8.3 The parabola  (Page 4/11)

Given a standard form equation for a parabola centered at ( h , k ), sketch the graph.

1. Determine which of the standard forms applies to the given equation: ${\left(y-k\right)}^2=4p\left(x-h\right)$ or ${\left(x-h\right)}^2=4p\left(y-k\right).$
2. Use the standard form identified in Step 1 to determine the vertex, axis of symmetry, focus, equation of the directrix, and endpoints of the latus rectum.
   1. If the equation is in the form ${\left(y-k\right)}^2=4p\left(x-h\right),$ then:
      • use the given equation to identify $h$ and $k$ for the vertex, $\left(h,k\right)$
      • use the value of $k$ to determine the axis of symmetry, $y=k$
      • set $4p$ equal to the coefficient of $\left(x-h\right)$ in the given equation to solve for $p.$ If $p>0,$ the parabola opens right. If $p<0,$ the parabola opens left.
      • use $h,k,$ and $p$ to find the coordinates of the focus, $\left(h+p,k\right)$
      • use $h$ and $p$ to find the equation of the directrix, $x=h-p$
      • use $h,k,$ and $p$ to find the endpoints of the latus rectum, $\left(h+p,k\pm 2p\right)$
   2. If the equation is in the form ${\left(x-h\right)}^2=4p\left(y-k\right),$ then:
      • use the given equation to identify $h$ and $k$ for the vertex, $\left(h,k\right)$
      • use the value of $h$ to determine the axis of symmetry, $x=h$
      • set $4p$ equal to the coefficient of $\left(y-k\right)$ in the given equation to solve for $p.$ If $p>0,$ the parabola opens up. If $p<0,$ the parabola opens down.
      • use $h,k,$ and $p$ to find the coordinates of the focus, $\left(h,k+p\right)$
      • use $k$ and $p$ to find the equation of the directrix, $y=k-p$
      • use $h,k,$ and $p$ to find the endpoints of the latus rectum, $\left(h\pm 2p,k+p\right)$
3. Plot the vertex, axis of symmetry, focus, directrix, and latus rectum, and draw a smooth curve to form the parabola.