Exponential Function
In short: An exponential function is written y = a·bˣ: the input sits up in the exponent, and every 1-step move in x multiplies the output by the same factor b.
Algebra 1Functions and GraphsF-LE.A.1
What is Exponential Function?
Whenever a quantity is multiplied by one fixed number at every step, its graph curves sharply, and the rule behind it is an exponential function with the variable in the exponent: \(y = a \cdot b^x\).
The constant \(a\) is the starting output at \(x = 0\), while the base \(b\), positive and different from 1, is the per-step factor. A base above 1 produces growth; a base between 0 and 1 produces decay. For equally spaced x-values, a steady ratio between neighboring outputs gives the pattern away.
How to spot it
- Outputs in a table with evenly spaced inputs keep getting multiplied by one constant.
- The story doubles, triples or sheds a fixed percent every period.
- An exponential function shows the variable in the exponent, like \(3 \cdot 2^x\), never in the base.
- A graph creeps up and then shoots upward, or plunges and then flattens near the x-axis.
The method
- Divide each output by the previous one for equally spaced inputs; a constant quotient is the factor b.
- Read the starting value a as the output at \(x = 0\).
- Write \(y = a \cdot b^x\) with those numbers.
- Check the rule against another row of the table before using it to predict.
Exponential Function: solved example
Problem
A new video channel gets weekly views as follows: week 0 has 5 views, week 1 has 15, week 2 has 45, week 3 has 135. Write an exponential function and predict week 5.
Solution
The ratios are \(\frac{15}{5} = \frac{45}{15} = \frac{135}{45} = 3\), so \(b = 3\), and \(a = 5\) at week 0. The rule is \(y = 5 \cdot 3^x\).
For week 5: \(y = 5 \cdot 3^5 = 5 \cdot 243 = 1{,}215\) views.
Try it
Question 1
Is \(y = 4x^2\) an exponential function?
Show the answer
No. The variable is in the base, not the exponent, so it is quadratic.
Question 2
Evaluate \(y = 7 \cdot \left(\frac{1}{2}\right)^x\) when \(x = 3\).
Show the answer
\(7 \cdot \frac{1}{8} = \frac{7}{8} = 0.875\).
Exponential Function vs. Linear Function
Straight-line rules add one amount per unit step in x, which draws a straight graph. The multiplying rule draws a curve that speeds up or levels off. Steady differences signal the line; steady ratios signal the curve.
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