Year 10: Quadratic turning point

What is a Quadratic turning point?

A quadratic equation in the form $y=ax\mathrm{²}+bx+c$ has a ‘turning point’ – also known as a vertex – on its parabola. This turning point represents the minimum or maximum value of the quadratic function.

Finding the turning point

1. Standard Form:$y=ax\mathrm{²}+bx+c$
2. x-coordinate of the turning point:$x=-b/2a$
3. Substitute 'x' back into the equation to find the y-coordinate.$y=a(-b/2a)\mathrm{²}+b(-b/2a)+c=c$

Using the vertex form

The vertex form of a quadratic equation is $y=a(x-h)\mathrm{²}+k$, where (h, k) are the coordinates of the turning point.

In this form, the turning point is always at the point (h, k).

Example

Let's say $y=2x\mathrm{²}-8x+6$.

1. $a=2$, $b=-8$, $c=6$
2. $x=-b/2a=-(-8)/(2\times 2)=8/4=2$
3. $y=2\left(2\right)\mathrm{²}-8\left(2\right)+6=8-16+6=-2$
4. The turning point is at (2, -2)

Remember to always consider whether the 'a' value is positive or negative, as this determines whether the parabola opens upwards (minimum) or downwards (maximum).