/*! 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 66 Graph each function and state th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 66

Graph each function and state the domain and range. $$y=x^{2}-2 x-3$$

Short Answer

Expert verified
Domain: \(\text{all real numbers}\); Range: \([ -4, \, \text{鈭瀩 )\).

Step by step solution

01

Identify the function type

Observe that the given function is a quadratic function of the form \(y = ax^2 + bx + c\). In this case, we have \(a = 1\), \(b = -2\), and \(c = -3\).
02

Find the vertex

The vertex of a quadratic function in standard form \(y = ax^2 + bx + c\) can be found using \(x = -\frac{b}{2a}\). Substituting the values, we get \(x = -\frac{-2}{2 \times 1} = 1\). Substitute \(x = 1\) back into the equation to get \(y = (1)^2 - 2(1) - 3 = -4\). So, the vertex is at \((1, -4)\).
03

Determine the direction of the parabola

Since the coefficient of \(x^2\) (which is \(a = 1\)) is positive, the parabola opens upwards.
04

Plot additional points

To graph the function accurately, calculate a few more points on either side of the vertex. For example, calculate \(y\) for \(x = 0\) and \(x = 2\). \(y = 0^2 - 2(0) - 3 = -3\), and \(y = 2^2 - 2(2) - 3 = -3\). Plot the points \((0, -3)\) and \((2, -3)\).
05

Sketch the graph

Using the vertex and additional points, sketch the shape of the parabola opening upwards.
06

State the domain and range

For the quadratic function \(y = x^2 - 2x - 3\), the domain is all real numbers since it is defined for every real number \(x\). The range is \([ -4, \, \text{鈭瀩 )\), starting from the vertex's y-value and extending to positive infinity.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

graphing quadratic functions
Graphing quadratic functions is an important skill in algebra. Quadratic functions have the form \(y = ax^2 + bx + c\). They create parabolic shapes on a graph. To graph a quadratic function, follow these simple steps:

  • Identify the coefficients (\

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

For each pair of functions, find \(\left(f^{-1} \circ f\right)(x)\) $$f(x)=3 x-9 \text { and } f^{-1}(x)=\frac{1}{3} x+3$$

You can graph the relation \(x=y^{2}\) by graphing the two functions \(y=\sqrt{x}\) and \(y=-\sqrt{x} .\) Try it and explain why this works.

Find \(f^{-1}\). Check that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\) Strategy for Finding \(f^{-1}\) by Switch-and Solve. $$f(x)=\frac{3}{x-4}$$

Discussion. Let \(f(x)=\sqrt{x}-4\) and \(g(x)=\sqrt{x} .\) Find the domains of \(f, g,\) and \(g \circ f\).

Find the proportionality constant and write a formula that expresses the indicated variation. See Example 3. \(A\) varies inversely as \(B,\) and \(A=10\) when \(B=3\).
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

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.