Math Problem Statement

Starting with the graph of a basic​ function, graph the following function using the techniques of​ shifting, compressing,​ stretching, and/or reflecting. Find the domain and range of the function. g left parenthesis x right parenthesis equals left parenthesis x plus 1 right parenthesis cubed plus 6 Question content area bottom left Part 1 Use the graphing tool to graph the equation.

Solution

The given function is g(x)=(x+1)3+6g(x) = (x + 1)^3 + 6. We need to analyze how this function can be graphed using techniques like shifting, compressing, stretching, and reflecting, and then find the domain and range.

Step-by-Step Graphing Process:

  1. Start with the basic function: The basic function is f(x)=x3f(x) = x^3. This is a cubic function with its characteristic shape (an "S"-like curve that passes through the origin). The domain of f(x)=x3f(x) = x^3 is (,)(-\infty, \infty), and its range is also (,)(-\infty, \infty).

  2. Horizontal Shift: The function has been modified to g(x)=(x+1)3+6g(x) = (x + 1)^3 + 6. The x+1x + 1 inside the cube indicates a horizontal shift. Specifically, the graph of f(x)=x3f(x) = x^3 is shifted 1 unit to the left. This means that instead of the curve passing through (0,0)(0,0), it now passes through (1,0)(-1, 0).

  3. Vertical Shift: The +6+6 outside the cubic term indicates a vertical shift. The graph is shifted 6 units upward, so the point (1,0)(-1, 0) moves to (1,6)(-1, 6).

Thus, the graph of g(x)=(x+1)3+6g(x) = (x + 1)^3 + 6 is the graph of f(x)=x3f(x) = x^3 shifted 1 unit left and 6 units up.

Domain and Range:

  • Domain: Since this is a cubic function, the domain remains (,)(-\infty, \infty) as the function is defined for all real values of xx.
  • Range: The cubic function x3x^3 has a range of (,)(-\infty, \infty), and since there is only a vertical shift by 6 units, the range of g(x)g(x) also remains (,)(-\infty, \infty).

Summary:

  • Horizontal shift: 1 unit to the left
  • Vertical shift: 6 units upward
  • Domain: (,)(-\infty, \infty)
  • Range: (,)(-\infty, \infty)

If you would like, I can help you graph the function visually or provide more detailed steps.

Would you like any details or further clarification?

Here are 5 questions to expand your understanding:

  1. How does a horizontal shift affect the function's behavior?
  2. What happens to the range if a cubic function is vertically stretched?
  3. Can the domain of a cubic function ever be restricted by transformations?
  4. How would the graph change if the function was (x1)3+6(x - 1)^3 + 6 instead?
  5. What role does the coefficient of xx play in stretching or compressing the cubic graph?

Tip: Horizontal shifts move the graph left or right, depending on whether the shift is positive or negative.

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Math Problem Analysis

Mathematical Concepts

Algebra
Graphing Transformations
Cubic Functions

Formulas

g(x) = (x + 1)^3 + 6

Theorems

Transformations of Functions
Domain and Range Theorems

Suitable Grade Level

Grades 10-12