/*! 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 47 The graph of a polynomial functi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 47

The graph of a polynomial function is given. From the graph, find (a) the \(x\) - and \(y\) -intercepts, and (b) the coordinates of all local extrema. $$P(x)=-x^{2}+4 x$$

Short Answer

Expert verified
x-intercepts: (0,0) and (4,0); y-intercept: (0,0); Local maximum at (2,4).

Step by step solution

01

Identify the Function

The polynomial function given is \(P(x)=-x^{2}+4x\). This is a quadratic function and can be expressed in the form \(P(x) = ax^2 + bx + c\) with \(a = -1\), \(b = 4\), and \(c = 0\).
02

Find the x-intercepts

To find the x-intercepts, set \(P(x) = 0\).\[-x^{2} + 4x = 0\]Factor the equation:\[x(-x + 4) = 0\]Thus, \(x = 0\) or \(x = 4\).The x-intercepts are (0,0) and (4,0).
03

Find the y-intercept

The y-intercept is the point where the graph crosses the y-axis. This occurs when \(x = 0\). Substitute \(x = 0\) into the function:\[P(0) = -(0)^2 + 4(0) = 0\]The y-intercept is at (0,0).
04

Calculate the Vertex

The vertex of a parabola in standard form \(P(x) = ax^2 + bx + c\) can be calculated using the formula \(x = -\frac{b}{2a}\).Here, \(a = -1\) and \(b = 4\), so:\[x = -\frac{4}{2(-1)} = 2\]Substitute \(x = 2\) back into the function to find \(y\):\[P(2) = -(2)^2 + 4(2) = -4 + 8 = 4\]The vertex, which is a local maximum for this downward opening parabola, is at (2,4).
05

List Local Extrema

Since the polynomial is a quadratic function opening downward, it has a single local maximum at the vertex. The local maximum occurs at (2,4).

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.

x-intercepts
The x-intercepts of a function are points where the graph crosses the x-axis. For a quadratic function like \( P(x) = -x^2 + 4x \), we find the x-intercepts by setting the function equal to zero. This is because at the x-intercept, the y-value is zero.
To solve \(-x^2 + 4x = 0\), you can factor out an \( x \):
  • \( x(-x + 4) = 0 \)
This gives us two solutions: \( x = 0 \) and \( x = 4 \). So, the x-intercepts are the points (0,0) and (4,0). These points tell us where the parabola touches the x-axis. Identifying x-intercepts is crucial for understanding the roots of the function.
y-intercepts
The y-intercept of a graph is the point where it crosses the y-axis. This occurs when \( x = 0 \). For our quadratic function \( P(x) = -x^2 + 4x \), we can find the y-intercept by substituting \( x = 0 \) into the function.
So,
  • \( P(0) = -(0)^2 + 4(0) = 0 \)
Thus, the y-intercept is at the origin (0,0). This means that our parabola touches both the x-axis and y-axis at this single point. Understanding the y-intercept helps when drawing the graph, as it gives a starting point on the y-axis.
local extrema
Local extrema in a function are points where the function reaches a local minimum or maximum. For a quadratic function like \( P(x) = -x^2 + 4x \), the parabola opens downwards since the leading coefficient \( a \) is negative. Therefore, this function has a local maximum.
A short guide to identify local extrema:
  • Differentiate the function to find the critical points.
  • Use the first or second derivative test to determine if these points are minima or maxima.
In this quadratic, it matches the vertex, which is found by other methods as shown in the next section.
vertex of a parabola
The vertex of a quadratic function is an important feature since it is the highest or lowest point on the graph, known as the parabola's peak. For a standard quadratic function in the form \( ax^2 + bx + c \), the vertex can be calculated using the formula:
  • \( x = -\frac{b}{2a} \)
For \( P(x) = -x^2 + 4x \), with \( a = -1 \) and \( b = 4 \), we find:
  • \( x = -\frac{4}{2(-1)} = 2 \)
Substitute \( x = 2 \) in the function to find \( y \):
  • \( P(2) = -(2)^2 + 4(2) = 4 \)
Thus, the vertex is at (2,4). Since our parabola opens downward, this vertex represents a local maximum. The vertex aids in sketching the parabola accurately and understanding the graph鈥檚 behavior.

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

Graph the rational function \(f\), and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{-x^{3}+6 x^{2}-5}{x^{2}-2 x}, \quad g(x)=-x+4$$

Recall that the symbol \(\bar{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. $$\bar{z}+\bar{w}=\overline{z+w}$$

Evaluate the expression and write the result in the form \(a+b i\) $$(2-3 i)^{-1}$$

Find all solutions of the equation and express them in the form \(a+b i\) $$2 x^{2}+3=2 x$$

Evaluate the radical expression and express the result in the form \(a+b i\) $$(3-\sqrt{-5})(1+\sqrt{-1})$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied 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.