/*! 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 21 Find the standard form of the eq... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 21

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: \(\left(0, \frac{1}{2}\right)\)

Short Answer

Expert verified
The standard form equation of the parabola with the provided characteristics is \( y = x^2 \).

Step by step solution

01

Determine Position of Parabola

First, understand that the focus lies on the axis of the parabola. Since the focus is \( (0, \frac{1}{2}) \), with the vertex at the origin, the axis of the parabola must be parallel to the y-axis, and so the parabola opens either upwards or downwards. Because the y-coordinate of the focus is positive, this particular parabola opens upwards.
02

Use Vertex Form of Equation

As the parabola opens upwards, use the standard form for a parabola that opens up or down, which is \( y = a(x - h)^2 + k \). In this case, the vertex (h,k) is at the origin (0,0), so the formula simplifies to \( y = ax^2 \).
03

Use the Focus to Determine 'a'

Next, determine the value of exact 'a' using the given focus. The distance from the vertex to the focus is \( \frac{1}{4a} \), so \( \frac{1}{2} = \frac{1}{4a}\). Solving for 'a', you get \( a = 1 \).
04

Write Final Form of Equation

Substitute the value of 'a' into the equation \( y = ax^2 \), giving the final standard form equation of the parabola: \( y = x^2 \).

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

Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(9 x^{2}+4 y^{2}-54 x+40 y+37=0\)=

Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(9 x^{2}+4 y^{2}+36 x-24 y+36=0\)

Find the standard form of the equation of the ellipse with the given characteristics. Foci: (0,0),(0,8)\(;\) major axis of length 16

Find an equation of the tangent line to the parabola at the given point, and find the \(x\) -intercept of the line. \(x^{2}=2 y,\left(-3, \frac{9}{2}\right)\)

The concept of ________ is used to measure the ovalness of an ellipse.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.