Page 74 - Calculus for the Clueless, Calc II

Vertex Focus Directrix Equation Graph Comment

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(0,0) (0,c) y =-c x = 4cy The original derivation

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(0,0) (0,-c) y = c x = -4cy y replaced by -y

(0,0) (c,0) x =-c y2= 4cx x,y interchange in 1

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(0,0) (-c,0) x = c y -4cx x replaced by -x in 3

If you relate the picture to the original equation, the sketching will be easy.

Example 3—

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Given y = -7x, sketch. Label the vertex, focus, and directrix.
From the chart, we know the sketch is picture 4. Now let 4c = 7 (ignore the minus sign); c = 7/4. The vertex is
(0,0). The focus is (-7/4,0), because it is on x-axis to the left of the origin. The directrix is y = 7/4; y, a vertical
line, = +7/4, because it is to the right of the origin.

Example 4—

Sketch (y- 3) = -7(x + 2).
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To understand the following, we need only note the difference between x + y = 25 and (x- 3) + (y + 6) = 25.
Has the shape changed? No. Has the radius changed? No. What has changed? The center. Instead of being at
the point (0,0), the center is at the point (3,-6).