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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 40a
Chapter 1, Problem 40a

The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>
a. ƒ(x+1)ƒ( x + 1)

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1
Identify the transformation: The function \( f(x + 1) \) represents a horizontal shift of the graph of \( f(x) \).
Determine the direction of the shift: Since the transformation is \( f(x + 1) \), the graph of \( f(x) \) will shift to the left by 1 unit.
Sketch the new graph: Take each point on the original graph of \( f(x) \) and move it 1 unit to the left to obtain the graph of \( f(x + 1) \).
Check key points: Verify that key points on the graph, such as intercepts and turning points, have been shifted correctly.
Ensure continuity and shape: Make sure the overall shape and continuity of the graph are preserved after the transformation.

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Key Concepts

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

Function Transformation

Function transformation refers to the changes made to the graph of a function based on modifications to its equation. Common transformations include translations, reflections, stretches, and compressions. For instance, adding a constant to the input of a function, such as ƒ(x + 1), results in a horizontal shift of the graph to the left by one unit.
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Intro to Transformations

Horizontal Shift

A horizontal shift occurs when the graph of a function is moved left or right along the x-axis. Specifically, for a function ƒ(x + c), where c is a positive constant, the graph shifts to the left by c units. Conversely, if c is negative, the graph shifts to the right. This concept is crucial for accurately graphing transformed functions.
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Intro to Transformations

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visually represent the relationship between the input (x-values) and output (y-values) of a function. Understanding how to graph functions, including their transformations, is essential for interpreting their behavior and characteristics, such as intercepts, asymptotes, and overall shape.
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Graph of Sine and Cosine Function
Related Practice
Textbook Question

Solve the following equations.

sin3x=22,0≤x<2π\(\sin\)3x=\(\frac{\sqrt{2}\)}{2},0\(\leq{x}\]\lt{2\pi}\)

Textbook Question

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


f(2(x−1))f(2(x - 1))

Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

F(y⁴)

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Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

F(F(x))

Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4, g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

ƒ (√(x+4))

Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

F(g(y))