/*! 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 63 Graph each function. Be sure to ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 63

Graph each function. Be sure to label any intercepts. [Hint: Notice that each function is half a hyperbola.] \(f(x)=\sqrt{16+4 x^{2}}\)

Short Answer

Expert verified
The graph is half a hyperbola, symmetric about the y-axis, with a y-intercept at (0, 4).

Step by step solution

01

- Identify the Function Type

Recognize that the given function is a square root function with the form \(f(x) = \sqrt{a + bx^2}\). In this case, \(a = 16\) and \(b = 4\). This indicates that the function describes half of a hyperbola.
02

- Determine Domain

Derive the acceptable range of x-values (domain) for which the function is defined. Here, since the expression under the square root \(16 + 4x^2\) is always non-negative, \(x\) can be any real number. Thus, the domain of the function is \(-\infty < x < \infty\).
03

- Find the Intercepts

To find the intercepts, substitute \x = 0\ into the function. Calculate \(f(0) = \sqrt{16 + 4(0)^2} = \sqrt{16} = 4\). Therefore, the y-intercept is (0, 4). Note that there are no x-intercepts because \f(x)\ never equals zero.
04

- Analyze Shape and Symmetry

Recognize the symmetry of the hyperbola about the y-axis. This means \(f(-x) = f(x)\). Therefore, the graph is symmetric with respect to the y-axis.
05

- Plot Key Points

Choose additional key points to plot the curve. For instance, calculate \(f(1) = \sqrt{16 + 4(1)^2} = \sqrt{20} \approx 4.47\) and \(f(2) = \sqrt{16 + 4(2)^2} = \sqrt{32} \approx 5.66\). Similarly, find corresponding points for negative x-values using symmetry. Plot points like \( (1, 4.47)\), \((2, 5.66)\), \((-1, 4.47)\), and \((-2, 5.66)\).
06

- Draw the Graph

Using the y-intercept and the calculated points, draw the curve smoothly through these points. Make sure it reflects the symmetry about the y-axis.

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.

Square Root Function
A square root function has the basic form \( f(x) = \sqrt{ \text{expression} } \). In our exercise, the function is \(f(x) = \sqrt{16 + 4x^2} \). The expression inside the square root, \(16 + 4x^2\), determines the shape of our graph. The square root function always gives non-negative values because the square root of any non-negative number is also non-negative. This affects the range and behavior of our graph. When graphing, look for key points and intercepts to guide you on how to draw the curve accurately.
Hyperbola
A hyperbola is a type of curve seen in algebra and geometry. In the given square root function, the expression inside the square root, \(16 + 4x^2\), represents a form of half a hyperbola. Normally, a hyperbola has two branches, but our function shows only one side. By graphing \(f(x) = \sqrt{16 + 4x^2} \), we notice the shape resembles half of a hyperbola. Understanding this can help when predicting the general shape of our function as we graph it.
Function Symmetry
Function symmetry helps in graphing, as it shows how one side mirrors another. Our function, \(f(x) = \sqrt{16 + 4x^2} \), is symmetric about the y-axis. This means that \(f(-x) = f(x)\). For instance, if you know the point \((1, 4.47)\), you also automatically know the point \((-1, 4.47)\). This y-axis symmetry simplifies plotting points and gives more accuracy in drawing the graph correctly.
Function Domain
The domain of a function represents all the possible \(x\) values that can be input into the function. For \(f(x) = \sqrt{16 + 4x^2} \), the expression inside the square root \(16 + 4x^2\) must be non-negative. Here, the square root is always defined because \(16 + 4x^2\) can never be negative for any real number \(x\). Hence, the domain is \(-\infty < x < \infty\), which means all real numbers.
Intercepts
Intercepts are points where the graph crosses the axes. For the y-intercept, set \(x = 0\) in \(f(x) = \sqrt{16 + 4x^2} \). This gives \(f(0) = \sqrt{16 + 0} = \sqrt{16} = 4\), so the y-intercept is \((0, 4)\). The function has no x-intercepts because the square root function \sqrt{16 + 4x^2} \ is never zero. This means the graph does not touch the x-axis anywhere.
Plotting Points
Plotting points involves calculating several key values of \(x\) and finding their corresponding \(y\) values. For \(f(x) = \sqrt{16 + 4x^2} \), some points are: when \(x = 1\), \(f(1) = \sqrt{16 + 4(1)^2} = \sqrt{20} \approx 4.47\); when \(x = 2\), \(f(2) = \sqrt{16 + 4(2)^2} = \sqrt{32} \approx 5.66\). Due to symmetry, you also plot points for negative \(x\) values, such as \((-1, 4.47)\) and \((-2, 5.66)\). Finally, connect these points smoothly to reflect the curve's shape.

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

Find the center, transverse axis, vertices, foci, and asymptotes, Graph each equation. \(\frac{(x-2)^{2}}{4}-\frac{(y+3)^{2}}{9}=1\)

Use the fact that the orbit of a planet about the Sun is an ellipse, with the Sun at one focus. The aphelion of a planet is its greatest distance from the Sun, and the perihelion is its shortest distance. The mean distance of a planet from the Sun is the length of the semimajor axis of the elliptical orbit. The mean distance of Mars from the Sun is 142 million miles. If the perihelion of Mars is 128.5 million miles, what is the aphelion? Find an equation for the orbit of Mars about the Sun.

A racetrack is in the shape of an ellipse 100 feet long and 50 feet wide. What is the width 10 feet from a vertex?

Challenge Problem Show that the graph of an equation of the form \(A x^{2}+C y^{2}+D x+E y+F=0 \quad A \neq 0, C \neq 0\) where \(A\) and \(C\) are opposite in sign, (a) is a hyperbola if \(\frac{D^{2}}{4 A}+\frac{E^{2}}{4 C}-F \neq 0\) (b) is two intersecting lines if \(\frac{D^{2}}{4 A}+\frac{E^{2}}{4 C}-F=0\).

Extracorporeal shock wave lithotripsy is a procedure that uses shockwaves to fragment kidney stones without the need for surgery. Using an elliptical reflector, a shock wave generator is placed at one focus and the kidney stone is positioned at the other focus. Shock waves pass through a water column and disintegrate the stone, allowing it to be passed out of the body naturally. If the equation of the ellipse formed by the reflector is \(\frac{x^{2}}{324}+\frac{y^{2}}{100}=1,\) how far from the kidney stone does the shock wave generator need to be placed? (Units are in centimeters)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.