Key takeaway: An exponential function has the variable in the exponent, f(x) = a·b^x, so its output is multiplied by the same factor b every time x increases by 1.
Grade 9 – Algebra 1F-LE.A.1Functions and Graphs
Definition of Exponential Function
An exponential function changes by repeated multiplication rather than repeated addition. Take a rule shaped like \(f(x) = a \cdot b^{x}\): its output at \(x = 0\) is \(a\), and each unit step multiplies by the base \(b\). If \(b > 1\) the outputs climb faster and faster; if \(0 < b < 1\) they shrink toward zero.
Compare a table with a constant difference (linear) to one with a constant ratio (exponential). In an exponential table, dividing each output by the one before it gives the same number every time, and that number is \(b\). The picture is a smooth curve that starts on the vertical axis at height \(a\) and stays on one side of the horizontal axis forever.
Formula and notation
An exponential function is written
\[f(x) = a \cdot b^{x}\]
where \(a\) is the start and \(b\) is the multiplier for each unit of \(x\); the base must be positive and different from 1. For a percent change, change the rate into a decimal: the multiplier is one more than it for growth and one less for decay.
Key facts
- The y-intercept of f(x) = a·b^x is the point (0, a).
- Equal steps in x multiply the output by equal factors.
- The x-axis is a horizontal asymptote that the graph approaches but never reaches.
- Eventually an exponential function with b greater than 1 outgrows any linear function.
Exponential Function example
A baker keeps a sourdough starter that weighs 3 ounces on day 0 and doubles every day. Write the function and find the weight on day 5.
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The starting amount is \(a = 3\) and the factor is \(b = 2\), so \(f(x) = 3 \cdot 2^{x}\).
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Substitute \(x = 5\): \(f(5) = 3 \cdot 2^{5}\).
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Compute \(2^{5} = 32\), then \(3 \times 32 = 96\).
The starter weighs 96 ounces on day 5.
Common mistake
Watch out
Gus treats the rule as \(f(x) = 3 \cdot 2x\) and finds \(f(5) = 3 \cdot 10 = 30\) ounces.
Why it is wrong
He multiplied by \(2x\) instead of using \(x\) as an exponent. In \(2^{x}\), the 2 is multiplied by itself 5 times (\(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32\)). His version grows by 6 ounces a day, but a doubling starter grows by larger amounts every day.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 1 vocabulary · Functions and Graphs