Can the shift be negative
Properties of exponential functions
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Properties of the exponential function
The graph of an exponential function with, ≠ contains the points and. So you can quickly determine the function term of an exponential function with the help of the graph.
The graph contains the points and .function term:
The domain of an exponential function is ℝ which is. So exponential functions have none. However, the function values approach zero as closely as desired. The x-axis or the straight line is the horizontal of the exponential function. Exponential functions with are monotonically increasing, exponential functions with are monotonically decreasing. The graphs of the exponential functions and are symmetrical to one another with respect to the y-axis.
with and with
The general exponential function
You know the normal exponential function. By using you can change the equation, e.g. to describe or model various exponential growth processes. In general, the equation then has the form: The parameter is also called the stretching factor, because that of the normal exponential function is stretched or compressed. If negative, the curve is also mirrored on the x-axis. The graphs contain the points and. For is that, for is. So the graphs have none. However, the function values approach zero as closely as desired. The x-axis or the straight line is the horizontal of the exponential function.
with with with
Shift in y-direction
In the function equation, the parameter causes a shift of the function graph in the y-direction. For the shift is upwards, for downwards. In the case of the shift, the value range changes to. The asymptote is shifted to. The shift downwards adds a zero.
with with with
Shift in the x direction
In the function equation, the parameter causes a shift in the x-direction. For is shifted to the left, for to the right. The shift does not change the range of values.
with with with
There are also functions of the shape, because a displacement in the x-direction can also be described as stretching or compressing. For with, the shift by units to the left corresponds to a stretching with the factor, because. The shift by units to the right corresponds to a compression with the factor, because.
Shifting the exponential curve by 3 units to the left corresponds to a stretching by a factor of 8.
The compression of the exponential curve with the factor corresponds to a shift by two units to the right.
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