What is an Exponential Function?. Discover the key difference between exponential functions and polynomials. Learn the definition, identify what makes a function exponential, and understand the restrictions on the base.
Key Characteristics of f(x) = cˣ. Explore the properties shared by all exponential functions: y-intercept, horizontal asymptote, domain and range. Learn why every exponential graph passes through (0, 1) and (1, c).
Growth vs. Decay. Understand the crucial difference between growth functions (c > 1) and decay functions (0 < c < 1). Learn how the base determines whether values increase or decrease, and discover the connection between them.
Finding the Equation of an Exponential Function. Master the step-by-step process for finding exponential equations from given information. Learn the k-a-c strategy and work through detailed examples with verification.
Finding the Equation from a Graph. Learn how to read exponential equations directly from graphs. Practice identifying the asymptote, y-intercept, and second point to determine k, a, and c values.
Inside the lesson — a free preview
The Big Idea
You've already worked with exponents before — things like 2^3 = 8 or 5^2 = 25. In those cases, the exponent was a fixed number. What makes exponential functions different and exciting is that the exponent is now the variable.
In plain English. Instead of asking "what is 2 to the power of 3?", we're now asking "what is 2 to the power of x?" — where x can be anything.
The Definition
Exponential Function. An exponential function is a continuous function of the form:
f(x) = c^x
Where.
What does "continuous" mean?. The graph has no breaks, holes, or jumps in it — it flows smoothly from left to right without lifting your pencil.
Why Do We Use "c" Instead of "b"?
You might wonder why we use the letter c for the base instead of b. Later in this unit, we'll use the parameters a, b, h, and k to describe transformations of functions. Since b already has a job in that notation, we use c here to avoid confusion.
Worked example: ✅ Examples That ARE Exponential
f(x) = 3^x
x is the exponent, 3 is the fixed base
f(x) = ((1)/(2))^x
x is the exponent, (1)/(2) is the fixed base
f(x) = 7^x
x is the exponent, 7 is the fixed base
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