Exponential Functions transformations

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 $$x$$ is the exponent.

Stretching an exponential function means to increase its steepness, typically by multiplying the function by a factor greater than 1.

Shrinking an exponential function means to decrease its steepness, typically by multiplying the function by a factor between 0 and 1.

A vertical translation shifts the graph of the function up or down without changing its shape.

A horizontal translation shifts the graph of the function left or right without changing its shape.

This transformation stretches or shrinks the graph vertically by a factor of $$a$$ , shifts it horizontally by $$h$$ units, and shifts it vertically by $$k$$ units.

The parent function of exponential functions is $$f(x) = b^x$$ , where $$b > 0$$ and @@b eq 1@@.

In transformations, 'factor' refers to the multiplier that affects the steepness or width of the graph.

A negative sign in front of an exponential function reflects the graph across the x-axis.

This transformation indicates a horizontal shift of the graph of $$f(x) = 2^x$$ to the right by 3 units.