Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 80

Chapter 3, Problem 80

Begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = 2√(x+1)-1

Verified step by step guidance

Start by graphing the parent function f(x) = √x. This is the basic square root function, which starts at the origin (0, 0) and increases gradually as x increases. The domain is x ≥ 0, and the range is y ≥ 0.

Next, analyze the transformation g(x) = 2√(x+1) - 1. The term (x+1) inside the square root indicates a horizontal shift. Specifically, the graph of f(x) = √x is shifted 1 unit to the left because of the +1 inside the square root.

The coefficient 2 in front of the square root represents a vertical stretch. This means that the y-values of the graph will be multiplied by 2, making the graph steeper compared to the parent function.

The -1 outside the square root represents a vertical shift downward by 1 unit. This means that the entire graph will be shifted 1 unit lower along the y-axis.

Combine all these transformations: Start with the graph of f(x) = √x, shift it 1 unit to the left, stretch it vertically by a factor of 2, and then shift it 1 unit downward. Plot the resulting graph to visualize g(x) = 2√(x+1) - 1.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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 base for applying transformations in the given problem.

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For example, adding a constant inside the function shifts the graph horizontally, while adding outside shifts it vertically. In the case of g(x) = 2√(x+1)-1, the transformations include a horizontal shift left by 1 unit, a vertical stretch by a factor of 2, and a downward shift by 1 unit.

Graphing Techniques

Graphing techniques involve plotting points and understanding the behavior of functions to create accurate representations. For the function g(x), one must first graph f(x) = √x, then apply the identified transformations step-by-step. This process helps visualize how the original function is altered, leading to a correct graph of the transformed function.

Recommended video:

Guided course

02:16

Graphs and Coordinates - Example

Related Practice

Textbook Question

Express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x).

h(x) = |2x-5|

Textbook Question

Find the value of y if the line through the two given points is to have the indicated slope. (3, y) and (1, 4), m = −3

Textbook Question

Express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x).

h(x) = 1/(2x-3)

Textbook Question

In Exercises 77–92, use the graph to determine a. the function's range; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.

Textbook Question

Use the graph to determine a.the x-intercepts, if any; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.

Textbook Question

In Exercises 77–92, use the graph to determine a. the function's domain; b. the function's range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.

views