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GB-19-04-001 · Completing the square

The curve has equation y=x2−8x+15y=x^{2} - 8x + 15.

(a) Write y in completed-square form.

(b) Write down the coordinates of its turning point.

Part (a) Completing the Square (a = 1) & Turning Point Booklet p. 14
Method
1) Complete the square
2) Half the x coefficient inside the bracket
3) Subtract the added square
4) Read the turning point (−p, q)
Worked Example (GB-19-04-001 · Part (a))
Write y in completed-square form.
1)
Half the coefficient of x inside the bracket: 1[(x−4)2−(−4)2]+151[(x - 4)^2-(-4)^2] + 15.
2)
Simplify: y=(x−4)2−1y=(x - 4)^2 - 1.
Ans = y=(x−4)2−1y=(x - 4)^2 - 1
Part (b) Completing the Square (a = 1) & Turning Point Booklet p. 14
Method
1) Complete the square
2) Half the x coefficient inside the bracket
3) Subtract the added square
4) Read the turning point (−p, q)
Worked Example (GB-19-04-001 · Part (b))
Write down the coordinates of its turning point.
1)
Half the coefficient of x inside the bracket: 1[(x−4)2−(−4)2]+151[(x - 4)^2-(-4)^2] + 15.
2)
Simplify: y=(x−4)2−1y=(x - 4)^2 - 1.
3)
The squared term is zero at x=4x=4, giving turning point (4,−1)(4,-1).
4)
Since the coefficient of the square is positive, this is a minimum.
Ans = (4,−1)(4,-1)