(a) Express y = -x² + 2x + 3 in the form a(x + p)² + q. (b) Hence, write down the coordinates of the turning point on the graph of y = -x² + 2x + 3.

(a) To express the quadratic equation y = − x 2 + 2 x + 3 in the form a ( x + p ) 2 + q , we will use the method of completing the square:

Start with the original equation: y = − x 2 + 2 x + 3

Factor out − 1 from the terms involving x : y = − ( x 2 − 2 x ) + 3

Complete the square for the expression x 2 − 2 x :

Take half of the coefficient of x (which is − 2 ), square it, and add and subtract this inside the parentheses.
( − 2/2 ) 2 = 1
So we rewrite the expression as: y = − ( x 2 − 2 x + 1 − 1 ) + 3


Simplify the expression:

Combine the perfect square trinomial and the constant on the inside: y = − (( x − 1 ) 2 − 1 ) + 3


Distribute the − 1 and simplify: y = − ( x − 1 ) 2 + 1 + 3 y = − ( x − 1 ) 2 + 4


The equation in the form a ( x + p ) 2 + q is: y = − ( x − 1 ) 2 + 4
(b) The coordinates of the turning point (vertex) of a quadratic equation in the form a ( x + p ) 2 + q are given by ( − p , q ) .

From the equation y = − ( x − 1 ) 2 + 4 , we have: a = − 1 , ( x + p ) = ( x − 1 ) , q = 4
Here p = − 1 (since x + p = x − 1 , thus p = − 1 )



Thus, the coordinates of the turning point are: ( 1 , 4 )

Answered by danjohnbrain | 2025-07-21