Polynomial and Other Equations

Main Topic: Graphing Polynomial and Rational Functions 

Subtopic: Use the technique of completing the square to interpret a quadratic function as a transformation of x2

Which transformation is needed to be used on x2, to get the graph of f(x) = 2x2 – 12x + 22?

A: Shift right by 3 units, stretch vertically by a factor 2 and then shift upward by 13 units

B: Shift left by 3 units, stretch vertically by a factor 2 and then shift upward by 4 units

C: Shift right by 3 units, stretch vertically by a factor 2 and then shift upward by 4 units - Correct option

D: Shift right by 3 units and shift upwards by 4 units

Answer: C

Rationale

First step is to use the technique of completing a square.

2x2 – 12x + 22

= 2 (x2 – 6x) + 22

= 2 (x2 – 6x + 9 - 9) + 22

= 2 (x2 – 6x + 9) – 18 + 22

= 2 (x – 3)2 + 4

Next step is to decide the transformation on x2.

First transformation is: shift graph of x2 to right side by 3 units to get a graph of (x - 3)2.

Then stretch the graph of (x - 3)2 vertically by factor 2 to get a graph of 2(x - 3)2.

Finally shift the graph of 2(x - 3)2 upwards by 4 units to get a graph of 2(x - 3)2 + 4.

Answer option A is incorrect. Check your calculations for completing the square method.

Answer option B is incorrect. Check your calculations for completing the square method. Also check the transformation from f(x) to f(x - c).

Answer option D is incorrect. Check your calculations for completing the square method. Also check the transformation from f(x) to c*f(x).