6.1.1

Exponential Function

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Exponential Functions

Functions that are linear show growth or decay that occurs at a steady rate. The exponential function models growth or decay with an increasing rate.

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Definition

  • For any real number xx, an exponential function is a function with the form"
    • f(x)=abxf(x) = ab^x
  • Where aa is a non-zero real number called the initial value and bb is any positive real number such that b1b\ne1.
  • We call bb the base of the exponential function.
    • f(x)=3xf(x) = 3^x has base 3.
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Exponential growth

  • Exponential growth means that the original value increases by the same percentage over equal increments found in the domain.
    • As the value grows, so does the percentage of the value.
  • Linear growth means the original value increases by the same amount over equal increments found in the domain.
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Exponential growth

  • The table shows the values for which the exponential function of base 2 is larger than linear and quadratic functions.
  • In general, the exponential function will always eventually outgrow any polynomial.
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Graph

  • The yy-intercept of an exponential function is (0,a)(0,a).
    • For a function with no initial value such as f(x)=2xf(x)=2^x, this intercept is always equal to 1.
  • The graph does not cross the xx-axis, instead getting closer toward it for very large and negative values of xx.
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Properties

  • The domain of f(x)=abxf(x) = ab^x is all of the real numbers.
  • The range is all positive real numbers if a>0a>0.
  • The range is all negative real numbers if a<0a<0.

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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