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EXTRA

Unit 58: Vertex Form of a Parabola

This is a formula-based topic that the SAT will test you on.

The standard form of a parabola is 

$$y={ax}^2+bx+c$$

The vertex form of a parabola is 

$$boldsymbol yboldsymbol=boldsymbol aboldsymbol(boldsymbol xboldsymbol-boldsymbol hboldsymbol)^{mathbf2}boldsymbol+boldsymbol k$$

Just like in the standard form, a controls whether the parabola opens up (is positive) or down (a is negative).  

h and k are the coordinates for the vertex of the equation.  This might be written as (h,k).

An example:

What is the vertex of the following parabola: (y=3(x+3)^2-4) ?

Strategy:

  • Know the formula and fully understand all components of it.

Another example:

Which of the following is true about the parabola with the equation (y=a{(x-b)}^2+c) where (a>0)? 

A) The vertex is ((b,c)) and the graph opens upward. 
B) The vertex is ((b,c)) and the graph opens downward. 
C) The vertex is ((-b,c)) and the graph opens upward. 
D) The vertex is ((-b,c)) and the graph opens downward. 

This question is asking if you understand how the vertex form of a parabola equation works. 

Since a is positive, we know the parabola opens upward, so can immediately rule out B and D.  Because the standard formula is (– h), and here we have (x – b), we know it is just b for our x coordinate, giving us answer A.  If it has been (xb), then we would have selected answer C.