Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Transformations

Calculus

0. Functions

Transformations

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1:27 minutes

Problem 53e

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

Verified step by step guidance

Start with the original function g(x). To stretch the graph vertically, you will multiply the entire function by a constant factor.

Identify the factor by which you want to stretch the graph vertically. In this case, the factor is 5.

Multiply the function g(x) by the factor 5. This means you will write the new function as 5 * g(x).

Understand that multiplying the function by 5 will make all the y-values of the graph five times larger, effectively stretching the graph vertically.

The equation for the vertically stretched graph is y = 5 * g(x). This represents the transformation applied to the original graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vertical Stretch

A vertical stretch occurs when the output values of a function are multiplied by a factor greater than 1. For example, if the function g(x) is transformed to 5g(x), every point on the graph of g is moved away from the x-axis by a factor of 5, effectively stretching the graph vertically. This transformation increases the height of the graph while maintaining the same x-coordinates.

Function Transformation

Function transformations involve altering the graph of a function through shifts, stretches, or reflections. These transformations can be represented mathematically by modifying the function's equation. Understanding how to apply these transformations allows one to predict the new position and shape of the graph based on the original function.

Graph of a Function

The graph of a function visually represents the relationship between the input (x-values) and output (y-values) of the function. Each point on the graph corresponds to a pair (x, g(x)). Analyzing the graph helps in understanding the behavior of the function, including its intercepts, slopes, and overall shape, which are crucial when applying transformations like stretching or shifting.

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Textbook Question

Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed.

$g (x) = - 3 x^{2}$

Textbook Question

Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed.

$h (x) = - 4 x^{^{2}} - 4 x + 12$

Textbook Question

The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>

a. $f with hook open paren x plus 1 close paren$

Textbook Question

The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>

$f (2 (x - 1))$

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

Textbook Question

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

a. 𝔂 = ƒ(x - 5)

Textbook Question

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

d. 𝔂 = ƒ(2x + 1)

Textbook Question

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

e. 𝔂 = ƒ( x ) - 4

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