Take Note Key Concept: Transformations of a Parabola

Problem 4 Analyzing a Parabola

What are the vertex, focus, and directrix of the parabola with equation y = x 2 − 4 x + 8 ? y equals , x squared , minus 4 x plus 8 question mark

First, complete the square to get the equation in vertex form.

y = x 2 − 4 x + 8  Standard form  y = a x 2 + b x + c y = ( x 2 − 4 x + 4 ) + 8 − 4 Add  ( 1 2 · − 4 ) 2 inside parentheses; subtract it outside. y = ( x − 2 ) 2 + 4 Vertex form  y = 1 4 c ( x − h ) 2 + k table with 3 rows and 3 columns , row1 column 1 , y , column 2 equals , x squared , minus 4 x plus 8 , column 3 cap standardform . y equals , eh x squared , plus b x plus c , row2 column 1 , y , column 2 equals open , x squared , minus 4 x plus 4 close plus 8 minus 4 , column 3 cap add . open . 1 half , middle dot negative 4 . close squared . insideparenthesessemicolonsubtractitoutside. , row3 column 1 , y , column 2 equals . open x minus 2 close squared . plus 4 , column 3 cap vertexform . y equals , fraction 1 , over 4 c end fraction . open , x minus h , close squared . plus k , end table

Note that, in this case, 1 4 c = 1 , fraction 1 , over 4 c end fraction , equals 1 comma  so c = 0.25 . c equals , 0.25 , .

The vertex (h, k) is (2, 4).

The focus (h, k + c) is (2, 4.25).

The directrix y = k − c y equals k minus c  is y = 3.75 . y equals , 3.75 , .

✓ Got It?

4. What are the vertex, focus, and directrix of the parabola with equation y = x 2 + 8 x + 18 ? y equals , x squared , plus 8 x plus 18 question mark