Question

Find an equation for a rational function 'f(x)" that satisfies the following:• Vertical asymptote at x = – 7 and x = -8X-intercepts at (2,0) and (3,0)1y-intercept at (0,12)f(x) =help (formulas)

Answer 1

The rational function will have the next form:

`f(x)=(a(x-x_1)(x-x_2))/((x-x_3)(x-x_4))`

where x1 and x2 are the x-coordinates of the x-intercepts, x3 and x4 are the vertical asymptotes and a is some coefficient.

The x-intercepts are (2,0) and (3,0), then x1 = 2, and x2 = 3

The vertical asymptotes are x = – 7 and x = -8, then x3 = -7 and x4 = -8.

Substituting this information we get:

`\begin{gathered} f(x)=(a(x-2)(x-3))/((x-(-7))(x-(-8))) \\ f(x)=(a(x-2)(x-3))/((x+7)(x+8)) \end{gathered}`

The y-intercept at (0,12) means when x = 0, f(x) = 12. Substituting this information into the previous formula and solving for a:

`\begin{gathered} 12=(a(0-2)(0-3))/((0+7)(0+8)) \\ 12=(a(-2)(-3))/(7\cdot8) \\ 12=(a\cdot6)/(56) \\ 12\cdot(56)/(6)=a \\ 112=a \end{gathered}`

Finally, the equation for the rational function is:

`f(x)=(112(x-2)(x-3))/((x+7)(x+8))`