Writing Exponential Functions from Data
Writing exponential functions from data is a Grade 11 Algebra 1 skill from enVision Chapter 6 that uses two key steps: find the base b by dividing consecutive y-values (b = y2/y1), then find the initial value a using any data point (a = y/b^x). For points (0, 3) and (1, 12): b = 12/3 = 4, a = 3, giving f(x) = 3 * 4^x. From a table with (1,8), (2,24), (3,72): b = 24/8 = 3, then a = 8/3. For non-adjacent points (2,50) and (4,200): b^2 = 200/50 = 4, so b = 2 and a = 50/4 = 12.5.
Key Concepts
To write an exponential function $f(x) = a \cdot b^x$ from data points or a table: 1. Find the constant ratio $b$ by dividing consecutive $y$ values: $b = \frac{y 2}{y 1} = \frac{y 3}{y 2}$ 2. Find the initial value $a$ using any point $(x, y)$: $a = \frac{y}{b^x}$.
Common Questions
How do you write an exponential function from two data points?
Find the base b by dividing consecutive y-values: b = y2/y1. Then find a using any point: a = y/b^x. Write f(x) = a * b^x.
Given points (0,3) and (1,12), what is the exponential function?
b = 12/3 = 4. Since x=0 gives a directly: a = 3. So f(x) = 3 * 4^x.
From a table with (1,8), (2,24), (3,72), what is f(x)?
b = 24/8 = 3. Using point (1,8): a = 8/3^1 = 8/3. So f(x) = (8/3) * 3^x.
How do you find b from non-consecutive points like (2,50) and (4,200)?
The x-values differ by 2, so b^2 = 200/50 = 4, giving b = 2. Then a = 50/2^2 = 12.5. So f(x) = 12.5 * 2^x.
How do you verify data is exponential before writing the function?
Check that the ratio between consecutive y-values is constant. If y2/y1 = y3/y2 = y4/y3, the data is exponential.
What does the initial value a represent in f(x) = a * b^x?
a is the y-value when x = 0. It is the starting amount of the exponential quantity before any growth or decay.