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Grade 10, Academic (MPM2D)

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Mathematics
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Determining the Maximum and Minimum of Quadratics

All quadratic equations have at least one minimum or maximum value. Think of a parabola, they either open upwards like a smile  or face downwards like a frown . What dictates the direction of opening is the a term.

  • If the a term is positive, it open upwards (a minimum).
  • If the a term is negative, it opens downwards (a maximum).

No matter what format your quadratic is in (shown below), you can always determine whether it will produce a minimum or maximum.

  1. General form: y = ax² + bx + c
  2. Vertex form: y = a(x + h)² + k
  3. Factored form:y = a(x – r)(x – s)

Unfortunately, not only do you need to know if the quadratic is a minimum or maximum, you need to know the vertex point – the exact point (x, y) where the parabola changes direction. Of the three formats above, the only one capable of giving this information directly is the vertex form (more on the vertex form here). For example, if we have the equation:

y = (x + 4)² + 3

  • The a term is 1, which suggests a maximum due to the negative
  • The vertex is (–4, 3)

You’re probably wondering, if information about the vertex can’t be obtained from a general form quadratic, are we trapped? Luckily there’s a technique to changing the general form quadratics into vertex form known as completing the square. The process of involves changing the first two terms of a quadratic relation of the for y = ax² + bx + c into a perfect square while maintaining the balance of the original relation.

The method is carefully explained in the video below. Be sure to watch both examples since example (2) is slightly more complicated given the a term is greater than 1.

If you don’t want to go through the hassle of completing the square, there’s a “trick” you can use to quickly find the x-coordinate of the vertex which involves using the formula shown below:

h=\frac{-b}{2a}

To use it, you need to make sure you can identify the a and b-term coefficient. The h in the formula represents the x-coordinate of the vertex; so, if you want the y-coordinate, you substitute h value back into the equation for y (also known as k when in vertex form).

Here’s a quick video summarizing the usage of the formula:

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