/*! 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 13 Let \(k(s)=\frac{1}{s}\). Constr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 13

Let \(k(s)=\frac{1}{s}\). Construct a new function \(j(s)\) that is the end result of the transformations of the graph of \(k(s)\) described in the following steps. Show your work for each transformation. a. First shift \(k(s)\) to the right by two units. b. Then compress your result by a factor of \(1 / 3\). c. Reflect across the \(s\) -axis. d. Finally, shift it up four units.

Short Answer

Expert verified
The final function is \(j(s) = -\frac{1}{3s - 2} + 4\).

Step by step solution

01

Shift Right by Two Units

The first transformation is to shift the function right by two units. If the original function is \(k(s) = \frac{1}{s}\), shifting it to the right by two units means replacing \(s\) with \(s - 2\). Therefore, the new function, after this transformation, is \(k(s) = \frac{1}{s-2}\).
02

Compress by a Factor of \(1/3\)

The next transformation is to compress the function horizontally by a factor of \(1/3\). This involves replacing \(s\) with \(3s\). Applying this to our result from Step 1, we get \(k(s) = \frac{1}{3s - 2}\).
03

Reflect Across the \(s\)-axis

Reflecting the function across the \(s\)-axis involves multiplying the entire function by \(-1\). So, applying this to our result from Step 2, we obtain \(k(s) = -\frac{1}{3s - 2}\).
04

Shift Up by Four Units

The final transformation is to shift the function up by four units. This means adding 4 to the entire function. Applying this to our result from Step 3, the final function is \(j(s) = -\frac{1}{3s - 2} + 4\).

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.

horizontal shift
A horizontal shift, sometimes called a translation, involves moving the entire graph of a function left or right along the x-axis. If we have a function like \(k(s) = \frac{1}{s}\), shifting it to the right by two units means substituting \(s\) with \(s - 2\).

This changes the function to \(k(s) = \frac{1}{s - 2}\). Essentially, every point on the original graph moves two units to the right. If you need to shift left, you would use \(s + a\) instead of \(s - a\).

This process does not alter the shape of the graph, only its position on the horizontal axis.
horizontal compression
Horizontal compression involves 'squeezing' the graph towards the y-axis. To compress the function \(k(s) = \frac{1}{s-2}\) by a factor of \(1/3\), we substitute \(s\) with \(3s\).

This gives us the new function \(k(s) = \frac{1}{3s - 2}\).

Horizontal compression makes the graph steeper because each unit change in \(s\) now affects the function value three times as much. Remember, horizontal compression by a factor of \(1/a\) means multiplying the variable by \(a\). In this case, \(a = 3\).
reflection
Reflecting a function over the s-axis flips it vertically across the s-axis. For our function \(k(s) = \frac{1}{3s - 2}\), reflecting across the s-axis involves multiplying the entire function by -1.

So, \(k(s)\) becomes \(k(s) = -\frac{1}{3s - 2}\).

This reflects each point on the graph to its 'mirror image' across the s-axis. Positive values become negative, and negative values become positive, giving the graph a flipped appearance.
vertical shift
The final transformation in our sequence is a vertical shift, which involves moving the graph up or down along the y-axis. If we need to shift our function \(k(s) = -\frac{1}{3s - 2}\) up by four units, we add 4 to the entire function.

This gives us the final function: \(j(s) = -\frac{1}{3s - 2} + 4\).

Vertical shifts move each point on the graph the same number of units up (for addition) or down (for subtraction). This translation affects the output values directly, altering the y-coordinates of each point without changing the input values.

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

A landlady currently rents each of her 50 apartments for \(\$ 1250\) per month. She estimates that for each \(\$ 100\) increase in rent, two additional apartments will remain vacant. a. Construct a function that represents the revenue \(R(n)\) as a function of the number of rent increases, \(n .\) (Hint: Find the rent per unit after \(n\) increases and the number of units rented after \(n\) increases.) b. After how many rent increases will all the apartments be empty? What is a reasonable domain for this function? c. Using technology, plot the function. From the graph, estimate the maximum revenue. Then estimate the number of rent increases that would give you the maximum revenue.

Find the \(x\) -intercepts for each of the following functions. Will the vertex lie above, below, or on the \(x\) -axis? Find the vertex and sketch the graph, labeling the \(x\) -intercepts. a. \(y=(x+2)(x+1)\) d. \(y=\frac{1}{2}(x)(x-5)\) b. \(y=3(1-2 x)(x+3)\) e. \(q(x)=2(x-3)(x+2)\) c. \(y=-4(x+3)^{2}\) f. \(f(x)=-2(5-x)(3-2 x)\)

A shot-put athlete releases the shot at a speed of 14 meters per second, at an angle of 45 degrees to the horizontal (ground level). The height \(y\) (in meters above the ground) of the shot is given by the function $$ y=2+x-\frac{1}{20} x^{2} $$ where \(x\) is the horizontal distance the shot has traveled (in meters). a. What was the height of the shot at the moment of release? b. How high is the shot after it has traveled 4 meters horizontally from the release point? 16 meters? c. Find the highest point reached by the shot in its flight. d. Draw a sketch of the height of the shot and indicate how far the shot is from the athlete when it lands.

Let \((h, k)\) be the coordinates of the vertex of a parabola. Then \(h\) equals the average of the two real zeros of the function (if they exist). For each of the following use this to find \(h,\) and then put the equations into the vertex form, \(y=a(x-h)^{2}+k\) a. A parabola with equation \(y=x^{2}+2 x-8\) b. A parabola with equation \(y=-x^{2}-3 x+4\)

Tom has a taste for adventure. He decides that he wants to bungee-jump off the Missouri River bridge. At any time \(t\) (in seconds from the moment he jumps) his height \(h(t)\) (in feet above the water level) is given by the function \(h(t)=20.5 t^{2}-123 t+190.5 .\) How close to the water will Tom get?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.