/*! 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 129 Sketch the graph of \(f(x)=x^{3}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 129

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) $$g(x)=2-x^{3}$$

Short Answer

Expert verified
The transformation from \(f(x)=x^{3}\) to \(g(x)=2-x^{3}\) is a reflection in the x-axis followed by a shift upward by 2 units.

Step by step solution

01

Sketching the graph of \(f(x)=x^{3}\)

In order to sketch the graph of \(f(x)=x^{3}\), it is important to remember the general shape of a cubic function, which is an 'S' shape. The function \(f(x)=x^{3}\) is a standard cubic function, hence, its graph is a standard 'S' shape.
02

Sketching the graph of \(g(x)=2-x^{3}\)

The function \(g(x)=2-x^{3}\) is a vertical reflection and vertical shift of the function \(f(x)=x^{3}\). This is because the \(x^{3}\) term is negated, which causes a reflection in the x-axis, and the graph is also shifted upward by 2 units (as indicated by the '+2' in the function). Therefore, the graph of \(g(x)=2-x^{3}\) should be an 'S' shape reflected in the x-axis but shifted upward by 2 units.
03

Describe the transformation from \(f\) to \(g\)

To transform the function \(f(x)=x^{3}\) into the function \(g(x)=2-x^{3}\), two steps were made: a reflection in the x-axis and a vertical shift upwards by 2 units. Therefore, we can say that the transformation from \(f\) to \(g\) is a reflection in the x-axis followed by a shift upwards by 2 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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cubic Functions
Cubic functions are mathematical functions of the form \(f(x) = ax^3 + bx^2 + cx + d\). These functions are characterized by the variable \(x\) being raised to the third power, which gives them a unique 'S' shaped graph. A standard cubic function, like \(f(x) = x^3\), has the following properties:
  • Symmetrical about the origin, which means if you rotate the graph 180 degrees around the origin, it looks the same.
  • Both ends of the graph move away from the origin, one going to positive infinity and the other to negative infinity.
  • A smooth curve with only one turning point, where the direction of the curve changes from increasing to decreasing or vice versa.
The basic cubic graph serves as the foundation for understanding how transformations affect it, helping us visualize changes such as stretching, compressing, and shifting performed through various transformations.
Reflection Transformation
A reflection transformation can be visualized as flipping a graph over a specific axis. In the case of the function \(g(x) = 2 - x^3\), the term \(-x^3\) indicates that \(f(x) = x^3\) is being reflected across the x-axis. This transformation inverts all the y-values of \(f(x)\).'s graph.Here's what happens during a reflection over the x-axis:
  • Every positive y-value becomes negative and vice versa.
  • The direction of the curves is effectively mirrored.
  • The graph maintains its 'S' shaped structure but points in the opposite vertical direction.
Reflections do not change the location horizontally; they only invert the graph vertically, which significantly alters the visual appearance and increases or decreases values' negativity.
Vertical Shift
A vertical shift involves moving a graph up or down on the coordinate plane. It modifies the position without altering the graph's shape.In the function \(g(x) = 2 - x^3\), the "+2" indicates a vertical shift upward by 2 units.Here's how a vertical shift functions:
  • All points on the graph move in the same direction and by the same distance.
  • The graph鈥檚 shape or orientation remains unchanged, with only the y-values being adjusted.
  • This kind of transformation makes it possible to adjust the baseline or equilibrium level of the function.
Applying a vertical shift helps position cubic functions where their critical features, like intercepts and turning points, show how adjustments interact to influence overall function behavior.
Graph Sketching
Graph sketching is a core method in mathematics, allowing us to describe the visual representation of functions. To sketch a graph, follow these steps:
  • Identify the function's general shape and any specific transformations applied.
  • Determine critical points, such as intercepts with the axes and turning points, by setting derivative equations to zero.
  • Apply transformations to the base graph systematically, noting reflections, shifts, and any other modifications.
When sketching \(f(x) = x^3\), start with its standard 'S' shape. For \(g(x) = 2 - x^3\), note the reflection over the x-axis and vertical shift upward by 2 units. By visualizing these transformations, the graph conveys how original points are relocated to new positions. Graph sketching helps make abstract mathematical concepts more tangible, providing insight into function dynamics and symmetry.

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

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side and then find the values of the other five trigonometric functions of \(\theta\) $$\cos \theta=\frac{3}{4}$$

(a) Use a graphing utility to complete the table. $$\begin{array}{|l|l|l|l|l|l|l|} \hline \theta & 0 & 0.3 & 0.6 & 0.9 & 1.2 & 1.5 \\ \hline \cos \left(\frac{3 \pi}{2}-\theta\right) & & & & & & \\ \hline-\sin \theta & & & & & & \\ \hline \end{array}$$ (b) Make a conjecture about the relationship between \(\cos \left(\frac{3 \pi}{2}-\theta\right)\) and \(-\sin \theta\).

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=-270^{\circ}$$

Write a study sheet that will help you remember how to evaluate the six trigonometric functions of any angle \(\theta\) in standard position. Include figures and diagrams as needed.

Determine whether the statement is true or false. Justify your answer. $$\cos \theta=-\sqrt{1-\sin ^{2} \theta} \text { for } 90^{\circ} < \theta < 180^{\circ}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Discrete 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.