Graphing Exponential Functions Graph the function, not by plotting points, but by starting from

step 1 We're going to use the figure at the left to graph the function, H of x is equal to 2 to the x m

Graphing Exponential Functions Graph the function, not by...

Key Concepts

Exponential Functions

An exponential function is one in which the variable appears in the exponent. These functions show rapid growth or decay and have the general form f(x) = a^(x), where the base a is a positive constant not equal to 1. Their graph is always smooth, continuous, and non-linear, and they are used to model various real-world phenomena such as population growth, radioactive decay, and interest compounding.

Function Transformations

Function transformations involve shifting, stretching, compressing, or reflecting the graph of a function. In the context of exponential functions, transformations such as horizontal shifts (moving the graph left or right) and vertical shifts (moving the graph up or down) can modify the location of key features like the asymptote. For example, in the function h(x) = 2^(x-4) + 1, the graph of the basic exponential function f(x)=2^x is shifted to the right by 4 units and upward by 1 unit.

Domain of Functions

The domain of a function is the complete set of possible input values (x-values) for which the function is defined. For exponential functions, the domain is typically all real numbers because an exponent can take any real value, ensuring that the function produces a valid output regardless of the input.

Range of Functions

The range of a function refers to the set of all possible output values (y-values) that the function can produce. For a standard exponential function, the range is usually all positive real numbers. However, when a vertical shift is applied, as in h(x)=2^(x-4)+1, the entire graph shifts upward, thereby changing the range to all real numbers greater than the value of the horizontal asymptote.

Asymptotes

An asymptote is a line that the graph of a function approaches but never actually touches. For exponential functions, a horizontal asymptote commonly exists because the function approaches a particular y-value as x tends to negative infinity (in the case of exponential growth) or as x tends to positive infinity (for exponential decay). In the function h(x)=2^(x-4)+1, the vertical shift moves the horizontal asymptote to y = 1, indicating that as x becomes very small, h(x) will get arbitrarily close to y = 1.

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