Writing Exponential Functions

Writing a linear equation from two points is routine: find the slope, then the intercept. Writing an exponential function from two points follows the same two-step pattern, with one change that makes all the difference — you divide instead of subtracting, because an exponential function changes by a constant factor rather than a constant amount. Once you can do this, you can build a model from any two measurements: two population counts, two account balances, two readings of a cooling cup of coffee.

What You Are Solving For

Every exponential function in standard form is

with the initial value (the y-intercept) and the growth factor. Writing the equation means finding these two numbers. Two unknowns need two conditions — usually two points on the graph.

Method 1: When One Point Is the y-Intercept

If you are given , you already have , because . Substitute the second point and solve for :

For and : , then , so and . The function is .

Method 2: How to Find an Exponential Equation With Two Points

When neither point is on the y-axis, write both equations and divide:

The cancels, leaving one equation in . Solve it, then substitute back to get :

Because must be positive, always take the positive root. If , then , not .

Method 3: From a Table

A table is just a list of points, but it often makes the factor obvious. If the x-values go up by , divide each y-value by the one before it — that ratio is . Read off the row where , or work it out backwards.

Each ratio is , so , , and .

Method 4: Exponential Equation From a Graph

To write an equation from a graph, read three things off it: the horizontal asymptote , the y-intercept, and one other clearly marked point.

• If the asymptote is , use the y-intercept and the second point exactly as in Method 1.
• If the asymptote is , the model is . Subtract from every y-value first — that turns the graph back into a plain — then find and as usual.

The graph below has asymptote and passes through and . Subtracting gives adjusted heights and , so , , and the equation is .

Example 1: Writing Exponential Functions From Two Points

The last row does not give a whole-number base, which is normal for real data. Compute the root with the key, and keep four or more decimal places — rounding too early gets magnified by the exponent.

Worked Example

Find the exponential function through and .

Step 1 — write both points into :

Step 2 — divide the second equation by the first so cancels:

Step 3 — simplify the ratio and take the cube root:

Step 4 — substitute into the first equation and solve for :

Step 5 — write the function and check the second point:

Since , this is exponential decay, which matches the output falling from to .

Common Mistakes to Avoid

• Subtracting the y-values. A constant difference belongs to a linear function. For an exponential, divide: the ratio of outputs is what stays constant.
• Forgetting the gap in x. From and the ratio is , not — the points are two steps apart.
• Using the first point’s y-value as when . is the value at . If the given point is , is , not .
• Measuring from the x-axis when the asymptote has moved. With asymptote , subtract from the y-values before dividing, or the ratio will be wrong.

Practice Problems

Work each one before opening the answer.

Problem 1. Write the exponential function through and .

Problem 2. Write the exponential function through and .

Problem 3. Write the exponential function through and .

Problem 4. Write the exponential function through and .

Problem 5. Find for the exponential function through and . Why is rejected?

Problem 6. A graph has horizontal asymptote and passes through and . Write its equation.

Many of the equations you write will describe something growing or shrinking by a percentage — a salary rising 3% a year, a medicine clearing 20% an hour. The exponential growth and decay lesson shows how to turn a rate like that straight into .