Math Problem Statement

y=2g(x+4) graph

Solution

To graph the function y=2g(x+4)y = 2g(x + 4), follow these steps:

1. Understand the Transformations:

The function y=2g(x+4)y = 2g(x + 4) represents a transformation of the base function y=g(x)y = g(x).

  • Horizontal Shift: The term x+4x + 4 shifts the graph of y=g(x)y = g(x) horizontally to the left by 4 units.
  • Vertical Stretch: The coefficient 2 vertically stretches the graph, making it twice as tall as the original graph of y=g(x)y = g(x).

2. Start with the Base Graph:

Begin by sketching the graph of the base function y=g(x)y = g(x). Since g(x)g(x) is not specified, treat it as a generic function.

3. Apply the Transformations:

  • Horizontal Shift: Shift the graph of y=g(x)y = g(x) 4 units to the left. This means every point (x,y)(x, y) on the graph of y=g(x)y = g(x) moves to (x4,y)(x - 4, y).
  • Vertical Stretch: Multiply the yy-values of the shifted graph by 2. This means if a point (x,y)(x, y) was on the graph after the horizontal shift, it now moves to (x,2y)(x, 2y).

4. Sketch the Final Graph:

Combine the transformations to produce the graph of y=2g(x+4)y = 2g(x + 4).

Example:

If y=g(x)y = g(x) is y=xy = \sqrt{x} (a basic square root function):

  • Base Graph: The graph of y=xy = \sqrt{x} starts at (0,0)(0, 0) and increases slowly, curving upwards.
  • Horizontal Shift: Shifting it left by 4 units, the graph starts at (4,0)(-4, 0).
  • Vertical Stretch: Multiply all yy-values by 2, so the graph becomes steeper, starting at (4,0)(-4, 0) but increasing more rapidly.

Would you like to see the graph for a specific base function g(x)g(x) or have any questions?


Here are some follow-up questions:

  1. What function would you like to use for g(x)g(x)?
  2. How does a vertical stretch affect the graph of a function?
  3. Can you explain the difference between a horizontal shift and a vertical shift?
  4. How would the graph change if the function were y=12g(x+4)y = \frac{1}{2}g(x + 4)?
  5. What if we had y=2g(x4)y = 2g(x - 4) instead?
  6. How do you apply multiple transformations in sequence?
  7. What does the graph look like if g(x)g(x) is a quadratic function?
  8. What impact does a negative coefficient (e.g., 2g(x+4)-2g(x + 4)) have on the graph?

Tip: When multiple transformations are applied to a graph, it's important to perform horizontal shifts before applying vertical stretches or compressions.

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Math Problem Analysis

Mathematical Concepts

Functions
Transformations

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12