G(x) = f(x) + 3

Definition

The equation g(x) = f(x) + 3 represents a vertical translation of the function f(x) by 3 units upwards. This means that for every point on the graph of f(x), the corresponding point on the graph of g(x) will be 3 units higher. This transformation is a key concept in understanding how the graphs of functions can be altered through shifts, which is essential for analyzing function behavior and relationships.

5 Must Know Facts For Your Next Test

1. The transformation g(x) = f(x) + 3 shifts the entire graph of f(x) upward by 3 units for every x value.
2. Vertical translations do not affect the horizontal position of the graph; they only change the y-values.
3. If f(x) has a maximum or minimum point, g(x) will have the same point, but elevated by 3 units.
4. This type of transformation can be combined with other transformations like horizontal shifts or reflections for more complex changes.
5. Understanding vertical shifts is crucial for solving real-world problems where a baseline value is adjusted, such as in economics or physics.

Review Questions

• How does the equation g(x) = f(x) + 3 specifically affect the y-values of the original function f(x)?
  • The equation g(x) = f(x) + 3 increases all y-values of the original function f(x) by 3 units. This means that for every point (x, y) on the graph of f(x), there is a corresponding point (x, y+3) on the graph of g(x). This vertical shift affects all points uniformly, moving the entire graph upward while keeping the horizontal positions unchanged.
• In what ways can you combine g(x) = f(x) + 3 with other transformations, and what would be an example?
  • You can combine g(x) = f(x) + 3 with other transformations like horizontal shifts or reflections. For example, if you have h(x) = f(x - 2) + 3, this represents a horizontal shift to the right by 2 units and a vertical shift up by 3 units. The combination of these transformations allows you to alter both positions of the graph in two dimensions, resulting in complex movements.
• Evaluate how vertical shifts like g(x) = f(x) + 3 can be applied in practical scenarios such as economics or physics.
  • In economics, vertical shifts such as g(x) = f(x) + 3 could represent an increase in fixed costs, where every output level now requires an additional cost of 3 units. Similarly, in physics, if a graph represents height over time, applying this transformation could signify raising the entire object's position by 3 units. These applications help illustrate how mathematical models can reflect real-world adjustments in conditions or constraints.

Related terms

Vertical Shift: A transformation that moves a graph up or down without changing its shape, determined by adding or subtracting a constant to the function.

Function Notation: A way to represent functions that specifies the input and output, typically written as f(x), g(x), etc., where the letter indicates the function name.

Graph Translation: The process of shifting a graph horizontally or vertically without altering its shape or orientation.