Math Problem Statement

The graph to the right was created by vertically shifting the graph of the basic function yequalsStartFraction 1 Over x EndFraction . Write a function that describes the graph to the right. -10 -5 5 10 -10 -5 5 10 x y f left parenthesis x right parenthesis

A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 10 to 10 in increments of 1. The graph of the function labeled as f(x) has two branches. The left branch approaches y equals negative 1 from below, as x approaches negative infinity, and the right branch approaches y equals negative 1 from above, as x approaches positive infinity. The left branch approaches x equals 0, as x approaches 0 from the left, and the right branch approaches x equals 0, as x approaches 0 from the right. Question content area bottom Part 1 The function that describes the graph is ​f(x)equals    enter your response here. ​(Simplify your​ answer.)

Solution

The parent function given is:
f(x)=1xf(x) = \frac{1}{x} From the description of the graph, the function has been vertically shifted downward by 1 unit because the horizontal asymptote has moved from y=0y = 0 to y=−1y = -1.

Transformation:

A vertical shift by cc units affects the function as:
f(x)=1x+cf(x) = \frac{1}{x} + c Since the asymptote is now at y=−1y = -1, we set c=−1c = -1.

Thus, the equation of the transformed function is:
f(x)=1x−1f(x) = \frac{1}{x} - 1

Would you like a step-by-step breakdown of how vertical shifts affect functions?

Related Questions:

  1. How does shifting f(x)=1xf(x) = \frac{1}{x} upward by 2 units change the function?
  2. What is the domain and range of f(x)=1x−1f(x) = \frac{1}{x} - 1?
  3. How does a horizontal shift affect the function f(x)=1xf(x) = \frac{1}{x}?
  4. What happens to the asymptotes if the function is multiplied by a constant?
  5. Can a rational function have more than one horizontal asymptote?

Tip:

A vertical shift only affects the y-values of a function, while a horizontal shift changes the x-values.

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Math Problem Analysis

Mathematical Concepts

Algebra
Transformations of Functions
Rational Functions

Formulas

f(x) = 1/x + c (vertical shift transformation)

Theorems

Transformation of Functions

Suitable Grade Level

Grades 9-12

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