/*! 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 53 Use the vertex and the direction... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 53

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$x=-4(y-1)^{2}+3$$

Short Answer

Expert verified
The parabola opens to the left. The vertex is (3,1). The domain of the given relation is \(-\infty, 3\] and the range is (-\infty, \infty). This relation is not a function.

Step by step solution

01

Identify the Direction of the Parabolic Opening

The parabola equation is given in the form \(x = a(y-k)^2 + h\), which indicates that the graph opens either to the left if \(a < 0\) or to the right if \(a > 0\). In the given equation, \(x = -4(y-1)^2 + 3\), 'a' is -4 which is less than 0, hence, the parabola opens to the left.
02

Calculate the Domain and Range

Since the parabola opens to the left and not up or down, the domain and range are different than usual. The domain (x-values) includes all real numbers less than or equal to the x-coordinate of the vertex because it opens to the left. Hence, the domain is \(-\infty, 3\]. The range (y-values), on the other hand, is all real numbers (-\infty, \infty) because the vertex's y-coordinate doesn't limit the graph.
03

Determine Whether the Relation is a Function

The relation is not a function. For every x-value in the domain, there are two corresponding y-values (above and below the vertex), invoking the 'vertical line test'. This violates the condition for a relation to be a function as each input (x-value) should have a unique output (y-value).

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

Graph \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) and \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\) in the same viewing rectangle for values of \(a^{2}\) and \(b^{2}\) of your choice. Describe the relationship between the two graphs.

Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. $$(x-2)^{2}=8(y-1)$$

How can you distinguish an ellipse from a hyperbola by looking at their equations?

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. $$y^{2}+10 y-x+25=0$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics 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.