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Chapter 6: Problem 38

Find the vertex, focus, and directrix of the parabola. Then sketch the parabola. $$\left(x+\frac{1}{2}\right)^{2}=4(y-1)$$

Short Answer

Expert verified
The vertex of the parabola is (-1/2, 1), the focus is (-1/2, 2), and the equation of the directrix is y = 0. The parabola opens upwards.

Step by step solution

01

Identify the Vertex

The vertex of the parabola is given by the point (h, k). By comparing the equation \((x+\frac{1}{2})^{2}=4(y-1)\) with the standard form, we can find the coordinates of the vertex to be (-1/2, 1).
02

Identify the Focus

The value of a can be found where 4a is the coefficient of y in the equation \((x+\frac{1}{2})^{2}=4(y-1)\), so a = 1. The focus lies a units along the axis of symmetry from the vertex. Here, since the parabola opens upwards, the focus will be (h, k+a) or (-1/2, 1+1) which gives the focus as (-1/2, 2).
03

Identify the Directrix

The directrix is a line that is a units in the opposite direction from the focus through the vertex. So, for a parabola opening upwards, the directrix will be y = k - a. This gives y = 1 - 1 = 0.
04

Sketch the Parabola

First plot the vertex (-1/2, 1). Then, plot the focus and draw the directrix. Since the parabola is vertically oriented and opens upwards, the parabola will be above the directrix and the vertex, enveloping the focus. Sketch the parabola, being careful to make the parabola symmetrical above the directrix.

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