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Chapter 6: Problem 37

Find the vertex, focus, and directrix of the parabola. Then sketch the parabola. $$(x+3)^{2}=4\left(y-\frac{3}{2}\right)$$

Short Answer

Expert verified
The Vertex is (-3,3/2), Focus is (-3,5/2), and Directrix is y=1/2.

Step by step solution

01

Identify the vertex

The standard form of a parabola is \( (x-h)^{2}=4a(y-k) \), where \( (h, k) \) is the vertex of the parabola. For our equation, by matching it with the standard parabolic equation, we identify that the vertex \( (h, k) = (-3, 3/2) \).
02

Identify the Focus

The coordinate of the focus is \( (h, k+a) \) where 'a' controls the 'openness' of the parabola. From our equation we can see that \( 4a = 4 \) so \( a = 1 \). Substituting these values, we get our focus as \( (-3, 3/2 + 1) = (-3, 5/2) \)
03

Identify the Directrix

The equation for the directrix is \( y = k - a \). Substituting the values we get from the equation, we get the directrix as \( y = 3/2 - 1 = 1/2 \)
04

Sketch the Parabola

Now, this point, the sketch can be drawn by plotting the vertex, focus and directrix and drawing the parabolic curve that touches the vertex and focus while remaining perpendicular to the directrix. As the value of a is positive, we know the parabola will open upwards.

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