/*! 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 25 Find the vertex, the focus, and ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 25

Find the vertex, the focus, and the directrix. Then draw the graph. $$x^{2}+2 x+2 y+7=0$$

Short Answer

Expert verified
Vertex: (-1, -3), Focus: (-1, -3.5), Directrix: y = -2.5.

Step by step solution

01

- Rewrite Equation in Standard Form

Rewrite the given equation in the form of a parabola equation. Start with the original equation: \(x^2 + 2x + 2y + 7 = 0\). Isolate the terms involving \(x\): \[x^2 + 2x = -2y - 7\]
02

- Complete the Square

Complete the square on the \(x\) terms. Take half of the coefficient of \(x\) (which is 2) and square it. Add and subtract this square inside the equation: \[x^2 + 2x + 1 = -2y - 7 + 1\]This simplifies to: \[(x + 1)^2 = -2y - 6\]
03

- Adjust Equation to Interpret Form

Rewrite the equation to make it clearer: \[(x + 1)^2 = -2(y + 3)\]This is the standard form of a parabola \((x - h)^2 = 4p(y - k)\) with \(h = -1\), \(k = -3\), and \(4p = -2\).
04

- Identify Vertex

From the standard form, identify the vertex \((h, k)\):The vertex is \((-1, -3)\).
05

- Calculate the Focus and Directrix

Calculate the value of \(p\) from \(4p = -2\): \[p = -\frac{1}{2}\]For a parabola that opens downwards, the focus is \((h, k + p)\):The focus is \((-1, -3 - \frac{1}{2}) = (-1, -3.5)\).The directrix is the line \(y = k - p\):The directrix is \(y = -3 + \frac{1}{2} = -2.5\).
06

- Graph the Parabola

Plot the vertex \((-1, -3)\), the focus \((-1, -3.5)\), and the directrix \(y = -2.5\). Draw the parabola opening downwards from the vertex toward the focus. The vertex is the highest point since the parabola opens downwards.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

standard form of a parabola
In mathematics, a parabola is often given in its standard form for easier analysis and graphing. The standard form of a parabola can vary depending on the orientation of the parabola. If the parabola opens upwards or downwards, the standard form is \((x-h)^2 = 4p(y-k)\). For parabolas opening left or right, the standard form becomes \((y-k)^2 = 4p(x-h)\). In both cases, \((h, k)\) represents the vertex of the parabola.
completing the square
Completing the square is a technique used to transform quadratic equations into a form that is easier to work with, particularly standard form. This method is especially useful in dealing with parabolas. To complete the square, follow these steps:
focus and directrix
The focus and directrix are crucial elements of a parabola. They help define its geometric properties. The focus is a point inside the parabola where all the reflected lines from any point on the parabola converge. The directrix is a line outside the parabola used together with the focus to construct the parabola.

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

Graph the system of inequalities. Then find the coordinates of the points of intersection of the graphs of the related equations. $$\begin{aligned} &y \geq x^{2}-3\\\ &y \leq 2 x \end{aligned}$$

Fill in the blank with the correct term. Some of the given choices will not be used. $$\begin{array}{ll}\text { piecewise function } & \text { ellipse }\\\ \text { linear equation } & \text { midpoint } \\ \text { factor } & \text { distance } \\ \text { remainder } & \text { one real-number } \\ \text { solution } & \text { solution } \\ \text { zero } & \text { two different real-number } \\\ x \text { -intercept } & \text { solutions } \\ y \text { -intercept } & \text { two different imaginary- } \\ \text { parabola } & \text { number solutions } \\ \text { circle } & \end{array}$$ A(n) __________________________________ is the set of all points in a plane the sum of whose distances from two fixed points is constant.

Solve. $$\begin{array}{l} a b-b^{2}=-4 \\ a b-2 b^{2}=-6 \end{array}$$

Solve. [ 5.5] $$\log (x-3)+\log x=1$$

Solve. $$\begin{aligned} &x^{2}+y^{2}+6 y+5=0\\\ &x^{2}+y^{2}-2 x-8=0 \end{aligned}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.