Converting an equation to a different form

Question

How would I express the equation y=x^2+2x+3 in y=a(x-h)^2+k form?

Answer 1

Stratton

y=x²+2x+3

Another method you could use here is graphical. If you plot this on Desmos or on a graphing calculator the coordinates of the vertex will be (h,k). The value of a will be the coefficient of x² in our expression for y (ie 1).

So if you plot this function you will find the vertex is (-1, 1), giving h=-1, k=1, and a(x-h)²+k = (x+1)²+1.

(you should check this by multiplying it out).

This form of the quadratic is known as the vertex form, which naturally suggests a graphical approach. I recommend practising visualising functions using a graphing calculator or Desmos.

Mike

Answer 2

y = x² + 2x + 3

To put the equation in the form y = a(x - h)² + k, complete the square:

y = (x² + 2x + 1) + 2

y = (x + 1)² + 2

This equation has the desired form with a = 1, h = -1 and k = 1.