/*! 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 11 Graphing Functions Sketch a grap... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 11

Graphing Functions Sketch a graph of the function by first making a table of values. $$g(x)=-(x+1)^{2}$$

Short Answer

Expert verified
The function graph is a downward-opening parabola with vertex at \((-1, 0)\).

Step by step solution

01

Understand the Function

The function given is \( g(x) = -(x + 1)^2 \). This is a quadratic function, which means its graph will be a parabola. Since the coefficient of \( x^2 \) is negative, the parabola will open downward.
02

Determine the Vertex

The function is in the form \( g(x) = -(x - h)^2 + k \). Here, \( h = -1 \) and \( k = 0 \). Thus, the vertex of the parabola is at \((-1, 0)\).
03

Create a Table of Values

Select some values of \( x \) to substitute into \( g(x) \) and calculate corresponding \( g(x) \) values:- For \( x = -3 \), \( g(-3) = -((-3 + 1)^2) = -(4) = -4 \).- For \( x = -2 \), \( g(-2) = -((-2 + 1)^2) = -(1) = -1 \).- For \( x = -1 \), \( g(-1) = -((-1 + 1)^2) = 0 \).- For \( x = 0 \), \( g(0) = -(0 + 1)^2 = -1 \).- For \( x = 1 \), \( g(1) = -(1 + 1)^2 = -4 \).This table helps visualize key points on the graph.
04

Plot the Points on a Graph

Using the values obtained in the table, plot the points:- \(( -3, -4 )\)- \(( -2, -1 )\)- \(( -1, 0 )\)- \(( 0, -1 )\)- \(( 1, -4 )\)Make sure the graph is a smooth curve passing through these points.
05

Sketch the Parabola

Draw a smooth curve through the plotted points, ensuring the vertex is at \(-1, 0\), and the parabola opens downward. This represents the function \( g(x) = -(x + 1)^2 \).

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.

Parabola
A parabola is the U-shaped curve that represents the graph of a quadratic function. Quadratic functions take the form of \[ f(x) = ax^2 + bx + c \]. In this particular exercise, the function is \( g(x) = -(x + 1)^2 \). When you look at this function, it's clear that it doesn't have the standard quadratic form because it's expressed in the vertex form: \( g(x) = a(x - h)^2 + k \).

The most important part to recognize about the parabolas is the direction they open. You determine this from the leading coefficient \( a \):
  • If \( a > 0 \), the parabola opens upward, forming a smile shape.
  • If \( a < 0 \), like in our example with \( g(x) = -(x +1)^2 \), the parabola opens downward.
Understanding these basics gives you a visual cue about where the function will lie relative to the x-axis.
Vertex
The vertex of a parabola is essentially its peak or valley. It is one of the fundamental characteristics of quadratic functions and gives you a lot of information about the graph.

In the vertex form of a quadratic function, \( y = a(x - h)^2 + k \), the vertex is located at \( (h, k) \). In our function \( g(x) = -(x + 1)^2 \), you'll notice it can be rewritten to \(-((x) - (-1))^2 + 0 \), identifying the vertex at \((-1, 0)\).

Why is the vertex important?
  • It tells you the highest or lowest point on the graph. For \( g(x) \), since the parabola opens downward, the vertex is the highest point.
  • The vertex also indicates the line of symmetry of the parabola, which means both sides of the curve are mirror images of each other about a vertical line passing through the vertex.
The vertex simplifies predicting how the graph will look, offering a critical starting point when you begin to plot the curve.
Table of Values
Creating a table of values is an effective way to plot the graph of a function, especially when graphing a quadratic function like a parabola.One of the advantages of using a table of values is that it helps pinpoint precise locations on the graph to ensure accuracy.

To create the table of values for \( g(x) = -(x + 1)^2 \),
  • Start by selecting some values for \( x \). In this case, \( x = -3, -2, -1, 0, 1 \) were chosen.
  • Next, plug each of these \( x \) values into the function to get the corresponding \( g(x) \) or \( y \) values.
  • These calculations show the outcome of the function at specific points, resulting in pairs of coordinates like \( (-3, -4) \), \( (-2, -1) \), \( (-1, 0) \), \( (0, -1) \), and \( (1, -4) \).
By plotting these points on a graph, you can see where the parabola will pass, making it easier to draw the whole curve smoothly. With these points, you form the backbone of the graph, allowing you to accurately depict the shape and direction of the parabola.

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

A print shop makes bumper stickers for election campaigns. If \(x\) stickers are ordered (where \(x<10,000\) ), the price per bumper sticker is \(0.15-0.000002 x\) dollars, and the total cost of producing the order is \(0.095 x-0.0000005 x^{2}\) dollars. Use the fact that $$\text{revenue \(=\) price per item \(\times\) number of items sold}$$ to express \(R(x),\) the revenue from an order of \(x\) stickers, as a product of two functions of \(x .\)

The table shows the number of DVD players sold in a small electronics store in the years 2003-2013. $$\begin{array}{|c|c|} \hline \text { Year } & \text { DVD players sold } \\ \hline 2003 & 495 \\ 2004 & 513 \\ 2005 & 410 \\ 2006 & 402 \\ 2007 & 520 \\ 2008 & 580 \\ 2009 & 631 \\ 2010 & 719 \\ 2011 & 624 \\ 2012 & 582 \\ 2013 & 635 \\ \hline \end{array}$$ (a) What was the average rate of change of sales between 2003 and 2013 ? (b) What was the average rate of change of sales between 2003 and 2004 ? (c) What was the average rate of change of sales between 2004 and 2005 ? (d) Between which two successive years did DVD player sales increase most quickly? Decrease most quickly?

Find a function whose graph is the given curve. The top half of the circle \(x^{2}+y^{2}=9\)

Determine whether the equation defines \(y\) as a function of \(x .\) (See Example 9.) $$x^{2}+(y-1)^{2}=4$$

DISCOVER: Graph of the Absolute Value of a Function (a) Draw graphs of the functions $$f(x)=x^{2}+x-6$$ $$\text { and } \quad g(x)=\left|x^{2}+x-6\right|$$ How are the graphs of \(f\) and \(g\) related? (b) Draw graphs of the functions \(f(x)=x^{4}-6 x^{2}\) and \(g(x)=\left|x^{4}-6 x^{2}\right| .\) How are the graphs of \(f\) and \(g\) related? (c) In general, if \(g(x)=|f(x)|,\) how are the graphs of \(f\) and \(g\) related? Draw graphs to illustrate your answer.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

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.