The general form of a vertically translated function is:

Where:

• g(x) is the original function.
• k is the vertical shift.

If k is positive, the graph shifts upward. If k is negative, the graph shifts downward.

Example: Consider the function g(x) = x². A vertical translation of 3 units upward results in:

The vertex moves from (0,0) to (0,3).

The general form of a horizontally translated function is:

Where:

• g(x) is the original function.
• h is the horizontal shift.

If h is positive, the graph shifts to the right. If h is negative, the graph shifts to the left.

Example: Consider the function g(x) = \sqrt{x}. A horizontal translation of 2 units to the left results in:

The starting point moves from (0,0) to (-2,0).

The general form of a function with both vertical and horizontal shifts is:

Where:

• h determines the horizontal shift.
• k determines the vertical shift.

Example: Take the function g(x) = \sin(x). Translating it π/4 units to the right and 2 units upward results in:

Vertical Translation: An upward shift lifts all points parallel to the y-axis, maintaining the shape and orientation of the graph.

Horizontal Translation: A right or left shift moves all points parallel to the x-axis, keeping the graph's shape intact.

Graphical Example: Consider the function g(x) = |x|. Applying a vertical shift of 3 units up and a horizontal shift of 2 units to the right transforms it to:

The vertex moves from (0,0) to (2,3), while the V-shape remains unchanged.

Physics: Translating the position-time graph of an object can help analyze its motion under different conditions.

Engineering: Adjusting supply and demand curves based on market changes relies on horizontal and vertical shifts of graphs.

Economics: Understanding shifts in consumer behavior can be modeled through graph translations.

Example: In physics, shifting the displacement-time graph of a projectile upwards can represent an increase in initial displacement.

Vertical Translation: To shift the graph vertically, add or subtract a constant from the function.

Given g(x), the translated function is:

Horizontal Translation: To shift the graph horizontally, shift the input variable.

Given g(x), the translated function is:

Combined Translations: Apply both transformations in the function's equation.

It's essential to apply translations in the correct order to achieve the desired graph position.

Vertical Inversion: To reverse a vertical shift of k, apply a shift of -k.

Horizontal Inversion: To reverse a horizontal shift of h, apply a shift of −h.

Example: If a graph is shifted downward by 4 units, its inverse transformation would be a shift upward by 4 units.

Domain and Range: Vertical shifts alter the range, while horizontal shifts modify the domain.

Intercepts: Translations change the positions of intercepts with the axes.

Symmetry: The inherent symmetry of functions like even and odd functions is preserved under vertical and horizontal translations.

Example: Shifting f(x) = x² vertically by 3 units up results in f(x) = x² + 3, altering the range from [0, ∞) to [3, ∞).

Scenario: A company's profit function is represented by P(x) = 2x - 5. If the company wants to model an increase in fixed costs of $10, the new profit function becomes:

This vertical translation shifts the profit graph downward by 10 units, reflecting increased costs.

Example: In projectile motion, adjusting the height from which an object is dropped would translate the displacement-time graph vertically, providing insights into its motion from different initial positions.

• Graph translations involve shifting functions vertically and horizontally without altering their shape.
• Vertical translations modify the range, while horizontal translations adjust the domain.
• Understanding translation equations is crucial for accurate graph manipulation.
• Translations have practical applications across various real-world scenarios.
• Inverse transformations allow reverting graphs to their original positions.