12.3 The parabola  (Page 6/11)

Key equations

Key concepts

• A parabola is the set of all points $\left(x,y\right)$ in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
• The standard form of a parabola with vertex $\left(0,0\right)$ and the x -axis as its axis of symmetry can be used to graph the parabola. If $p>0,$ the parabola opens right. If $p<0,$ the parabola opens left. See [link] .
• The standard form of a parabola with vertex $\left(0,0\right)$ and the y -axis as its axis of symmetry can be used to graph the parabola. If $p>0,$ the parabola opens up. If $p<0,$ the parabola opens down. See [link] .
• When given the focus and directrix of a parabola, we can write its equation in standard form. See [link] .
• The standard form of a parabola with vertex $\left(h,k\right)$ and axis of symmetry parallel to the x -axis can be used to graph the parabola. If $p>0,$ the parabola opens right. If $p<0,$ the parabola opens left. See [link] .
• The standard form of a parabola with vertex $\left(h,k\right)$ and axis of symmetry parallel to the y -axis can be used to graph the parabola. If $p>0,$ the parabola opens up. If $p<0,$ the parabola opens down. See [link] .
• Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. See [link] .

Section exercises

Verbal

Algebraic

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.

For the following exercises, rewrite the given equation in standard form, and then determine the vertex $(V),$ focus $(F),$ and directrix $(d)$ of the parabola.