Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

3. Functions

Transformations

College Algebra

3. Functions

Transformations

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1:41 minutes

Problem 102

Textbook Question

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.
(b) y = ƒ(x-2)

Verified step by step guidance

Identify the original function f(x) from the graph. Note the key points: (-8, 0), (-4, 8), (0, 0), (8, -4), and (16, 0).

Understand that y = f(x-2) represents a horizontal shift of the function f(x) to the right by 2 units.

Shift each key point of the original function f(x) to the right by 2 units. For example, (-8, 0) becomes (-6, 0), (-4, 8) becomes (-2, 8), and so on.

Plot the new points on the graph: (-6, 0), (-2, 8), (2, 0), (10, -4), and (18, 0).

Sketch the graph of y = f(x-2) by connecting the new points, maintaining the same shape and orientation as the original function.

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Key Concepts

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

Function Transformation

Function transformations involve changing the position or shape of a graph based on specific rules. In this case, the transformation y = ƒ(x-2) indicates a horizontal shift of the graph of ƒ to the right by 2 units. Understanding how transformations affect the graph is crucial for accurately sketching the new function.

Horizontal Shifts

A horizontal shift occurs when the input variable of a function is altered by adding or subtracting a constant. For y = ƒ(x-2), the '-2' indicates that every point on the graph of ƒ will move 2 units to the right. This concept is essential for predicting the new coordinates of points on the transformed graph.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane based on the function's output for various inputs. To sketch the graph of y = ƒ(x-2), one must take the original points from the graph of ƒ and apply the horizontal shift. This skill is fundamental in visualizing how functions behave and interact with one another.