/*! 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 54 Use the vertex and the direction... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 54

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$x=-3(y-1)^{2}-2$$

Short Answer

Expert verified
The vertex is at (-2, 1); the parabola opens to the left. The domain is \((- \infty, -2]\) and the range is \((- \infty, \infty)\). The relation is not a function.

Step by step solution

01

Identifying Vertex and Direction of Opening

The parabola equation is given in vertex form as \(x=a(y-k)^{2}+h\). If a is less than 0, the parabola opens to the left. If a is greater than 0, the parabola opens to the right. Here, a is equal to -3, which is less than 0, so the parabola opens to the left. Vertex is at point(h, k). In our equation, h is -2 and k is 1, which gives us the vertex at (-2, 1).
02

Determining the Domain

For a parabola that opens to the left or right, the domain is an interval. Since our parabola opens to the left (meaning there is a maximum x-value), the domain is \((- \infty, h]\), in other words, \((- \infty, -2]\).
03

Determining the Range

The range of a parabola that opens to the left or right is all real numbers because there are no minimum or maximum y-values. Therefore, the range is \((- \infty, \infty)\).
04

Is it a Function?

A relation is a function if every input has exactly one output. In this equation, for every x in the domain, there can be two corresponding values of y as the graph is a downward parabola. Therefore, it is not a function.

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

Use a graphing utility to graph the parabolas in Exercises 77-78. Write the given equation as a quadratic equation in \(y\) and use the quadratic formula to solve for \(y .\) Enter each of the equations to produce the complete graph. $$y^{2}+10 y-x+25=0$$

Radio towers \(A\) and \(B, 200\) kilometers apart, are situated along the coast, with \(A\) located due west of \(B\). Simultaneous radio signals are sent from each tower to a ship, with the signal from \(B\) received 500 microseconds before the signal from \(A\) a. Assuming that the radio signals travel 300 meters per microsecond, determine the equation of the hyperbola on which the ship is located. b. If the ship lies due north of tower \(B\), how far out at sea is it?

In \(1992,\) a NASA team began a project called Spaceguard Survey, calling for an international watch for comets that might collide with Earth. Why is it more difficult to detect a possible "doomsday comet" with a hyperbolic orbit than one with an elliptical orbit?

In Exercises 79-80, write each equation as a quadratic equation in \(y\) and then use the quadratic formula to express \(y\) in terms of \(x\). Graph the resulting two equations using a graphing utility. What effect does the \(xy\)-term have on the graph of the resulting parabola? $$16 x^{2}-24 x y+9 y^{2}-60 x-80 y+100=0$$

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. \(9 x^{2}-16 y^{2}-36 x-64 y+116=0\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

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.