Question

find the formula for a rational function that satisfies all of the following. its x-intercept is (-3,0). it has a vertical asymptotes, namely x=-1 and x=0. it has one horizontal asymptote, namely y=3

Answer 1

Let that be

$\\ \rm\Rightarrow y=\dfrac{a(f(x))}{g(x)}$

Two vertical asymptotes at -1 and 0

$\\ \rm\Rightarrow y=\dfrac{a(f(x)}{x(x+1)}$

If we simply

$\\ \rm\Rightarrow y=\dfrac{a(f(x)}{x^2+1}$

• Denominator has degree 2
• Numerator should have degree as 2 and coefficient as 3 inorder to get horizontal asymptote y=3 means the quadratic equation should contain 3x²
• But there should be a x intercept at -3 so one zeros should be -3

Find a equation

• 3x²+9x

Find zeros

• 3x²+9x=0
• 3x(x+3)=0
• x=0,-3

Horizontal asymptote

• 3x²/x²
• 3

So our equation is

$\\ \rm\Rightarrow y=\dfrac{3x^2+9x}{x(x+1)}$

Graph attached