/*! 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 9 If you have the graph of \(y=f(x... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 9

If you have the graph of \(y=f(x),\) how do you obtain the graph of \(y=f(3 x) ?\)

Short Answer

Expert verified
Answer: The transformation applied is a horizontal compression by a factor of 3.

Step by step solution

01

Identify the transformation

The given function is \(y = f(3x)\). This indicates a horizontal compression (or "squishing") of the graph of \(y=f(x)\). The horizontal compression is determined by the coefficient of \(x\) inside the function, which is 3 in this case.
02

Apply horizontal compression

To obtain the graph of \(y = f(3x)\), we need to compress the original graph of \(y = f(x)\) horizontally by a factor of 3. This means that each point \((x, y)\) on the graph of \(y=f(x)\) transforms to the point \(\left(\frac{x}{3}, y\right)\) on the graph of \(y=f(3x)\).
03

Plot the transformed points

Now, plot the transformed points \(\left(\frac{x}{3}, y\right)\) on the graph. Connect these points to create the curve of \(y=f(3x)\). This new curve represents the horizontally compressed graph. To summarize, the graph of \(y=f(3x)\) is obtained by horizontally compressing the graph of \(y=f(x)\) by a factor of 3, which involves transforming each point \((x, y)\) on the original graph to the point \(\left(\frac{x}{3}, y\right)\) on the new graph.

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 Compression
Horizontal compression occurs when we modify a function by multiplying the variable inside its function argument by a factor greater than one. For example, in the function \( y = f(3x) \), the factor 3 causes the graph to compress horizontally. Imagine you have a rubber band stretched between your fingers, with marks at regular intervals. A horizontal compression would be like pushing your fingers closer together. The marks get closer, similarly to how points on a graph move closer along the x-axis.
  • The effect of multiplying by 3 inside the function means "compress the graph by a factor of 1/3".
  • This is because you are stretching or squeezing the function horizontally to fit the same x-shape into a space that鈥檚 only a third as wide.
The key idea is that horizontal compression brings points of the function graph together along the x-axis, compacting it width-wise without altering their y-coordinates.
Function Graph
A function graph visually expresses the relationship between input values (x) and output values (y). It's like a roadmap showing all the points of a function. For example, the graph of \( y = f(x) \) shows all the solutions of y for each given x. When drawing the graph of \( y = f(x) \), we observe the behavior of y depending on x. Whether it peaks or dips lets us understand the function's changes.
  • The graph of any function serves as a powerful visual tool to comprehend the function's overall trend and distinctive characteristics, such as maxima or minima.
  • Transformations, like moving or squeezing the graph, help us focus on certain features or behavior changes.
Understanding the function graph allows us to see operations and transformations graphically, rather than just as mathematical formulas. This is crucial when we transform functions, such as applying horizontal compression.
Coordinate Transformations
Coordinate transformations are changes applied to each point in a graph to create a new graph. By altering the coordinates of each point in a graph, we reshape it. In the example of \( y = f(3x) \), the points \((x, y)\) from \( y = f(x) \) change to \(\left( \frac{x}{3}, y \right)\). Here鈥檚 how it works:
  • Look at each point on the original graph \((x, y)\).
  • Calculate the new x-coordinate by dividing the original x by 3, which is \( \frac{x}{3} \).
  • The y-coordinate stays the same since it's unaffected by horizontal compression.
Applying these transformations allows us to redraw the function graph in a new form. Coordinate transformations are powerful tools for exploring how different functions relate and display on a graph. By understanding them, you can confidently modify graphs and predict how changes to the function will reflect in its graph.

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

Simplify the difference quotients \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\) by rationalizing the numerator. $$f(x)=\sqrt{1-2 x}$$

A boat approaches a 50 -ft-high lighthouse whose base is at sea level. Let \(d\) be the distance between the boat and the base of the lighthouse. Let \(L\) be the distance between the boat and the top of the lighthouse. Let \(\theta\) be the angle of elevation between the boat and the top of the lighthouse. a. Express \(d\) as a function of \(\theta\) b. Express \(L\) as a function of \(\theta\)

Determine whether the following statements are true and give an explanation or a counterexample. a. All polynomials are rational functions, but not all rational functions are polynomials. b. If \(f\) is a linear polynomial, then \(f \circ f\) is a quadratic polynomial. c. If \(f\) and \(g\) are polynomials, then the degrees of \(f \circ g\) and \(g \circ f\) are equal. d. To graph \(g(x)=f(x+2),\) shift the graph of \(f\) two units to the right.

Draw a right triangle to simplify the given expressions. $$\tan \left(\cos ^{-1} x\right)$$

The height of a baseball hit straight up from the ground with an initial velocity of \(64 \mathrm{ft} / \mathrm{s}\) is given by \(h=f(t)=\) \(64 t-16 t^{2},\) where \(t\) is measured in seconds after the hit. a. Is this function one-to-one on the interval \(0 \leq t \leq 4 ?\) b. Find the inverse function that gives the time \(t\) at which the ball is at height \(h\) as the ball travels upward. Express your answer in the form \(t=f^{-1}(h)\). c. Find the inverse function that gives the time \(t\) at which the ball is at height \(h\) as the ball travels downward. Express your answer in the form \(t=f^{-1}(h)\). d. At what time is the ball at a height of \(30 \mathrm{ft}\) on the way up? e. At what time is the ball at a height of \(10 \mathrm{ft}\) on the way down?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.