/*! 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 20 Use the vertex and intercepts to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 20

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)=(x-3)^{2}+2$$

Short Answer

Expert verified
The vertex of the quadratic function \(f(x)=(x-3)^{2}+2\) is at \((3,2)\). The axis of symmetry is given by \(x=3\). The domain is all real numbers, or \(x \in (-\infty, +\infty)\), and the range is \(y \geq 2\) or \([2, +\infty)\).

Step by step solution

01

Identify the Vertex

The given function is in vertex form \(f(x) = a(x-h)^2 + k\). Compare the given function \(f(x)=(x-3)^{2}+2\) with the vertex form to find that the vertex is at \((h, k) = (3,2)\).
02

Sketch the Graph

Plot the vertex at point \((3,2)\). Because the coefficient of \(x^2\) is positive, the parabola opens upwards. The graph is a U-shaped curve that has its lowest point, the vertex, at \((3,2)\).
03

Identify the Axis of Symmetry

The axis of symmetry for a parabola in vertex form is expressed by the equation \(x=h\). Thus, for this given function, the equation of the axis of symmetry is \(x=3\).
04

Identify the Domain

The domain of a quadratic function is all real numbers, as there is no restriction on the input values for the function. Therefore, the domain of the given function is \(x \in (-\infty, +\infty)\).
05

Identify the Range

The range of a quadratic function depends on the direction in which the parabola opens. Because our parabola opens upwards, the range is all y-values greater than or equal to the y-coordinate of the vertex, so the range of the given function is \(y \geq 2\) or \([2, +\infty)\).

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

One's intelligence quotient, or IQ, varies directly as a person's mental age and inversely as that person's chronological age. A person with a mental age of 25 and a chronological age of 20 has an IQ of \(125 .\) What is the chronological age of a person with a mental age of 40 and an IQ of \(80 ?\)

The average number of daily phone calls, \(C\), between two cities varies jointly as the product of their populations, \(P_{1}\) and \(P_{2}\) and inversely as the square of the distance, \(d\), between them. a. Write an equation that expresses this relationship. b. The distance between San Francisco (population: \(777,000\) ) and Los Angeles (population: \(3,695,000\) ) is 420 miles. If the average number of daily phone calls between the cities is \(326,000,\) find the value of \(k\) to two decimal places and write the equation of variation. c. Memphis (population: \(650,000\) ) is 400 miles from New Orleans (population: \(490,000\) ). Find the average number of daily phone calls, to the nearest whole number, between these cities.

Use the four-step procedure for solving variation problems given on page 424 to solve Exercises 1-10. \(y\) varies directly as \(x . y=65\) when \(x=5 .\) Find \(y\) when \(x=12 .\)

Use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$\frac{1}{(x-2)^{2}}>0$$

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies inversely as \(x . y=12\) when \(x=5 .\) Find \(y\) when \(x=2\).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.