Question

plot all of the existing five features of the following rational function (some may not be needed). if you get a fraction or decimal then plot as close to the true location as possible.
$f(x) = \frac{3x - 10}{4x + 10}$
plot rational function
vertical asymptote horizontal asymptote
x - intercept y - intercept hole
click on a feature then drag it into place.

Explanation:

Step1: Find Vertical Asymptote

The vertical asymptote of a rational function \( f(x)=\frac{N(x)}{D(x)} \) occurs where \( D(x) = 0 \) (and \( N(x)
eq0 \) at that point). For \( f(x)=\frac{3x - 10}{4x+10} \), set denominator \( 4x + 10=0 \). Solving \( 4x=- 10\), we get \( x =-\frac{10}{4}=-\frac{5}{2}=-2.5 \).

Step2: Find Horizontal Asymptote

For a rational function \( f(x)=\frac{ax + b}{cx + d} \) (degree of numerator and denominator is 1), the horizontal asymptote is \( y=\frac{a}{c} \). Here, \( a = 3 \), \( c = 4 \), so horizontal asymptote is \( y=\frac{3}{4}=0.75 \).

Step3: Find x - Intercept

x - intercept occurs where \( f(x)=0 \), i.e., numerator \( 3x - 10 = 0 \). Solving \( 3x=10 \), we get \( x=\frac{10}{3}\approx3.33 \).

Step4: Find y - Intercept

y - intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into \( f(x) \): \( f(0)=\frac{3(0)-10}{4(0)+10}=\frac{- 10}{10}=-1 \).

Step5: Check for Hole

A hole occurs when there is a common factor in numerator and denominator. Here, numerator \( 3x - 10 \) and denominator \( 4x + 10 \) have no common factors, so there is no hole.

Answer:

• Vertical Asymptote: \( x = - 2.5 \)
• Horizontal Asymptote: \( y=0.75 \)
• x - Intercept: \( x=\frac{10}{3}\approx3.33 \) (point \( (\frac{10}{3},0) \))
• y - Intercept: \( y=-1 \) (point \( (0, - 1) \))
• Hole: None