Transformations of exponential functions

Exponential functions and logarithms: Exponential functions

Transformations of exponential functions

Transformation of an exponential By horizontal and vertical translation, and by vertical multiplication of the graph of the exponential function \(f(x)=\e^x\) the graph of the function \[g(x)=a\cdot \e^{b(x+c)} + d\quad\text{with parameters }a,b,c,d\text.\] can be created. An arbitrary exteded exponential relationship \(y=a\cdot g^{x+c}\) with base \(g=\e^b\) can be constructed in this way.

We speak of an extended exponential relationship when \(d\neq0\). For \(d=0\) we have a 'normal' exponential relationship of the form \(y=a\cdot g^{x+c}\) for certain constants \(a\), \(g\), and \(c\). Using the calculation rules for powers you can even limit the number of constants needed to two: \(y=a\cdot g^c \cdot g^{x}\) and therefore we can write the relationship as \(y =\alpha\cdot g^x\) for a certain constant \(\alpha\) and \(g\).

To illustrate transformations of exponential functions, we consider the horizontal and vertical translation as well as the vertical scaling of the graph of the exponential function \(f(x)=e^x\).

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Vertical translation

We translate the graph of \(f(x)=\e^x\) vertically over a distance \(\blue{d}\).

The new function becomes \[g(x)=\e^x+\blue{d}\] The asymptote of \(g\) becomes the line \(y=\blue{d}\). The range of \(g\) becomes \((\blue{d},\infty)\).

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Horizontal translation

We translate the graph of \(f(x)=\e^x\) horizontally to the left over a distance \(\blue{c}\).

The new function becomes \[g(x)=e^{x+\blue{c}}\] The asymptote of \(g\) is the same as the one of the exponential function, i.e., the line \(y=0\). Also, the range of \(g\) is equal to the range of the exponential function, i.e., \((0,\infty)\).

In fact this transformation is according to calculation trules of exponential functions the same as vertical scaling by \(e^c\).

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Vertical scaling

We multiply the graph of \(f(x)=\e^x\) with respect to the \(x\)-axis by \(\blue{a}\).

The new function becomes \[g(x)=\blue{a}\cdot \e^x\] The asymptote of \(g\) is the same as the one of the exponential function, i.e., the line \(y=0\). Also, the range of \(g\) is equal to the range of the exponential function, viz. \((0,\infty)\), for positive values of \(\blue{a}\). For negative values of \(\blue{a}\), the range of \(g\) becomes the interval \((-\infty,0)\).

GeoGebra

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Horizontal scaling

We multiply the graph of \(f(x)=\e^x\) with respect to the \(y\)-axis by \(\blue{\frac{1}{b}}\) for some \(\blue {b}\neq 0\).

The new function becomes \[g(x)= \e^{\blue{b}x}\] The asymptote of \(g\) is the same as the one of the exponential function, i.e., the line \(y=0\). Also, the range of \(g\) is equal to the range of the exponential function, i.e., \((0,\infty)\).

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Interactive example You can combine the above transformations to construct any extended exponential relationship. In the interactive example below you can explore via the sliders the effect of the parameters \(a\), \(b\), \(c\), and \(d\) that lead to the graph of the function \[g(x)=\color {red}{a}\cdot \e^{\color{green}{b}\cdot(x+\color{blue}{c})+\color{magenta}{d}} \] We have initialised these sliders so that a graph for limited exponential growth is shown.

GeoGebra