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

Common Functions

College Algebra

3. Functions

Common Functions

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2:25 minutes

Problem 15

Textbook Question

In Exercises 1-16, use the graph of y = f(x) to graph each function g.
g(x) = −ƒ( x/2) +1

Verified step by step guidance

Identify the transformation components in g(x) = -f(x/2) + 1: horizontal stretch by a factor of 2, reflection over the x-axis, and vertical shift up by 1.

Apply the horizontal stretch: For each x-coordinate of f(x), multiply by 2. The points (1, -3) and (4, -3) become (2, -3) and (8, -3).

Apply the reflection over the x-axis: Multiply the y-coordinates by -1. The points (2, -3) and (8, -3) become (2, 3) and (8, 3).

Apply the vertical shift: Add 1 to each y-coordinate. The points (2, 3) and (8, 3) become (2, 4) and (8, 4).

Plot the transformed points (2, 4) and (8, 4) and draw the horizontal line segment connecting them to graph g(x).

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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 altering the graph of a function through shifts, stretches, or reflections. In this case, the function g(x) = -f(x/2) + 1 represents a horizontal stretch by a factor of 2, a reflection across the x-axis, and a vertical shift upwards by 1 unit. Understanding these transformations is crucial for accurately graphing the new function based on the original function f(x).

Horizontal Stretch

A horizontal stretch occurs when the input of a function is multiplied by a factor less than 1. For g(x) = -f(x/2), the x-values of f(x) are effectively doubled, which means that the graph of f(x) will be stretched horizontally. This transformation affects the spacing of points on the graph, making them appear further apart along the x-axis.

Reflection Across the X-Axis

A reflection across the x-axis occurs when the output of a function is multiplied by -1. In the function g(x) = -f(x/2), this reflection means that all y-values of f(x) are inverted. Consequently, points that were above the x-axis will now be below it, and vice versa, which is essential for accurately plotting the transformed function.