/*! 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 71 Begin by graphing the square roo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 71

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=-\sqrt{x+2}$$

Short Answer

Expert verified
The function \(h(x)=-\sqrt{x+2}\) is a reflection of the square root function \(f(x)=\sqrt{x}\) over the x-axis, and it is also shifted 2 units to the left. The transformation includes a reflection across the x-axis and a horizontal translation to the left by 2 units.

Step by step solution

01

Understanding the base function \(f(x)=\sqrt{x}\)

The square root function \(f(x)=\sqrt{x}\) is defined for all non-negative x, and its graph starts at the origin (0,0) and increases slowly, bending upwards towards \(+\infty\). The shape of its graph is half of a parabola turned 90 degrees clockwise.
02

Identifying the transformations

The function \(h(x)\) can be obtained by transforming \(f(x)\) in two ways: 1) Reflection across the x-axis: The negative sign in front of the square root is indicative of a reflection or an upside down flip over the x-axis. 2) Horizontal translation to the left by 2 units: The term inside the square root function, \(x+2\), indicates a shift or translation of the graph to the left by 2 units. The '+' sign inside the function actually indicates a 'left shift' in the graph.
03

Applying the transformation on \(f(x)=\sqrt{x}\)

Applying the transformations to the graph of \(f(x)=\sqrt{x}\) involves two parts: 1) Reflect the graph of \(f(x)\) across the x-axis. This means that the graph will now descend from the y-axis. 2) Shift the reflected graph to the left by two units. This will move your entire graph 2 units leftwards.
04

Graphing \(h(x)=-\sqrt{x+2}\)

Now it's time to plot \(h(x)\) using the transformed graph. Begin at the point (-2,0). This is the 'new' starting point after transformation. The graph will decline downwards from (-2,0) because it has been reflected and should look like an inverted version of the basic square root function, displaced to the left by two units.

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

determine whether each statement makes sense or does not make sense, and explain your reasoning. The graph of \((x-3)^{2}+(y+5)^{2}=-36\) is a circle with radius 6 centered at \((3,-5)\)

If you are given a function's equation, how do you determine if the function is even, odd, or neither?

Furry Finances A pet insurance policy has a monthly rate that is a function of the age of the insured dog or cat. For pets whose age does not exceed \(4,\) the monthly cost is \(\$ 20\). The cost then increases by \(\$ 2\) for each successive year of the pet's age. $$ \begin{array}{cc} \text { Age Not Exceeding } & \text { Monthly Cost } \\ \hline 4 & \$ 20 \\ 5 & \$ 22 \\ 6 & \$ 24 \end{array} $$ The cost schedule continues in this manner for ages not exceeding \(10 .\) The cost for pets whose ages exceed 10 is S40. Use this information to create a graph that shows the monthly cost of the insurance, \(f(x)\), for a pet of age \(x,\) where the function's domain is \([0,14]\)

Sketch the graph of \(f\) using the following properties. (More than one correct graph is possible.) \(f\) is a piecewise function that is decreasing on \((-\infty, 2), f(2)=0, f\) is increasing on \((2, \infty),\) and the range of \(f\) is \([0, \infty)\)

Explain how to find the difference quotient of a function \(f\) \(\frac{f(x+h)-f(x)}{h},\) if an equation for \(f\) is given.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

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.