Lesson 4

Working Backwards: Expanding and Then Simplifying the Vertex Form

You have been verifying whether two functions written in the forms y = ax2 + bx + c and y = a(x − p)2 + q represent the same function by graphing the two functions to see whether their graphs are identical.

Another method to verify your work is to work backwards by expanding and then simplifying the vertex form of the quadratic function to see if the vertex form matches the standard form.

Try This 3

1. Take the vertex form you derived in Self-Check 2 question 1.a., y = 2(x − 2)2 + 2. Expand the function by squaring the binomial term. Then simplify the polynomial on the right side of the function to a trinomial. This will show that the function is equivalent to the original form, y = 2x2 − 8x + 10.
   
   
2. Now show that y = 4(x −3)2 + 6 is not equivalent to y = 4x2 + 24x + 30.
   
   
   1. Square the binomial term and simplify the polynomial to a trinomial.
      
      
   2. Complete the square of the quadratic function in standard form.

If you feel you have a solid understanding of how to determine the characteristics of a quadratic function in the form y = ax2 + bx + c and how to determine whether two functions written in the forms y = ax2 + bx + c and y = a(x − p)2 + q represent the same function, go to Connect. If you need a bit more practice, complete Self-Check 3.