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Question: Graph of an exponential function \(y=A{{e}^{-x}}\) is shown in the figure. Find the value of A....

Graph of an exponential function y=Ae−xy=A{{e}^{-x}} is shown in the figure. Find the value of A.

Explanation

Solution

Let us find any point on the graph where the x-coordinates and the y-coordinates are known. The equation of the graph satisfies the point, as the graph is meeting the point and passing through the point. Substitute the x and y coordinates in the equation, we will get the constant value.

Complete answer:
Let us write down the values and the equations given,
y=Ae−xy=A{{e}^{-x}}
As we see the graph, it is given that the point (0,4 ) lies on the graph.
It means that, the graph equation satisfies the point,
If we substitute the point given in the equation, we get,
x=0,y=4 ⇒y=Ae−x ⇒4=Ae0 ⇒A=4,e0=1 \begin{aligned} & x=0,y=4 \\\ & \Rightarrow y=A{{e}^{-x}} \\\ & \Rightarrow 4=A{{e}^{0}} \\\ & \Rightarrow A=4,{{e}^{0}}=1 \\\ \end{aligned}

Therefore, we can say the value of A in the above exponential equation is 4.

Additional information:
The exponential function is one of the most important functions in mathematics. To form an exponential function, we let the independent variable be the exponent. Exponential functions look somewhat similar to functions you have seen before, in that they involve exponents, but there is a big difference, in that the variable is now the power, rather than the base. The graph equation satisfies all the points through which the graph will be passing. The constant will not change for all the points through which the graph passes. If we want to know whether the graph is passing through the point, the constant obtained from the known point in the graph must be equal to the new one. If it's not, then we need to conclude that the graph will not pass through that point.

Note:
In the exponential equations, the bases are usually constant and the variables will be in the power terms. The constant term will be the same throughout the graph and it will not change for points through which the graph is passing. If we want to know if a graph is passing through a point or not, equate the constant obtained with the constant obtained when the new point is substituted in the equation.