/*! 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 18 Use the vertex and intercepts to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 18

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. $$f(x)=(x-1)^{2}-2$$

Short Answer

Expert verified
The vertex is at (1, -2), the x-intercepts are \(x = 1 - 鈭�2\) and \(x = 1 + 鈭�2\), and the y-intercept is -1. After plotting these on a graph, a parabolic curve can be drawn. It can be observed that both x and y are all real numbers, hence the domain is \(-鈭�, 鈭瀄), and the range is \([-2, + 鈭�)\). The axis of symmetry is \(x = 1\).

Step by step solution

01

Identify the Vertex

The quadratic function is in the form \(f(x) = (x - a)^2 + b\), where the vertex is at the point (a, b). Hence, for \(f(x) = (x - 1)^2 - 2\), the vertex is at the point (1, -2).
02

Identify the Intercepts

To find the x-intercepts, set \(f(x) = 0\), and solve for \(x\). \n 0 = (x - 1)^2 - 2 \n (x - 1)^2 = 2 \n x - 1 = 卤鈭�2 \n x = 1 卤鈭�2. \n The x-intercepts are \(x = 1 - 鈭�2\) and \(x = 1 +鈭�2\). The y-intercept is found by setting \(x = 0\). \n f(0) = (0 - 1)^2 - 2 = -1. Hence, the y-intercept is -1.
03

Sketch the Graph

Identify the vertex and intercepts on the graph. Because this is a quadratic function, it is known it will have a U-shape (a parabola). Sketch the graph to include the vertex at (1, -2), x-intercepts at \(1 - 鈭�2\) and \(1 + 鈭�2\), and the y-intercept at -1.
04

Identify the Domain and Range

From the graph, it can be seen that the function's domain, which is the set of all permissible x-values, is all real numbers or \(-鈭�, 鈭瀄). The range of a function refers to all the possible values of y, based on the values of x. As you can see from the graph, y reaches a minimum value at y = -2 (the y coordinate of the vertex) and goes up to +鈭�. Hence, the range is \([-2, + 鈭�)\).
05

Determine the Equation of the Axis of Symmetry

In a quadratic function, the axis of symmetry is a vertical line that passes through the vertex of the parabola. It can be represented by the equation \(x = a\), where a is the x-coordinate of the vertex. Hence, for this quadratic function, the axis of symmetry is \(x = 1\).

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

Write a rational inequality whose solution set is \((-\infty,-4) \cup[3, \infty)\).

Determine whether cach statement is true or false If bhe statement is false, make the necessary change(s) to produce a true statement. It is possible to have a rational function whose graph has no \(y\) -intercept.

Will help you prepare for the material covered in the next section. a. If \(y=k x^{2},\) find the value of \(k\) using \(x=2\) and \(y=64\) b. Substitute the value for \(k\) into \(y=k x^{2}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\)

What is a rational function?

If you are given the equation of a rational function, explain how to find the vertical asymptotes, if there is one, of the function's graph.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.