Holographic Diffraction Gratings. Typical holographic diffraction gratings are produced on soda lime and fused silica glass substrates from high quality ...

Vertical translationequation example

The domain is [latex]\left(-\infty ,\infty \right)[/latex], the range is [latex]\left(-3,\infty \right)[/latex], and the horizontal asymptote is [latex]y=-3[/latex].

Vertical translationdown

Polarization of light is a property that applies to turning waves that shows the geometrical blooming of the oscillations. In a turning wave, the way of the ...

Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the parent function [latex]f\left(x\right)={b}^{x}[/latex] without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.

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Vertical translationrule

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Translating exponential functions follows the same ideas you’ve used to translate other functions. Add or subtract a value inside the function argument (in the exponent) to shift horizontally, and add or subtract a value outside the function argument to shift vertically.

The domain is [latex]\left(-\infty ,\infty \right)[/latex], the range is [latex]\left(3,\infty \right)[/latex], and the horizontal asymptote is y = 3.

Vertical translationgraph

This microscope is composed of a light source, a substage condenser, a stage where the specimen slide is placed, 3-4 objective lenses and ocular lenses. These ...

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The first transformation occurs when we add a constant d to the parent function [latex]f\left(x\right)={b}^{x}[/latex] giving us a vertical shift d units in the same direction as the sign. For example, if we begin by graphing a parent function, [latex]f\left(x\right)={2}^{x}[/latex], we can then graph two vertical shifts alongside it using [latex]d=3[/latex]: the upward shift, [latex]g\left(x\right)={2}^{x}+3[/latex] and the downward shift, [latex]h\left(x\right)={2}^{x}-3[/latex]. Both vertical shifts are shown in the figure below.

Watch the following video for more examples of the difference between horizontal and vertical shifts of exponential functions and the resulting graphs and equations.

Vertical translationexamples

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Online graphing calculators automatically calculate points of interest including intersections. Essentially, you are looking for the intersection of two functions. Click on the point of intersection, and you will see the the x and y values for the point.

Vertical translationmath

Use an online graphing calculator to plot the function [latex]f\left(x\right)={2}^{x-1}+3[/latex]. State domain, range, and asymptote.

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Vertical translationand horizontaltranslation

The next transformation occurs when we add a constant c to the input of the parent function [latex]f\left(x\right)={b}^{x}[/latex] giving us a horizontal shift c units in the opposite direction of the sign. For example, if we begin by graphing the parent function [latex]f\left(x\right)={2}^{x}[/latex], we can then graph two horizontal shifts alongside it using [latex]c=3[/latex]: the shift left, [latex]g\left(x\right)={2}^{x+3}[/latex], and the shift right, [latex]h\left(x\right)={2}^{x - 3}[/latex]. Both horizontal shifts are shown in the graph below.

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Vertical translationequation

Shift the graph of [latex]f\left(x\right)={b}^{x}[/latex] left 1 unit and down 3 units. The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(-3,\infty \right)[/latex]; the horizontal asymptote is [latex]y=-3[/latex].

For any constants c and d, the function [latex]f\left(x\right)={b}^{x+c}+d[/latex] shifts the parent function [latex]f\left(x\right)={b}^{x}[/latex]

Graphing can help you confirm or find the solution to an exponential equation. An exponential equation is different from a function because a function is a large collection of points made of inputs and corresponding outputs, whereas equations that you have seen typically have one, two, or no solutions.  For example, [latex]f(x)=2^{x}[/latex] is a function and is comprised of many points [latex](x,f(x))[/latex], and [latex]4=2^{x}[/latex] can be solved to find the specific value for x that makes it a true statement. The graph below shows the intersection of the line [latex]f(x)=4[/latex] and [latex]f(x)=2^{x}[/latex]. You can see they cross at [latex]y=4[/latex].

We have an exponential equation of the form [latex]f\left(x\right)={b}^{x+c}+d[/latex], with [latex]b=2[/latex], [latex]c=1[/latex], and [latex]d=-3[/latex].