/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Find the vertex, focus, and dire... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 39

Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. $$(y+3)^{2}=12(x+1)$$

Short Answer

Expert verified
The vertex is \((-1, -3)\), the focus of the parabola is \((2, -3)\) and the equation of the directrix of the parabola is \(x = -4\). The parabola opens to the right.

Step by step solution

01

Determining the vertex

Compare the given equation \((y+3)^2=12(x+1)\) with standard form \((y-k)^2=4p(x-h)\). From the comparison, it can be inferred that the vertex \((h, k)\) of the parabola is \((-1, -3)\).
02

Calculating the value of p

The coefficient of \(x\) in the given equation is \(12 = 4p\). Solving this equation for \(p\) gives \(p = 3\). Note that since p > 0, the parabola opens to the right.
03

Finding the focus of the parabola

By definition, the focus \((h',k')\) of the parabola is \((h+p, k) = (-1+3, -3) = (2, -3)\).
04

Calculating the equation of directrix

Equation of directrix is \(x=h-p = -1-3 = -4\).
05

Sketching the parabola

Plot the vertex, focus and the directrix of the parabola. Draw a smooth curve passing through the vertex that bends and opens towards the focus while keeping away from the directrix.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$\left\\{\begin{array}{c}4 x^{2}+y^{2}=4 \\\x+y=3\end{array}\right.$$

Convert each equation to standard form by completing the square on \(x\) or \(y .\) Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. $$y^{2}-2 y+12 x-35=0$$

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((-5,0) ;\) Directrix: \(x=5\)

Describe how to graph \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)

Describe one similarity and one difference between the graphs of \(y^{2}=4 x\) and \((y-1)^{2}=4(x-1)\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.