How do you obtain the graph of y=f(x+2)y=f\left(x+2\right)y=f(x+2) from the graph of y=f(x)y=f\left(x\right)y=f(x)?

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 1 - Functions

Problem 9

Chapter 1, Problem 9

How do you obtain the graph of $y=f$\left$(x+2$\right$)$ from the graph of $y=f$\left$(x$\right$)$ ?

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Understand that the expression y = f(x + 2) represents a horizontal transformation of the function y = f(x).

Recognize that adding a positive number inside the function's argument, such as x + 2, results in a horizontal shift to the left.

To graph y = f(x + 2), take each point (x, y) on the graph of y = f(x) and shift it 2 units to the left, resulting in the new point (x - 2, y).

Ensure that all points on the original graph are shifted consistently to maintain the shape of the graph.

Check the new graph to confirm that it accurately represents the function y = f(x + 2) by verifying a few key points and their transformations.

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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 process of altering the graph of a function through shifts, stretches, or reflections. In this case, the function y = f(x + 2) represents a horizontal shift of the original function y = f(x) to the left by 2 units. Understanding how these transformations affect the graph is crucial for accurately sketching or interpreting the new function.

Horizontal Shifts

Horizontal shifts occur when the input variable of a function is adjusted by adding or subtracting a constant. For the function y = f(x + 2), the addition of 2 inside the function's argument indicates a shift to the left. This concept is essential for predicting how the graph will move in relation to the original function.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visually represent the relationship between the input (x) and output (y) values. To graph y = f(x + 2), one must first understand the original graph of y = f(x) and then apply the horizontal shift. This skill is fundamental in calculus for analyzing and interpreting functions.

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Evaluating functions from graphs Assume ƒ is an odd function and that both ƒ and g are one-to-one. Use the (incomplete) graph of ƒ and g the graph of to find the following function values. <IMAGE>

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Textbook Question

Taxicab fees A taxicab ride costs \$3.50 plus \$2.50 per mile. Let m be the distance (in miles) from the airport to a hotel. Find and graph the function c(m) that represents the cost of taking a taxi from the airport to the hotel. Also determine how much it will cost if the hotel is 9 miles from the airport.

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Textbook Question

Decide whether $f$ , $g$ , or both represent one-to-one functions. <IMAGE>