/*! 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 77 Sketch the graph of the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 77

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the real zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. $$f(t)=\frac{1}{4}\left(t^{2}-2 t+15\right)$$

Short Answer

Expert verified
The graph of the given function opens upwards (as indicated by the Leading Coefficient Test), and it does not cross the \( t \) axis (as there are no real zeros). Points (-1, 16), (0, 15), (1, 14) lie on the graph, which is a smooth continuous curve running through these points, extending infinitely upwards.

Step by step solution

01

Apply Leading Coefficient Test

The leading term of the polynomial in \(f(t) = \frac{1}{4}(t^{2} - 2t + 15)\) is \( \frac{1}{4}t^{2}\). Since the leading coefficient, \( \frac{1}{4}\), is positive and the degree, 2, is even, the graph opens upwards and is wider than the standard parabola \( y = x^{2} \). This indicates that as \( t \to 卤鈭� \), \( f(t) \to 鈭� \).
02

Find real zeros of the polynomial

To find real zeros of the polynomial, we set \( f(t) = 0 \) and solve the equation: \( 0 = \frac{1}{4}(t^2 - 2t +15), \) which simplifies to \( 0 = t^2 - 2t + 15 \). Solving this equation gives complex roots (as the discriminant \( b^{2} - 4ac = 2^{2} - 4*15 < 0 \)), hence there are no real zeros.
03

Plot sufficient solution points

Since we do not have real zeros, we find other points by choosing values of \( t\) and computing corresponding \( f(t)\) values. Common choices are \( t = -1, 0, 1 \) and data points obtained are: \( (-1,16), (0,15), (1,14) \) respectively.
04

Draw continuous curve through points

Now plot all these points (including imaginary zeros as well) and draw a smooth, continuous curve to form the graph of the given function. Since the graph opens upwards as indicated by the Leading Coefficient Test and has a minimum point (the vertex of the parabola), the curve will touch the \( t \) axis at the lowest point and then extend upwards on either side.

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!

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

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. $$f(x)=x^{4}+6 x^{2}-27$$

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{4}-4 x^{3}+16 x-16\) (a) Upper: \(x=5\) (b) Lower: \(x=-3\)

Geometry You want to make an open box from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. (a) Let \(x\) represent the side length of each of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume \(V\) of the box as a function of \(x .\) Determine the domain of the function. (c) Sketch the graph of the function and approximate the dimensions of the box that will yield a maximum volume. (d) Find values of \(x\) such that \(V=56 .\) Which of these values is a physical impossibility in the construction of the box? Explain.

Use the information in the table to answer each question. $$\begin{array}{|c|c|} \hline \text { Interval } & \text { Value of } f(x) \\\\\hline(-\infty,-2) & \text { Positive } \\\\\hline(-2,1) & \text { Negative } \\\\\hline(1,4) & \text { Negative } \\\\\hline(4, \infty) & \text { Positive } \\\\\hline\end{array}$$ (a) What are the three real zeros of the polynomial function \(f ?\) (b) What can be said about the behavior of the graph of \(f\) at \(x=1 ?\) (c) What is the least possible degree of \(f ?\) Explain. Can the degree of \(f\) ever be odd? Explain. (d) Is the leading coefficient of \(f\) positive or negative? Explain. (e) Sketch a graph of a function that exhibits the behavior described in the table.

Write the polynomial as the product of near factors and list all the zeros of the function. $$f(x)=x^{4}+29 x^{2}+100$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Discrete 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.