Math Problem Statement

What transformations are needed to graph the function g(x) = 1/3 √(x - 5)? Choose the correct answer from the multiple-choice options.

Solution

We are tasked with analyzing the graph of the function g(x)=13x5g(x) = \frac{1}{3} \sqrt{x - 5} using transformations of the basic square root function f(x)=xf(x) = \sqrt{x}.

Let's break down the transformations in g(x)g(x):

  1. Horizontal Shift: The term x5x - 5 indicates a horizontal shift. The graph of f(x)=xf(x) = \sqrt{x} will shift to the right by 5 units because the general form of a horizontal shift is f(xh)f(x - h), which shifts the graph right by hh units.

  2. Vertical Stretch/Shrink: The coefficient 13\frac{1}{3} outside the square root indicates a vertical shrink. A number less than 1 (in absolute value) multiplies the output of the function, thus compressing the graph vertically.

Combining both transformations:

  • A horizontal shift to the right by 5 units.
  • A vertical shrink by a factor of 13\frac{1}{3}.

Now, let's review the multiple-choice answers:

  • Option A: Horizontal shift left by 5 units and horizontal stretch by 13\frac{1}{3} – Incorrect.
  • Option B: Horizontal shift right by 5 units and vertical shrink by 13\frac{1}{3} – Correct.
  • Option C: Horizontal shift left by 5 units and vertical shrink by 13\frac{1}{3} – Incorrect.
  • Option D: Horizontal shift right by 5 units and horizontal stretch by 13\frac{1}{3} – Incorrect.

Correct Answer: B.

Would you like further details on how these transformations affect the graph? Here are some related questions to consider:

  1. How do horizontal shifts impact the graph of any function?
  2. What is the effect of vertical stretches and shrinks on a graph?
  3. How would the graph change if the 13\frac{1}{3} coefficient was negative?
  4. Can you explain the difference between horizontal and vertical transformations?
  5. How do transformations differ for other types of functions, like quadratics?

Tip: When analyzing transformations, always address horizontal and vertical changes separately for clarity.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Functions
Transformations
Square Root Functions

Formulas

g(x) = 1/3 √(x - 5)

Theorems

Transformation rules for functions
Horizontal and vertical shifts

Suitable Grade Level

Grades 9-11