A function of the form $f(x) = ab^x$, where $a$ is a constant, $b$ is the base, and $x$ is the exponent.

The base, $b$, is a positive real number not equal to 1 in the exponential function $f(x) = ab^x$.

A shift of a function's graph left or right along the x-axis.

A stretch or compression of a function's graph in the vertical direction (y-axis).

A stretch or compression of a function's graph in the horizontal direction (x-axis).

An exponent that indicates the reciprocal of the base raised to the positive value of the exponent.

An exponent in the form of 1/n, representing the nth root of the base.

The reciprocal of a number x is 1/x.

A transformation where the graph of a function is flipped over the y-axis.

A change in the position, shape, or size of a function's graph.

When multiplying exponential terms with the same base, add the exponents: $b^m * b^n = b^{m+n}$.

When raising an exponential term to a power, multiply the exponents: $(b^m)^n = b^{mn}$.

A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent: $b^{-n} = \frac{1}{b^n}$.

An exponential unit fraction, like $b^{1/k}$, represents the kth root of b. The value of $b^{1/k}$ is the kth root of b, when it exists.

A horizontal translation of an exponential function, $f(x) = b^{x+k}$, is equivalent to a vertical dilation, $f(x) = ab^x$, where $a = b^k$.

A horizontal dilation of an exponential function, $f(x) = b^{cx}$, is equivalent to changing the base, $f(x) = (b^c)^x$.

Reflecting the graph of $f(x) = b^x$ over the y-axis gives the graph of $f(x) = b^{-x} = \frac{1}{b^x}$.

If the base were negative, the function would oscillate between positive and negative values. If the base were 1, the function would be constant. If the base were 0, the function would be undefined for negative exponents.

Understanding transformations helps analyze and predict the behavior of exponential functions under various conditions, which is crucial for modeling real-world scenarios.

The square root of a negative number is not a real number because no real number multiplied by itself yields a negative result.

Add the exponents: $2^{2x + (x-1)} = 2^{3x-1}$.

Multiply the exponents: $4^{2(x+1)} = 4^{2x+2}$.

Use the negative exponent property: $5^{-x} = \frac{1}{5^x}$.

Find the square root: $9^{1/2} = \sqrt{9} = 3$.

Use the product property: $3^{x-2} = 3^x cdot 3^{-2} = \frac{1}{9} cdot 3^x$.

Use the power property: $16^{3x} = (16^3)^x = 4096^x$.

Horizontal translation 3 units to the left.

Vertical stretch by a factor of 2.

Reflection over the y-axis.

Reflection over the y-axis.