Covers the basic definition, components, and graphical representation of exponential functions, including the concepts of growth and decay. This stack is the starting point for understanding functions that model rapid increase or decrease.
What is the general form of an exponential function?
The general form is \(f(x) = ab^x\), where \(a \neq 0\), \(b > 0\), and \(b \neq 1\).
In the exponential function \(f(x) = ab^x\), what does the coefficient \(a\) represent?
\(a\) represents the initial value of the function, which is the value of the function when \(x=0\). It is also the y-intercept of the graph.
In the exponential function \(f(x) = ab^x\), what does the base \(b\) represent?
\(b\) is the growth or decay factor. It's the constant ratio by which the function's value is multiplied for each unit increase in \(x\).
What condition on the base \(b\) indicates exponential growth?
Exponential growth occurs when the base \(b\) is greater than \(1\) (\(b > 1\)).
What condition on the base \(b\) indicates exponential decay?
Exponential decay occurs when the base \(b\) is between \(0\) and \(1\) (\(0 < b < 1\)).
What is the y-intercept of the graph of \(f(x) = ab^x\)?
The y-intercept is at the point \((0, a)\), since \(f(0) = ab^0 = a \cdot 1 = a\).
What is the horizontal asymptote of the graph of \(f(x) = ab^x\) (assuming no vertical shift)?
The horizontal asymptote is the x-axis, which is the line \(y = 0\).
What is the domain of a standard exponential function \(f(x) = ab^x\)?
The domain is all real numbers, which can be written as \((-\infty, \infty)\).
What is the range of a standard exponential function \(f(x) = ab^x\) where \(a > 0\)?
The range is all positive real numbers, which can be written as \((0, \infty)\).
Describe the general shape of an exponential decay function's graph.
The graph starts high on the left, passes through the y-intercept \((0, a)\), and decreases towards the horizontal asymptote \(y=0\) as \(x\) moves to the right.
What is the natural base, denoted by the letter \(e\)?
\(e\) is an irrational mathematical constant, approximately equal to \(2.71828\). It is the base of the natural logarithm and arises naturally in contexts of continuous growth.
What is the natural exponential function?
The natural exponential function is \(f(x) = e^x\), an exponential function with the natural base \(e\).
Describe the general shape of an exponential growth function's graph.
The graph starts close to the horizontal asymptote \(y=0\) on the left, passes through the y-intercept \((0, a)\), and increases rapidly as \(x\) moves to the right.
