Graph of an Exponential Function
An exponential function is a mathematical function of the form:
f(x) = a * b^x
where ‘a’ and ‘b’ are constants and ‘x’ is the input variable
An exponential function is a mathematical function of the form:
f(x) = a * b^x
where ‘a’ and ‘b’ are constants and ‘x’ is the input variable. The base ‘b’ is a positive number greater than 1, which determines the rate at which the function grows or decays.
To graph an exponential function, you can follow these steps:
1. Choose values for ‘x’: Start by selecting a few values for ‘x’. It’s a good idea to include both positive and negative values and pick them symmetrically around the y-axis. For simplicity, let’s consider ‘x’ values of -3, -2, -1, 0, 1, 2, and 3.
2. Evaluate ‘f(x)’: Substitute each chosen ‘x’ value into the equation f(x) = a * b^x to find the corresponding ‘y’ values.
3. Plot the points: Plot the points (x, f(x)) on a coordinate plane. For example, if f(1) = 3, you would plot the point (1, 3).
4. Connect the points: Draw a smooth curve that passes through the plotted points. Since exponential functions can grow or decay rapidly, it is helpful to plot additional points if needed between the chosen ‘x’ values.
5. Determine the behavior of the graph: Exponential functions can either increase or decrease. If the base ‘b’ is greater than 1, the graph will increase as ‘x’ gets larger. If ‘b’ is between 0 and 1, the graph will decrease as ‘x’ gets larger. The y-axis serves as the asymptote for the graph.
6. Label the axes and add any necessary information: Label the x- and y-axes, and provide a title for the graph if needed. Include any other relevant information, such as the value of ‘a’ or ‘b’ if specified.
It’s important to note that the specific shape of the graph depends on the values of ‘a’ and ‘b’. For example, if ‘a’ is positive, the graph will start in the positive y-direction. If ‘a’ is negative, the graph will start in the negative y-direction.
Remember, practice makes perfect! The more you work with exponential functions and graph them, the better you will become at visualizing their behavior.
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