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Question

A student thinks that y= 2x^2 - 12x + 15 is equivalent to y= 2 (x-3)^2 + 3. Prove algebraically that they are not equivalent.

Answer 1

 The function y = 2x² - 12x + 15

Convert the above function to vertex form.

Standard form of parabola y  = ax ² + bx + c

Vertex form of parabola is y  = a (x - h )² + k

Here x² coefficient is 2 , for perfect square make x² coefficient 1 by dividing each side by 2.

y/2 = (2x² - 12x + 15)/2

y/2 = x² - 6x + (15/2)

To complete the square add (half coefficient of x) ² to each side of the expression.

x coefficient = - 6 then (half the x coefficient)² is (-3)² = 9.

y/2 + 9 = x² - 6x + (15/2) + 9

y/2 = (x² - 6x + 9) + 15/2 - 9

y/2 = (x - 3)² - 3/2

y = 2 (x - 3)² - (3/2)2

y = 2 (x - 3)² - 3

y = 2 ( x - 3)² - 3

Therefore, y = 2x² - 12x + 15 is equivalent to y = 2 (x - 3)² - 3.

y = 2x² - 12x + 15 is not equivalent to y = 2 (x - 3)² + 3.