/*! 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 15 Begin by graphing the square roo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 15

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 graph of \(h(x)=-\sqrt{x+2}\) is the graph of the basic square root function \(f(x)=\sqrt{x}\), shifted 2 units to the left and reflected in the x-axis.

Step by step solution

01

- Graph the Basic Function \(f(x)=\sqrt{x}\)

Start by graphing the function \(f(x)=\sqrt{x}\). This function gives a curve that starts from the point (0,0) and extends towards positive infinity without ever touching the x-axis. The y-values increase as the x-values increase, but at a decreasing rate so the curve gets flatter as x increases.
02

- Understand the Transformations

The transformations to get from \(f(x)=\sqrt{x}\) to \(h(x)=-\sqrt{x+2}\) are a reflection in the x-axis (caused by the negative sign) and a shift 2 units to the left (caused by the +2 inside the square root).
03

- Apply the Transformations to the Graph

Apply the transformations to the graph in Step 1. First shift the graph 2 units to the left: every point on \(f(x)=\sqrt{x}\) is moved 2 units to the left to get the graph of \(\sqrt{x+2}\). Then apply the reflection in the x-axis: every point on the curve is mirrored in the x-axis to get the graph of \(-\sqrt{x+2}\)

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.

Graph Transformations
Graph transformations consist of various modifications that alter a graph's position, shape, or orientation in a coordinate plane. Transformations can include shifts, reflections, stretches, and compressions. Understanding these individual transformations helps in predicting how a graph will change its appearance.

Specifically, when transforming a function like the square root function, we often perform a series of alterations:
  • Horizontal Shifts: Moves the graph left or right based on the addition or subtraction of a constant within the function.
  • Vertical Shifts: Moves the graph up or down and involves adding or subtracting a constant to the entire function.
  • Reflections: Reflects the graph over an axis, often altering the direction the graph opens.
These transformations are especially important in adapting basic functions to fit specific equations, like changing the basic square root function into a more complex one.
Reflection in the x-axis
A reflection in the x-axis involves flipping the graph of a function over the x-axis. In simpler terms, what was above the x-axis moves below it, and vice versa. This is accomplished by multiplying the function by -1.

For example, consider the transformation from \(f(x) = \sqrt{x}\) to \(-\sqrt{x+2}\). The negative sign in front of the square root indicates a reflection in the x-axis. Here鈥檚 how the reflection changes the function:
  • The graph remains the same in terms of shape, but it is vertically flipped, meaning what was once above the x-axis is now below.
  • The y-coordinates of every point on the graph are inverted. If a point was at (x, y), it is now at (x, -y).
This transformation is crucial when translating the appearance of a graph and understanding how outputs are affected by reflections in equations.
Shifting Graphs
Shifting graphs is a transformation that translates the graph horizontally or vertically across the plane. A horizontal shift is controlled by changes within the function itself, often affecting the input variable.

For the function given in the exercise, \(-\sqrt{x+2}\), the \(+2\) indicates a horizontal shift to the left by 2 units.

Here鈥檚 what happens with a horizontal shift:
  • Leftward Shift: Adding a constant within the radical or parentheses results in moving the graph to the left; i.e., \(\sqrt{x+2}\) moves \(\sqrt{x}\) two units left.
  • Variables鈥� Movement: For every x-coordinate on the graph, it is adjusted by subtracting the constant within the function. Thus, point (x, y) becomes (x-2, y).
This understanding assists in manually graphing complex equations derived from simpler parent functions, helping visualize how the graph's position changes on the coordinate plane.
Basic Function Graphing
Basic function graphing is the foundation for any graph-related task. Understanding and drawing the base graph is the first step before any transformations are applied. The function \(f(x) = \sqrt{x}\) is a common example, representing a positive square root graph.

Key characteristics of the \(\sqrt{x}\) graph include:
  • Starting Point: The graph originates at (0,0), as the square root of 0 is 0.
  • Shape: It forms a curve that continually rises towards the right, gradually flattening as x increases.
  • Domain and Range: The domain includes all non-negative numbers, as you cannot take the square root of a negative number in basic real number math. The range also starts from zero and only includes non-negative values.
By mastering the graphing of such basic functions, students can easily adapt this knowledge to graph and comprehend more complex transformations.

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

Then use the TRACE feature to trace along the line and find the coordinates of two points. Use these points to compute the line's slope. Check your result by using the coefficient of \(x\) in the line's equation. $$y=-\frac{1}{2} x-5$$

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)=-|x+4|+1 $$

Group members should consult an almanac, newspaper, magazine, or the Internet to find data that lie approximately on or near a straight line. Working by hand or using a graphing utility, construct a scatter plot for the data. If working by hand, draw a line that approximately fits the data and then write its equation. If using a graphing utility, obtain the equation of the regression line. Then use the equation of the line to make a prediction about what might happen in the future. Are there circumstances that might affect the accuracy of this prediction? List some of these circumstances.

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$ h(x)=(x-2)^{2}+1 $$

During a particular year, the taxes owed, \(T(x),\) in dollars, filing separately with an adjusted gross income of \(x\) dollars is given by the piecewise function $$ T(x)=\left\\{\begin{array}{ll} 0.15 x & \text { if } 0 \leq x<17,900 \\ 0.28(x-17,900)+2685 & \text { if } 17,900 \leq x<43,250 \\ 0.31(x-43,250)+9783 & \text { if } x \geq 43,250 \end{array}\right. $$ In Exercises \(89-90,\) use this function to find and interpret each of the following. 90\. \(T(70,000)\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

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.