Define exponential function.

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

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Define exponential function.

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

What is the base of an exponential function?

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

Define horizontal translation.

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

Define vertical dilation.

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

Define horizontal dilation.

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

Define negative exponent.

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

What is a unit fraction exponent?

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

Define reciprocal.

The reciprocal of a number x is 1/x.

Define reflection over the y-axis.

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

What is a transformation of a function?

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

Simplify $2^{2x} * 2^{x-1}$ .

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

Simplify $(4^{x+1})^2$ .

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

Simplify $5^{-x}$ .

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

Evaluate $9^{1/2}$ .

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

Rewrite $f(x) = 3^{x-2}$ in the form $a cdot 3^x$ .

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

Rewrite $g(x) = 16^{3x}$ in the form $a^x$ .

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

Describe the transformation from $y = 2^x$ to $y = 2^{x+3}$ .

Horizontal translation 3 units to the left.

Describe the transformation from $y = 3^x$ to $y = 2 cdot 3^x$ .

Vertical stretch by a factor of 2.

Describe the transformation from $y = 4^x$ to $y = 4^{-x}$ .

Reflection over the y-axis.

Describe the transformation from $y = 5^x$ to $y = (1/5)^x$ .

Reflection over the y-axis.

What does a steeper slope in the graph of $f(x) = b^x$ indicate?

A larger value for $b$ , indicating faster exponential growth.

How does a horizontal shift affect the y-intercept of $f(x) = b^x$ ?

A leftward shift increases the y-intercept, while a rightward shift decreases it.

What does a reflection over the y-axis do to the graph of $f(x) = b^x$ ?

It transforms the graph into that of $f(x) = (1/b)^x$ , changing growth to decay.

How does vertical dilation affect the y-intercept of $f(x) = b^x$ ?

It multiplies the y-intercept by the dilation factor.

How does horizontal dilation affect the graph of $f(x) = b^x$ ?

It changes the rate of growth/decay; a compression increases the rate, and a stretch decreases it.

What does the graph of $f(x) = b^{-x}$ look like?

It is a decreasing exponential function, decaying towards zero as x increases.

What does the graph of $f(x) = b^{x+k}$ look like?

It is the graph of $f(x) = b^x$ shifted horizontally by k units.

What does the graph of $f(x) = ab^x$ look like?

It is the graph of $f(x) = b^x$ vertically stretched by a factor of a.

What does the graph of $f(x) = (b^c)^x$ look like?

It is the graph of $f(x) = b^x$ horizontally compressed by a factor of c.

What does the graph of $f(x) = -b^x$ look like?

It is the graph of $f(x) = b^x$ reflected about the x-axis.