Question

Explain the key features of the exponential function y= a*b^x including the asymptote, key points on the graph, domain and range

Answer 1

ANSWER

• Domain: all real values

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• Range: all real positive values

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• Asymptote: y = 0

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• y-intercept: (0, a)

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• The graph is continuous and smooth

EXPLANATION

The domain of this exponential function is all real values because x can take any real value without creating discontinuities.

The range, on the other hand, is all real positive values, not including zero. This is because y = 0 is a horizontal asymptote - i.e. an exponential function such as this cannot be zero.

We can also find the y-intercept, which is the point where the graph crosses the y-axis. This occurs when x = 0,

$y=a\cdot b^0=a\cdot1=a$

So the graph contains the point (0, a), which is the y-intercept.

Since there are no discontinuities or "jumps", the graph is continuous and smooth.

The end-behavior is given by the value of b. If b is between 0 and 1, the graph is decreasing and, if b is greater than 1, the graph is increasing.