/*! 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 77 Use the vertex and the direction... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 77

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

Short Answer

Expert verified
The given equation \(y=-x^{2}+4 x-3\) represents a parabola that opens downwards with the vertex at (2, -7). It has a domain of all real numbers and a range of \(y \le -7\). The given relation is a function.

Step by step solution

01

Identify Parameters of the Quadratic Equation

Given a quadratic equation in the form of \(y = ax^2 + bx + c\), we have \(a= -1, b= 4, c= -3\) in the equation \(y=-x^{2}+4 x-3\). The factor \(a\) tells us about the direction the parabola is opening. Here \(a<0\), meaning the parabola opens downwards.
02

Calculate Vertex of the Parabola

The vertex of the parabola can be calculated using the formula \(h = - \frac{b}{2a}\) for \(x\) coordinate and \(k = c - \frac{b^2}{4a}\) for \(y\) coordinate. Substituting the known values, we get:\(h = - \frac{4}{2(-1)} = 2\) and \(k = -3 - \frac{4^2}{4*(-1)} = -3 - 4 = -7\). So, the vertex is at \((2, -7)\).
03

Determine Domain and Range

Since a quadratic equation represents a parabola, the domain, or set of possible \(x\)-values is all real numbers. The range is determined by where the parabola opens and the \(y\)-coordinate of the vertex. The parabola opens downwards, so the maximum \(y\)-value will be at the vertex. Hence, the range is \(y \le -7\).
04

Determine if the Relation is a Function

A relation is a function if and only if each \(x\)-value in the domain corresponds to exactly one \(y\)-value in the range. In other words, no two ordered pairs have the same \(x\)-values. For a parabola, every \(x\) value is related to exactly one \(y\) value, hence the given equation is indeed 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 the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=(y-2)^{2}-4$$

Explain how to use \(x=y^{2}+8 y+9\) to find the parabola's vertex.

Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola. Then graph the conic section. $$x=(y-1)^{2}-4$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of \(x=a(y-k)+h\) is a parabola with vertex at \((h, k)\)

Will help you prepare for the material covered in the next section. Solve by the substitution method: $$\left\\{\begin{array}{c}4 x+3 y=4 \\\y=2 x-7\end{array}\right.$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

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.