HW Key Features of Exponential functions

An exponential function is a mathematical function of the form $$f(x) = a \times b^x$$ , where $$a$$ is a constant, $$b$$ is a positive real number, and @@b eq 1@@.

The general form of an exponential function is $$f(x) = a \times b^x$$ , where $$a$$ is the initial value and $$b$$ is the base of the exponential.

An exponential function is increasing if the base $$b > 1$$ , meaning as $$x$$ increases, $$f(x)$$ also increases.

An exponential function is decreasing if the base $$0 < b < 1$$ , meaning as $$x$$ increases, $$f(x)$$ decreases.

The y-intercept of an exponential function $$f(x) = a \times b^x$$ is the point where $$x = 0$$ , which is $$f(0) = a$$ .

An asymptote is a line that the graph of a function approaches but never touches. For exponential functions, the horizontal asymptote is typically $$y = 0$$ .

The average rate of change between two points $$(x_1, f(x_1))$$ and $$(x_2, f(x_2))$$ is calculated as $$\frac{f(x_2) - f(x_1)}{x_2 - x_1}$$ .

The base determines the growth or decay rate of the function. If $$b > 1$$ , the function grows; if $$0 < b < 1$$ , the function decays.

The y-intercept is $$(0, 3)$$ because $$f(0) = 3(4)^0 = 3$$ .

The y-intercept is $$(0, 2)$$ because $$f(0) = 2(1/4)^0 = 2$$ .