Question

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.
$$ f ( x ) = \frac { 3 x - 7 } { 2 x + 3 } $$
plot rational function
vertical asymptote horizontal asymptote
x - intercept y - intercept hole
click on a feature then drag it into place.

Explanation:

Step1: Find the vertical asymptote

Set the denominator equal to zero: \(2x + 3=0\). Solve for \(x\): \(x =-\frac{3}{2}\).

Step2: Find the horizontal asymptote

Since the degree of the numerator and denominator are equal (both degree 1), the horizontal asymptote is \(y=\frac{3}{2}\) (ratio of leading coefficients).

Step3: Find the \(x\) - intercept

Set \(y = 0\) (numerator equal to zero): \(3x-7 = 0\). Solve for \(x\): \(x=\frac{7}{3}\).

Step4: Find the \(y\) - intercept

Set \(x = 0\): \(y=\frac{3(0)-7}{2(0)+3}=-\frac{7}{3}\).

Step5: Check for a hole

Since there is no common factor in the numerator and denominator (after factoring \(3x - 7\) and \(2x+3\) which are both linear and non - factorable over integers), there is no hole.

Answer:

Vertical Asymptote: \(x =-\frac{3}{2}\), Horizontal Asymptote: \(y=\frac{3}{2}\), \(x\) - intercept: \((\frac{7}{3},0)\), \(y\) - intercept: \((0,-\frac{7}{3})\), No Hole.