VIDEO: Transform vertex shape into factorized shape

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Vertex shape of a parabola - you should know that

  • Every quadratic function of the form y = ax² + bx + c can be converted into the so-called. Vertex shape y = a (x - xs) ² + ys reshaping, the easiest way to do this is with the square extension. This is always possible because every parabola has a vertex.
  • The vertex, i.e. the highest or lowest point of the parabola, can easily be read from the vertex shape, namely S (xs / ys).

Factored form - what is it?

  • This is a so-called. Linear factorization of the quadratic function.
  • In this case, the parabolic equation is represented with two simple brackets and has the form y = a (x - x1)*(x-x2).
  • This is x1 and x2 around the two zeros (points of intersection with the x-axis) of the parabola, which can be different but also identical.
    • Calculate the vertex coordinates of a parabola - this is how it's done

    Parabolas are the graphical representation of quadratic functions. …

  • The factored form only exists, of course, if the parabola has at least one root. Parabolasthat are completely above or below the x-axis cannot be displayed in factored form.

Bringing vertex shape into factored shape - here's how you can go about it

Depending on the task at hand, there are several ways of converting the parabola from the vertex form to the factored form, provided, of course, that it exists (see above).

  • Perhaps not the simplest, but a computationally feasible option is to use the vertex form to determine the zeros x1 and x2 to calculate.
  • To do this, simply set the vertex shape equal to zero (after all, you want to calculate the zeros), bring ys as well as a on the other side and extract the square root of both sides of the equation. Notice that there is both a negative and a positive root, from which you get the two zeros.
  • Now you have to find the result for x1 and x2 only in the factored form (s. o.).

Find factorized form - a calculated example

You have the parabola in the vertex form y = 1/2 (x - 3) ² -1. Incidentally, the vertex of this function is at S (3 / -1) (pay attention to the sign!).

  1. Set the vertex shape to zero and you get 0 = 2 (x - 3) ² -1.
  2. Do the math +1 and then times 2 and you get 2 = (x - 3) ².
  3. Now pull the square root (use TR) on both sides of the equation and get ± 1.41 (rounded for square root 2) = x - 3.
  4. From this you calculate the two zeros x1 = 4.41 and x2 = 1,59.
  5. The factorized form of this parabola is therefore y = 1/2 (x - 4.41)*(x-1.59).
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