Exponential Parameters: the 'h' Value

Algebra-1

1. Fundamental Concepts

Definition: The 'h' value in an exponential function of the form $$f(x) = a \cdot b^{(x-h)} + k$$ represents the horizontal shift of the graph.
Horizontal Shift: If 'h' is positive, the graph shifts right by 'h' units; if 'h' is negative, the graph shifts left by 'h' units.
Graphical Interpretation: The value of 'h' affects where the vertex of the exponential function's graph lies on the x-axis.

2. Key Concepts

Basic Rule: $$f(x) = a \cdot b^{(x-h)} + k$$

Shift Direction: Positive 'h' shifts the graph to the right; negative 'h' shifts the graph to the left.

Application: Used in modeling real-world phenomena such as population growth and radioactive decay.

3. Examples

Example 1 (Basic)

Problem: Identify the horizontal shift for the function $$f(x) = 2 \cdot 3^{(x-4)} + 1$$

Step-by-Step Solution:

The function is in the form $$f(x) = a \cdot b^{(x-h)} + k$$ with $$a = 2$$, $$b = 3$$, $$h = 4$$, and $$k = 1$$.
Since 'h' is positive, the graph shifts right by 4 units.

Validation: Substitute x=0 → Original: 2 \cdot 3^{-4} + 1; Simplified: 2 \cdot \frac{1}{81} + 1 ≈ 1.0247 ✓

Example 2 (Intermediate)

Problem: Write the equation of an exponential function that has been shifted 3 units to the left and has a vertical stretch factor of 5.

Step-by-Step Solution:

Start with the base function $$f(x) = b^x$$.
Apply the vertical stretch factor: $$f(x) = 5 \cdot b^x$$.
Apply the horizontal shift: $$f(x) = 5 \cdot b^{(x+3)}$$.

Validation: Substitute x=-3 → Original: 5 \cdot b^0 = 5; Simplified: 5 ✓

4. Problem-Solving Techniques

Visual Strategy: Graph the function before and after applying the shift to visualize the change.
Error-Proofing: Always check the sign of 'h' to determine the direction of the shift.
Concept Reinforcement: Practice with various values of 'h' to understand its impact on the graph.