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If you could fold a standard piece of paper in half 5050 times, its thickness would reach all the way from Earth past the Sun.
That staggering explosion comes from repeated multiplication, the defining engine of an exponential function.

The Parent Exponential Function

An exponential function is any function where the independent variable is in the exponent: f(x)=bxf(x) = b^x, where the base b>0b > 0 and b≠1b \neq 1.
We require b>0b > 0 so the outputs remain real numbers across all inputs, and b≠1b \neq 1 because 1x=11^x = 1 is simply a flat line.
For every parent function f(x)=bxf(x) = b^x, the domain is all real numbers (βˆ’βˆž,∞)(-\infty, \infty) and the range is (0,∞)(0, \infty).
A horizontal asymptote is a horizontal line y=cy = c that the graph approaches as xβ†’βˆžx \to \infty or xβ†’βˆ’βˆžx \to -\infty; for the parent function, it is always y=0y = 0.
What happens to these key landmarks when we shift, stretch, and flip the curve across the coordinate plane?

General Form and Transformation Model

Under the General Form and Transformation Model, exponential functions are expressed as f(x)=aβ‹…bxβˆ’h+kf(x) = a \cdot b^{x - h} + k or f(x)=aβ‹…ec(xβˆ’h)+kf(x) = a \cdot e^{c(x - h)} + k, where b>0b > 0, bβ‰ 1b \neq 1, and aβ‰ 0a \neq 0.
The parameter aa controls vertical stretching and reflection, hh provides the horizontal shift, and kk determines vertical shift and the asymptote.
The parameter aa controls vertical stretch and reflection, hh gives the horizontal shift, and kk sets the vertical shift.