Textbook Question
Shifting and Scaling Graphs
Suppose the graph of g is given. Write equations for the graphs that are obtained from the graph of g by shifting, scaling, or reflecting, as indicated.
e. Stretch vertically by a factor of 5
352
views
Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 54a
Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).
a. 𝔂 = ƒ(x - 5)
Verified step by step guidance1
Start with the graph of the function 𝔂 = ƒ(x). This is your base graph.
Identify the transformation applied to the function. In this case, the transformation is 𝔂 = ƒ(x - 5).
Recognize that the expression (x - 5) inside the function indicates a horizontal shift. Specifically, it is a shift to the right.
To apply this transformation, take each point (x, y) on the original graph of 𝔂 = ƒ(x) and move it 5 units to the right. This means you will replace each x-coordinate with (x + 5).
Plot the new points to obtain the graph of 𝔂 = ƒ(x - 5). The shape of the graph remains unchanged, but its position is shifted 5 units to the right.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformation
Function transformation refers to the process of altering the graph of a function through shifts, stretches, or reflections. These transformations can change the position and shape of the graph without altering its fundamental properties. Understanding how to manipulate the function's equation allows one to predict how the graph will change.
Recommended video:
5:25
Intro to Transformations
Horizontal Shifts
A horizontal shift occurs when the graph of a function is moved left or right along the x-axis. For the function 𝔶 = ƒ(x - 5), the graph is shifted 5 units to the right. This transformation is achieved by replacing x with (x - c), where c is the number of units shifted, affecting the input values of the function.
Recommended video:
5:25Intro to Transformations
Graphing Functions
Graphing functions involves plotting points that represent the output of a function for various input values. Understanding how to graph functions is essential for visualizing transformations. By knowing the original graph of 𝔶 = ƒ(x), one can easily apply transformations to see how the graph changes, enhancing comprehension of function behavior.
Recommended video:
5:53Graph of Sine and Cosine Function
Related Practice
Textbook Question
Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).
d. 𝔂 = ƒ(2x + 1)
333
views
Textbook Question
Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).
e. 𝔂 = ƒ( x ) - 4
3
241
views
Textbook Question
Shifting and Scaling Graphs
Suppose the graph of g is given. Write equations for the graphs that are obtained from the graph of g by shifting, scaling, or reflecting, as indicated.
f. Compress horizontally by a factor of 5
355
views
Textbook Question
For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.
cos 2θ + cos θ = 0
220
views
Textbook Question
In Exercises 55–58, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.
y = - √(1 + x/2)
211
views