/*! 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 59 For each polynomial function, ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 59

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$f(x)=-\left(x^{2}-1\right)(x-2)(x+3)$$

Short Answer

Expert verified
The function \(f(x)= -x^{2}\) has the same end behavior as \(f(x)=-\left(x^{2}-1\right)(x-2)(x+3)\). The x-intercepts are \(-3\), \(-1\), and \(2\), and the y-intercept is \(-6\). The function is positive for \(x < -3\) and \(x > 2\), and negative for \(-3 < x < -1\) and \(-1 < x < 2\).

Step by step solution

01

Determine the end behavior

The degree of the polynomial is 2 + 1 + 1 = 4, which is even, and the leading coefficient is negative. Therefore, as x goes to positive or negative infinity, \(f(x)\) approaches positive infinity. A quadratic function with the same end behavior is \(y=-x^{2}\).
02

Find the x- and y-intercepts

The x-intercepts can be found by setting \(f(x)\) equal to zero and solving for x: \(-\left(x^{2}-1\right)(x-2)(x+3)=0\). This leads to \(x=\pm1, 2, -3\). The y-intercept can be found by setting x equal to zero in \(f(x)\), which gives \(f(0)= -\left(0^{2}-1\right)(0-2)(0+3) = -6\)
03

Determine the intervals where the function is positive and negative

Knowing the x-intercepts, we can examine the intervals of \(x\) between and beyond these points. Choose test points in these intervals and substitute into \(f(x)\): For \(x < -3\), choose \(x= -4\), then \(f(-4) > 0\). For \(-3 < x < -1\), choose \(x= -2\), then \(f(-2) < 0\). For \(-1 < x < 2\), choose \(x= 0\), then \(f(0) < 0\). For \(x > 2\), choose \(x= 3\), then \(f(3) > 0\). Therefore, \(f(x)\) is positive for \(x < -3\) and \(x > 2\), and negative for \(-3 < x < -1\) and \(-1 < x < 2\).
04

Drawing the graph of \(f(x)\)

Plot the x and y intercepts onto a graph. The intercepts at x= -3, -1, 2 and y= -6. Sketch a curve passing through these points that reflects the determined end behavior and regions of positivity and negativity. The curve starts and ends up high (since \(f(x)\) approaches positive infinity as \(x\) approaches both positive and negative infinity), dips below the x-axis between \(x = -1\) and \(x = 2\) , and rises above the x-axis when \(x < -3\) or \(x > 2\) .

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.

End Behavior of Polynomials
The end behavior of a polynomial function describes how the function behaves as the input values, or x, approach positive or negative infinity. This behavior is largely determined by two main components: the degree of the polynomial and its leading coefficient.

For our given polynomial, the degree is 4, calculated as the sum of the exponents: 2, 1, and 1. This is an even degree, and since the leading coefficient is negative, the function's behavior is as follows:
  • As the input values approach positive infinity (\(x \to \infty\)), the value of the function (\(f(x)\)) approaches positive infinity.
  • Similarly, as the input values approach negative infinity (\(x \to -\infty\)), the value of the function remains consistently approaching positive infinity.
To match this behavior with a simpler quadratic function, we use \(y = -x^{2}\), exemplifying similar end behavior with an upside-down parabolic shape.
Graph Intercepts
Graph intercepts are the points where a polynomial function crosses the x-axis and y-axis. These points help in understanding the shape and position of the graph relative to the axes.

**X-intercepts:**
To find the x-intercepts, we solve the equation \(f(x) = 0\). In this case, the factored form of the polynomial gives us:
  • \(x^{2} - 1 = 0\), leading to \(x = \pm 1\).
  • \(x - 2 = 0\), resulting in \(x = 2\).
  • \(x + 3 = 0\), giving \(x = -3\).

**Y-intercept:**
The y-intercept is found by setting x to 0 in the function. Substituting gives: \(f(0) = -((-1)(-6)) = -6\). Therefore, the y-intercept is at \(y = -6\). These intercepts are crucial for plotting key points on the graph and envisioning its progression.
Intervals of Positivity and Negativity
Understanding where a polynomial is positive or negative involves examining the function's sign in different segments along the x-axis. The x-intercepts help define these segments as they divide the x-axis into intervals.

For the given polynomial, the intervals formed are:
  • \(x < -3\): Positive. Testing with \(x = -4\) shows \(f(-4) > 0\).
  • \(-3 < x < -1\): Negative. Testing with \(x = -2\) gives \(f(-2) < 0\).
  • \(-1 < x < 2\): Negative. Testing with \(x = 0\) provides \(f(0) < 0\).
  • \(x > 2\): Positive. Testing with \(x = 3\) results in \(f(3) > 0\).
These findings show where the polynomial lies above or below the x-axis, completing the picture needed to sketch the graph accurately. By determining these intervals, you can predict how the function behaves between and beyond its intercepts, aiding in constructing a reliable visual representation.

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

Solve the rational inequality. $$\frac{x+2}{x-1} \leq 0$$

Solve the rational inequality. $$\frac{2}{x+1}>\frac{1}{x-2}$$

Let \(p(x)=x^{5}+x^{3}-2 x\). (a) Show that \(p\) is symmetric with respect to the origin. (b) Find a zero of \(p\) by inspection of the polynomial expression. (c) Use a graphing utility to find the other zeros. (d) How do you know that you have found all the zeros of \(p ?\)

A gift box company wishes to make a small open box by cutting four equal squares from a 3 inch by 5 -inch card, one from each corner. (a) Let \(x\) denote the length of the square cut from each corner. Write an expression for the volume of the box in terms of \(x\). Call this function \(V(x) .\) What is the realistic domain of this function? (b) Write an expression for the surface area of the box in terms of \(x .\) Call this function \(S(x)\) (c) Write an expression in terms of \(x\) for the ratio of the volume of the box to its surface area. Call this function \(r(x)\) (d) Fill in the following table giving the values of \(r(x)\) for the given values of \(x\) (TABLE CANNOT COPY) (e) What do you observe about the ratio of the volume to the surface area as \(x\) increases? From your table, approximate the value of \(x\) that would give the maximum ratio of volume to surface area. (f) \(=\) Use a graphing utility to find the value of \(x\) that would give the maximum ratio of volume to surface area.

How much pure gold should be added to a 2-ounce alloy that is presently \(25 \%\) gold to make it \(60 \%\) gold?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Statistics

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.