Answered: a. y b. y 3+ 2- 1 -5-4-32- 1 2 3 -2-…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Find the vertex, focus, and directrix of the parabola with the given equation (x + 1)² = -4(y + 1). Then match the equation to one of the graphs that are shown and labeled (a)–(d).

a.
y
b.
y
3+
2-
1
-5-4-32-
1 2 3
-2-
-4-3 -2
-11,
1 2 3 4
-3-
-3-
с.
y
d.
y
5-
3-
2-
1+
4+
+
2-
3
1-
+
-5-4-3-2-11 1 2 3
++ x
12 3 4
-4-3 -2
-2-
-3-
-2.
-3-

Expert Solution

Step 1

Given:

The equation of a parabola is,

${(x + 1)}^{2} = - 4 (y + 1)$

The above equation can be converted into vertex form as,

$y = \frac{- 1}{4} {(x + 1)}^{2} - 1 (1)$

The vertex form a parabola whose vertex $(h, k)$ with concave down and the axis parallel to $y$ -axis is given as,

$y = - a {(x - h)}^{2} + k (2)$

Step 2

On comparing equation (1) and (2),

$a = \frac{1}{4} h = - 1 k = - 1$

Therefore, the focus of the given parabola is at $(- 1, - 1) .$

From the vertex form $y = - a {(x - h)}^{2} + k$ , the coordinates of the focus is at $(h, k - \frac{1}{4 a})$

$k - \frac{1}{4 a} = - 1 - \frac{1}{4 \times \frac{1}{4}} = - 2$

Hence, the focus is at $(- 1, - 2) .$

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