Math Problem Statement

graph 1/x-3

Solution

To graph the function f(x)=1x3f(x) = \frac{1}{x} - 3, we will follow these steps:

  1. Identify the basic graph: The basic graph of f(x)=1xf(x) = \frac{1}{x} is a hyperbola with two asymptotes:

    • The vertical asymptote is at x=0x = 0, since division by zero is undefined.
    • The horizontal asymptote is at y=0y = 0, because as xx becomes large (either positive or negative), 1x\frac{1}{x} approaches zero.
  2. Shift the graph: The given function f(x)=1x3f(x) = \frac{1}{x} - 3 shifts the basic graph of f(x)=1xf(x) = \frac{1}{x} vertically by 3 units downward. This means that the horizontal asymptote will shift from y=0y = 0 to y=3y = -3.

  3. Plot key points:

    • As x0+x \to 0^+, f(x)+f(x) \to +\infty, and as x0x \to 0^-, f(x)f(x) \to -\infty (the graph approaches the vertical asymptote at x=0x = 0 from either side).
    • As xx \to \infty, f(x)3f(x) \to -3, and similarly, as xx \to -\infty, f(x)3f(x) \to -3.
  4. Plot a few specific points: Choose values of xx and calculate f(x)f(x). For example:

    • f(1)=113=2f(1) = \frac{1}{1} - 3 = -2
    • f(2)=123=2.5f(2) = \frac{1}{2} - 3 = -2.5
    • f(1)=113=4f(-1) = \frac{1}{-1} - 3 = -4
    • f(2)=123=3.5f(-2) = \frac{1}{-2} - 3 = -3.5
  5. Sketch the graph: Now, plot these points and sketch the asymptotes. The graph will approach the asymptotes as xx moves away from zero, curving downward toward y=3y = -3.

Would you like me to create the graph visually for you? Let me know if you want more details or further explanations.

Here are 5 related questions you might find interesting:

  1. How do vertical and horizontal asymptotes impact the graph of rational functions?
  2. What happens if we change the constant term in f(x)=1x+cf(x) = \frac{1}{x} + c?
  3. How would the graph of f(x)=1x23f(x) = \frac{1}{x - 2} - 3 differ from the one we graphed?
  4. What are the domain and range of f(x)=1x3f(x) = \frac{1}{x} - 3?
  5. How do transformations affect the parent function f(x)=1xf(x) = \frac{1}{x}?

Tip: When graphing rational functions, always start by identifying asymptotes and a few key points to guide the shape of the graph!

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Graphing Functions
Asymptotes
Transformations

Formulas

f(x) = 1/x - 3

Theorems

Asymptote Theorem
Transformation of Functions

Suitable Grade Level

Grades 9-12