Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 22

Chapter 3, Problem 22

Use the graph of y = f(x) to graph each function g. g(x) = f(x + 1) − 2

Verified step by step guidance

Understand the problem: The given function g(x) = f(x + 1) − 2 is a transformation of the function f(x). The goal is to apply the transformations step by step to the graph of f(x).

Step 1: Identify the horizontal shift. The term (x + 1) inside the function indicates a horizontal shift. Specifically, it shifts the graph of f(x) 1 unit to the left. This is because adding a positive value inside the parentheses moves the graph in the negative x-direction.

Step 2: Apply the horizontal shift. Take the graph of f(x) and move every point on the graph 1 unit to the left. This gives you an intermediate graph of f(x + 1).

Step 3: Identify the vertical shift. The term −2 outside the function indicates a vertical shift. Specifically, it shifts the graph of f(x + 1) 2 units downward. This is because subtracting a value outside the function moves the graph in the negative y-direction.

Step 4: Apply the vertical shift. Take the intermediate graph of f(x + 1) and move every point on the graph 2 units downward. The resulting graph is the graph of g(x) = f(x + 1) − 2.

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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. In this case, g(x) = f(x + 1) - 2 involves a horizontal shift to the left by 1 unit and a vertical shift downward by 2 units. Understanding these transformations is crucial for accurately graphing the new function based on the original function's graph.

Horizontal Shifts

Horizontal shifts occur when the input variable of a function is altered, affecting the graph's position along the x-axis. For g(x) = f(x + 1), the '+1' indicates a shift to the left. This concept is essential for determining how the graph of f(x) will be adjusted to create g(x).

Vertical Shifts

Vertical shifts involve moving the graph of a function up or down along the y-axis. In the function g(x) = f(x + 1) - 2, the '-2' indicates a downward shift of 2 units. Recognizing how vertical shifts affect the overall graph is important for accurately representing the new function derived from f(x).

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