VIDEO: Calculate the stretch factor of a parabola

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Parable - you need to know that

A parabola is the graph of a quadratic function of the form f (x) = ax2+ bx + c. It has an apex and is open upwards or downwards depending on the sign of the stretching factor a.

  • If a> 0, then the opening of the parabola is directed upwards. For a <0 the opening of the parabola is directed downwards.
  • If the stretching factor a is between -1 and +1, then one speaks of stretching the parabola with respect to the x-axis. If a> +1 or a
  • It may also be that your parabola is in the vertex shape f (x) = a (x-d)2+ e is given. You can convert the general representation into the vertex form at any time by adding a square.

This is how you determine the stretching factor of the parabola

  • It is particularly easy, of course, if you have given the function equation of the parabola. All you have to do is read the a from your equation and have determined the stretch factor.
    • Setting up the vertex function - this is how you proceed

    A known problem - you have the vertex and one more point ...

  • It is a little more difficult when you have given a drawing. However, there are also various ways in which you can proceed here. You will find these in the next sections.

An example to calculate the stretch factor

Suppose you have given the graph of a parabola and you want to calculate the corresponding function. You can use the parabolic equation in the vertex form f (x) = a (x-d)2+ e specify.

  1. For example, if you now read S (1 | 2) for the vertex, then you can substitute the coordinates of the vertex in the above function. You get f (x) = a (x-1)2+2.
  2. Now you need one more point. Assume you read the further point P (2 | 3) of the parabola.
  3. Now do a point test for this point and you get 3 = a (2-1)2+2 <=> 3 = a + 2 <=> a = 1. So the stretch factor is 1.

Another way of calculating

If your parabola has two zeros, then you can find the parabola equation just as easily.

  1. Assume the zeros are N1(1 | 0) and N2(4|0). Then you can again state the functional equation of the parabola as a function of the stretching factor a. We have f (x) = a (x-1) (x-4).
  2. Now you need another point. For example, if you now read the vertex S (2.5 | 4.5), then you can carry out a point test for S once more.
  3. You get 4.5 = a (2.5-1) (2.5-4) <=> 4.5 = a (1.5) (- 1.5) <=> 4.5 = -2, 25a <=> a = -2. So the stretch factor is -2.

This is also how you can determine the factor

You can also determine the parabolic equation when you have read or given 3 points of the parabola. The parabola is in the form f (x) = ax2+ bx + c given.

  1. Now you need to do 3 point samples for your 3 points and solve the linear system of equations using the Gaussian algorithm to find the parameters a, b and c. Suppose your points are A (-1 | 1), B (0 | 0), C (2 | 4). For the 3 point samples you will receive the 3 Equations 1 = a-b + c, 0 = c, 4 = 4a + 2b + c.
  2. If you now insert equation 2 into the other two equations, this results in 1 = a-b and 4 = 4a + 2b.
  3. Solve the first of the two equations for a: a = 1 + b.
  4. Plug this into the second equation and you can determine b: 4 = 4 (1 + b) + 2b <=> 0 = 6b <=> b = 0.
  5. This results in equation 1: a = 1. So overall you have the parabolic equation f (x) = x2. It is the normal parabola with an aspect ratio of 1.

As you can see, there are different ways of determining the stretching factor of a parabola.

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