/*! 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 27 Sketch the graph of the inequali... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 27

Sketch the graph of the inequality. $$y

Short Answer

Expert verified
The solution to the inequality \(y<x^{2}+x+8\) is the area under the graph of the parabola (which is shaped like a 'U'), with its vertex at \(-0.5, 7.75\).

Step by step solution

01

- Determine the vertex of the parabola

The parabola described by \(y=x^{2}+x+8\) can be rewritten as \(y=(x+0.5)^{2}+7.75\). The vertex of this parabola is \(-0.5, 7.75\).
02

- Draw the graph of the parabola

Sketch the parabola with the vertex at \(-0.5, 7.75\). Note that the parabola opens upwards, as the coefficient of the \(x^{2}\) term is positive.
03

- Identify the solution to the inequality

Since the inequality is \(y<x^{2}+x+8\), the solution will be the area under the parabola. Shade this area on your graph.

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 a U-shaped curve that can open upwards or downwards. The general form of a parabolic equation is given by \( y = ax^2 + bx + c \). The sign of the coefficient \( a \) determines the direction in which the parabola opens:
  • If \( a > 0 \), the parabola opens upwards.
  • If \( a < 0 \), the parabola opens downwards.
In our example, the equation \( y = x^2 + x + 8 \) has \( a = 1 \), which means the parabola opens upwards. Parabolas are symmetrical about their vertex, and have only one turning point. Understanding the properties of parabolas is key to solving inequalities that involve them.
Vertex Form
The vertex form of a quadratic equation makes it easy to identify the vertex of the parabola. It is typically written as \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex. This form is helpful because it clearly indicates the location of the vertex, and how the graph is shifted horizontally and vertically from the origin.
In our example, the standard form \( y = x^2 + x + 8 \) is rewritten in vertex form as \( y = (x + 0.5)^2 + 7.75 \). This reveals the vertex to be at \( (-0.5, 7.75) \), which is crucial for graphing. Calculating the vertex might involve completing the square, which is a process of rewriting a quadratic into its vertex form.
Graph Sketching
Graph sketching involves drawing the graph based on its characteristics, such as its vertex and the direction in which it opens. When graphing a quadratic inequality like \( y < x^2 + x + 8 \), you first graph the equality \( y = x^2 + x + 8 \).
  • Start by plotting the vertex, \( (-0.5, 7.75) \).
  • Next, draw the parabolic curve through the vertex, opening upwards, since the coefficient of \( x^2 \) is positive.
  • Finally, to represent the inequality \( y < x^2 + x + 8 \), shade the region below the curve. This represents all the points \( (x, y) \) that satisfy the inequality.
Graph sketching provides a visual means to solve and understand inequalities and equations. It's important to accurately represent the vertex and curve, as this forms the basis for identifying the solution region.

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

In Exercises 25 and 26 , use the vertical motion model \(\boldsymbol{h}=-\mathbf{1 6 t}^{2}+\boldsymbol{v t}+\boldsymbol{s}(\mathbf{p} . \mathbf{5 3 5})\) and the following information. You and a friend are playing basketball. You can jump with an initial velocity of 12 feet per second. You need to jump 2.2 feet to dunk a basketball. Your friend can jump with an initial velocity of 14 feet per second. Your friend needs to jump 3.4 feet to dunk a basketball. Can you dunk the ball? Can your friend? Justify your answers.

(8) Home COMPUTER SALES The sales \(S\) (in millions of dollars) of home computers in the United States from 1988 to 1995 can be modeled by \(S=145.63 t^{2}+3327.56,\) where \(t\) is the number of years since \(1988 .\) Use this model to estimate the year in which sales of home computers will be 36,000 million dollars.

A boulder falls off the top of a cliff during a storm. The cliff is 60 feet high. Find how long it will take for the boulder to hit the road below. Write the falling object model for \(s=60\)

Solve the inequality and graph the solution. 2 \leq x<5

Write the prime factorization. (Skills Review, p. \(T T T\) ) $$24$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.