/*! 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 75 Use the catalog of basic functio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 75

Use the catalog of basic functions (page 170 ) and Section 3.4 to describe the graph of the given function. $$k(x)=\sqrt{x+4}-4$$

Short Answer

Expert verified
Question: Describe the graph of the function k(x) = 鈭�(x+4) - 4. Answer: The graph of the given function is a square root function that has been horizontally shifted 4 units to the left and vertically shifted 4 units down, with a domain of [-4, 鈭�) and a range of [-4, 鈭�).

Step by step solution

01

Identify the parent function

The parent function for the given function is the square root function, which has the general form: $$f(x) = \sqrt{x}$$
02

Identify the transformation

The given function is: $$k(x)=\sqrt{x+4}-4$$ Compare this with the parent function, we can see the transformation is: 1. Horizontal shift: 4 units to the left 2. Vertical shift: 4 units down
03

Determine the domain and range

Determining the domain: We know that with the square root function, the inside expression must be non-negative. So, for the given function: $$x+4 \geq 0$$ Solve for x: $$x \geq -4$$ So, the domain of the function is \([-4, \infty)\). Determining the range: Since the function has a vertical shift of 4 units down, the entire graph will be shifted down from its usual position. The range of the parent square root function is \([0, \infty)\). Thus, the range of the transformed function is: $$[0-4, \infty) = [-4, \infty)$$
04

Analyze and describe the graph

Now let's describe the graph of the function using the information we gathered from Steps 1-3: 1. Parent function: The graph of the given function is based on the square root function, $$\sqrt{x}$$. 2. Transformation: The graph has undergone a horizontal shift of 4 units to the left and a vertical shift of 4 units down. 3. Domain: The domain of the function is \([-4, \infty)\). 4. Range: The range of the function is \([-4, \infty)\). Combining all this information, we can describe the graph of the given function as a square root function that has been horizontally shifted 4 units to the left and vertically shifted 4 units down, with a domain of \([-4, \infty)\) and a range of \([-4, \infty)\).

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.

Parent Function
When talking about graph transformations, it's essential to start with the parent function. Think of the parent function as the simplest version of a function that forms the basis for creating more complex versions.
For a square root function, the standard parent function is written as:
  • \( f(x) = \sqrt{x} \)
This function forms a gentle curve starting from the origin (0,0) that continues to rise indefinitely along the x-axis.
All transformations, like shifts and stretches, are applied to this parent function to generate new graphs.
Knowing the parent function helps you identify how the graph is transformed.
Square Root Function
The square root function is a fundamental nonlinear function characterized by its unique curve.
It only exists on non-negative values of \(x\) due to the nature of square roots, which cannot have negative radicands without involving imaginary numbers.
The graph of the simplest square root function, \(f(x) = \sqrt{x}\), starts at the origin and gradually increases as \(x\) increases.
  • It has a domain of \([0, \infty)\)
  • And a range of \([0, \infty)\).
This means that both the input and output are non-negative.
Transformations adjust this graph's position or shape while maintaining its squareroot characteristics.
Domain and Range
Understanding the domain and range of a function is key in graph transformations.
The domain refers to all the possible input values (or \(x\)-values) that the function can accept.
For the function \(k(x)=\sqrt{x+4}-4\), we solve the inequality \(x + 4 \geq 0\) to determine its domain.
  • This gives us a domain of \([-4, \infty)\), since \(x\) must be greater than or equal to \(-4\).
The range, on the other hand, refers to all the possible output values (or \(y\)-values) that the function can produce.
Given the vertical shift of 4 units down from the parent function's range \([0, \infty)\), the range of \(k(x)\) becomes \([-4, \infty)\).
Understanding these values helps in sketching accurate graphs.
Horizontal Shift
Horizontal shifts in graph transformations occur when every point on the graph moves horizontally.
In the case of \(k(x)=\sqrt{x+4}-4\), there's a horizontal shift to the left.
  • This happens because of the \(+ 4\) inside the square root, which adjusts the root's starting point.
  • Every point on the parent graph \(\sqrt{x}\) slides 4 units to the left, altering how the graph is positioned but not its shape.
The horizontal shift affects the domain. Instead of beginning at 0, it now starts at \(-4\).
Recognizing these shifts is important for understanding how the graph's position changes along the x-axis.
Vertical Shift
Vertical shifts occur when the entire graph moves up or down along the y-axis.
For \(k(x)=\sqrt{x+4}-4\), a vertical shift is caused by the \(-4\) outside the square root.
  • This transformation moves the graph 4 units down.
Unlike horizontal shifts, vertical shifts affect the range.
While the parent function \(\sqrt{x}\) ranges from \(0\) to \(\infty\), moving the graph downwards results in a new range of \([-4, \infty)\).
Vertical shifts are crucial to learn because they alter the output values while the base function shape remains consistent.

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

Use the catalog of basic functions (page 170 ) and Section 3.4 to describe the graph of the given function. $$h(x)=\sqrt{x}-2$$

The number of digital devices (such as MP3 players, handheld computers, cell phones, and PCs) in the world was approximately .94 billion in 1999 and is growing at a rate of \(28.3 \%\) a year. (a) Find the rule of a function that gives the number of digital devices (in billions) in year \(x,\) with \(x=0\) corresponding to 1999 (b) Approximately how many digital devices will be in use in \(2010 ?\) (c) If this model remains accurate, when will the number of digital devices reach 6 billion?

According to one theory of learning, the number of words per minute \(N\) that a person can type after \(t\) weeks of practice is given by \(N=c\left(1-e^{-k t}\right),\) where \(c\) is an upper limit that \(N\) cannot exceed and \(k\) is a constant that must be determined experimentally for each person. (a) If a person can type 50 wpm (words per minute) after four weeks of practice and 70 wpm after eight weeks, find the values of \(k\) and \(c\) for this person. According to the theory, this person will never type faster than \(c\) wpm. (b) Another person can type 50 wpm after four weeks of practice and 90 wpm after eight weeks. How many weeks must this person practice to be able to type 125 wpm?

Determine whether an exponential, power, or logarithmic model (or none or several of these) is appropriate for the data by determining which (if any) of the following sets of points are approximately linear: $$\\{(x, \ln y)\\}, \quad\\{(\ln x, \ln y)\\}, \quad\\{(\ln x, y)\\}$$ where the given data set consists of the points \(\\{(x, y)\\}\) $$\begin{array}{|l|c|c|c|c|c|c|} \hline x & 1 & 3 & 5 & 7 & 9 & 11 \\ \hline y & 2 & 25 & 81 & 175 & 310 & 497 \\ \hline \end{array}$$

Simplify the expression without using a calculator. $$\sqrt{50}-\sqrt{72}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

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.