Problem 5 Writing an Equation of a Parabola

Multiple Choice Which is an equation of the parabola with vertex (3, 7) and focus (5, 7)?

1. x = 1 4 ( y − 7 ) 2 + 3 x equals , 1 fourth . open , y minus 7 , close squared . plus 3
2. y = 1 8 ( x − 7 ) 2 + 3 y equals , 1 eighth . open , x minus 7 , close squared . plus 3
3. x = 1 8 ( y − 7 ) 2 + 3 x equals , 1 eighth . open , y minus 7 , close squared . plus 3
4. x = 1 8 ( y + 7 ) 2 − 3 x equals , 1 eighth . open , y plus 7 , close squared . minus 3

The focus is to the right of the vertex, so the parabola is horizontal. Also, (h, k) is (3, 7) and (h + c, k) is (5, 7), so c = 2 . c equals 2 .  Substitute all this information into the equation for a horizontal parabola, x = 1 4 c ( y − k ) 2 + h , x equals , fraction 1 , over 4 c end fraction . open , y minus k , close squared . plus h comma  to get x = 1 8 ( y − 7 ) 2 + 3 . x equals , 1 eighth . open , y minus 7 , close squared . plus 3 .  The correct answer is C.

✓ Got It?

5. What is an equation of the parabola with vertex (1, 4) and focus (1, 6)?

✓ Lesson Check

Do you know HOW?

Write an equation of a parabola with the given information.

1. vertex (0, 0), focus ( 0 , 1 2 ) open . 0 comma , 1 half . close
2. vertex (3, 2), focus (4, 2)

Find the vertex, focus, and the directrix of each parabola.

3. x = 1 16 y 2 x equals , 1 sixteenth , y squared
4. y = x 2 + 6 x + 5 y equals , x squared , plus 6 x plus 5

Do you UNDERSTAND?

5. Vocabulary If the vertex of a parabola is 3 units from the focus, how far is the focus from the directrix?
6. Error Analysis The vertex of a parabola is at the origin, one unit away from the focus. A student concludes that the equation is y = 1 4 x 2 . y equals , 1 fourth , x squared , .  Identify at least two ways in which the student's equation might be in error.