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 are the differences between horizontal and vertical transformations?

Horizontal: Affect the x-values, inside the function. | Vertical: Affect the y-values, outside the function.

What are the differences between growth and decay exponential functions?

Growth: Base > 1, function increases as x increases. | Decay: 0 < Base < 1, function decreases as x increases.

What are the differences between product and power properties?

Product: Adding exponents when multiplying same bases. | Power: Multiplying exponents when raising a power to a power.

Compare $b^x$ and $b^{-x}$ .

$b^x$ : Exponential growth if b > 1, decay if 0 < b < 1. | $b^{-x}$ : Exponential decay if b > 1, growth if 0 < b < 1.

Compare horizontal translation and vertical stretch.

Horizontal translation: Shifts the graph left or right. | Vertical stretch: Changes the steepness of the graph.

Compare horizontal and vertical dilations.

Horizontal: Affects the x-axis, compression or stretch. | Vertical: Affects the y-axis, stretch or compression.

Compare $f(x) = b^{x+k}$ and $f(x) = b^x + k$ .

$f(x) = b^{x+k}$ : Horizontal shift. | $f(x) = b^x + k$ : Vertical shift.

Compare $f(x) = b^{cx}$ and $f(x) = cb^x$ .

$f(x) = b^{cx}$ : Horizontal dilation. | $f(x) = cb^x$ : Vertical dilation.

Compare reflection about the x-axis and y-axis.

x-axis: Changes the sign of the output. | y-axis: Changes the sign of the input.

Compare $b^{1/2}$ and $(b)^{2}$ .

$b^{1/2}$ : Square root of b. | $(b)^{2}$ : Square of b.

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.

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