Step3: Find x - Intercept

Set \( f(x) = 0 \), so \( \frac{3x - 7}{4x + 14}=0 \). The numerator must be zero (denominator non - zero at this \( x \)): \( 3x - 7 = 0 \), \( 3x=7 \), \( x=\frac{7}{3} \).

Step4: Find y - Intercept

Set \( x = 0 \) in \( f(x) \): \( f(0)=\frac{3(0)-7}{4(0)+14}=\frac{-7}{14}=-\frac{1}{2} \).

Step5: Check for Hole

A hole occurs when there is a common factor in numerator and denominator. The numerator \( 3x - 7 \) and denominator \( 4x + 14 = 2(2x + 7) \) have no common factors, so no hole.

Answer:

Horizontal Asymptote: \( y=\frac{3}{4} \)
Vertical Asymptote: \( x = -\frac{7}{2} \)
x - Intercept: \( (\frac{7}{3},0) \)
y - Intercept: \( (0,-\frac{1}{2}) \)
Hole: No hole