\( \def\dfrac#1#2{\displaystyle\frac{#1}{#2}} \def\solve#1{\begin{array}{rcl}#1\end{array} } \)

Home / 06 Rational Functions / 04 Combining And Graphing RationalFunctions

Use the graphs below to answer the following questions:

\(f(x)\) \(g(x)\) \(h(x)\)
f(x) = -(x-3)(x+2);{color:black}\ (0,f(0)); {color:black, showLabel:true} g(x) = 2(x+4)(x-1);{color:black}\ (0,g(0)); {color:black, showLabel:true} h(x) = -5x+15;{color:black}\ (0,h(0)); {color:black, showLabel:true}

  1. Determine the equation for each of the functions above.

  2. Find all asymptotes and intercepts for \(m(x)=\dfrac{f(x)}{g(x)}\)

  3. Find all asymptotes and intercepts for \(q(x)=\dfrac{{1}}{(h(x))^2}\)

Solution

  1. Equations:
    1. \(f(x) = -(x-3)(x+2)\)
    2. \(g(x) = 2(x+4)(x-1)\)
    3. \(h(x) = -5x+15\)
  2. Features of \(m(x)\):
    1. Vertical Intercept at \((0,-\frac{{3}}{{4}})\)
    2. Horizontal Intercepts at \((3,0)\) and \((-2,0)\).
    3. Vertical Asymptotes at \(x=-4\) and \(x=1\).
    4. Horizontal Asymptote at \(y=-\frac{{1}}{{2}}\)
  3. Features of \(q(x)\):
    1. Vertical Intercept at \((0,\frac{{1}}{{225}})\)
    2. Horizontal Intercepts: None.
    3. Vertical Asymptotes at \(x=3\).
    4. Horizontal Asymptote at \(y=0\)