Parabolas are the graphs of quadratic functions. All parabolas are “U” shaped and have a highest or lowest point that is called the vertex. Vertex is the highest point if the parabola is opened down and is the lowest point if the parabola is opened up. Parabolas always have a single $y$ - intercept and may or may not have $x$ - intercept. Every parabola has an axis of symmetry which divided the graph in two parts which are mirror image of each other.

                                       Parabola

The standard form of a Parabola equation is a quadratic equation $f(x)= ax^{2} + bx + c$

The vertex form of the Parabola equation is given by

                       $y = ax - h^{2} + k$

Where $(h, k)$ is the vertex of the parabola,

           $(x, y)$ is a point on parabola,

          and ‘a’ is the stretch/compression of the equation.
  • Parabolas open upwards or downwards.
  • The “zero” or “x – intercept” or “root” of a parabola is where the graph crosses the x – axis.
  • The point where the graph crosses the y – axis is y-intercept of a parabola.
  • The axis of symmetry divides the parabola into two equal halves mirror image to each other.
  • The point where the axis of symmetry and the parabola meet is called the vertex of a parabola and It is at this point where the parabola obtains its maximum or minimum value.
  • The value of the y co-ordinate of the vertex is called the optimal value.
Below you could see the steps

Step 1: Identify the form in which the given Parabolic equation is.
Step 2: If the equation is given in standard form use any of the three methods mentioned below to solve for the parabola.
  • By factorization
  • By completing the squares
  • By using quadratic formula.
Step 3: If the equation is given in Vertex form then find the x-intercept by plugging $y = 0$ to solve for parabola.
Step 4: Find the vertex of the parabola.
  • If the given equation is in the vertex form $y = ax - h^{2} + k$  then $(h,k)$ is the vertex of the parabola.
  • If the given equation is in standard form  $f(x)= ax^{2} + bx + c$ then $h =$ $\frac{-b}{2a}$ and $k = f h$.
Step 5: If the value of $a > 0$, the parabola opens up and if $a < 0$, then the parabola opens downwards.
Step 6:
Find the axis of symmetry which is given by $x = h$ or $x =$ $\frac{-b}{2a}$.