Determine the Exponential Function Whose Graph Is Given
An exponential function is a mathematical relationship where the variable appears in the exponent, typically written as f(x) = abˣ, where a is the initial value and b is the base. So when given a graph of an exponential function, determining its equation involves identifying key features such as the y-intercept and another point on the graph. This process allows you to calculate the values of a and b, forming the complete function. Below is a step-by-step guide to solving this problem.
Steps to Determine the Exponential Function
Step 1: Identify the Y-Intercept
The y-intercept occurs when x = 0. For an exponential function in the form f(x) = abˣ, substituting x = 0 gives f(0) = a. Because of this, the y-coordinate of the y-intercept is the value of a.
Step 2: Use Another Point to Solve for the Base
Select a second point on the graph, preferably with an integer x-value for simplicity. Substitute the coordinates of this point into the equation f(x) = abˣ. Since a is already known from the y-intercept, you can solve for b Not complicated — just consistent..
Step 3: Verify the Function
Plug in additional points from the graph into your derived equation to ensure consistency. If discrepancies arise, recheck your calculations or consider alternative forms of the exponential function (e.g., using base e) Simple, but easy to overlook..
Example Problems
Example 1: Exponential Growth
Problem: A graph passes through the points (0, 3) and (1, 6). Determine the exponential function Took long enough..
Solution:
- Find a: The y-intercept is (0, 3), so a = 3.
- Solve for b: Substitute (1, 6) into f(x) = 3bˣ:
6 = 3b¹
b = 6 ÷ 3 = 2 - Final Function: f(x) = 3(2)ˣ
Verification: Test (2, 12) on the graph:
f(2) = 3(2)² = 3(4) = 12 ✓
Example 2: Exponential Decay
Problem: A graph passes through (0, 8) and (2, 2). Determine the function.
Solution:
- Find a: The y-intercept is (0, 8), so a = 8.
- Solve for b: Substitute (2, 2) into f(x) = 8bˣ:
2 = 8b²
b² = 2 ÷ 8 = 0.25
b = √0.25 = 0.5 - Final Function: f(x) = 8(0.5)ˣ
Verification: Test (1, 4) on the graph:
f(1) = 8(0.5)¹ = 4 ✓
Key Notes
- The base b must be positive and ≠ 1 for a valid exponential function.
- If the graph shows **exponential
