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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1:24 minutes

Problem 68

Textbook Question

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √x + 1

Verified step by step guidance

Graph the basic square root function \( f(x) = \sqrt{x} \). This graph starts at the origin (0,0) and increases slowly, forming a curve that moves to the right.

Identify the transformation needed to graph \( g(x) = \sqrt{x} + 1 \). This function is a vertical shift of the basic square root function.

Recognize that the \( +1 \) outside the square root indicates a vertical shift upwards by 1 unit.

Take each point on the graph of \( f(x) = \sqrt{x} \) and move it up by 1 unit to obtain the graph of \( g(x) = \sqrt{x} + 1 \).

Plot the new points to complete the graph of \( g(x) = \sqrt{x} + 1 \), ensuring the shape remains the same as \( f(x) = \sqrt{x} \) but shifted upwards.

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Key Concepts

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

Square Root Function

The square root function, f(x) = √x, is defined for x ≥ 0 and produces non-negative outputs. Its graph is a curve that starts at the origin (0,0) and increases gradually, reflecting the relationship between x and its square root. Understanding this function is crucial as it serves as the foundation for applying transformations.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting the graph of a function. In this case, the transformation applied to f(x) = √x to obtain g(x) = √x + 1 is a vertical shift upwards by 1 unit. Recognizing how these transformations affect the graph is essential for accurately sketching the new function.

Vertical Shift

A vertical shift occurs when a constant is added to or subtracted from a function's output. For g(x) = √x + 1, the '+1' indicates that every point on the graph of f(x) = √x is moved up by one unit. This concept is vital for understanding how the original function's graph is altered to create the new function.