Product property formula.
$b^m * b^n = b^{m+n}$
Power property formula.
$(b^m)^n = b^{mn}$
Negative exponent property formula.
$b^{-n} = \frac{1}{b^n}$
Exponential unit fraction formula.
$b^{\frac{1}{k}} = \sqrt[k]{b}$
Formula for horizontal translation of k units to the left.
$f(x+k)$
Formula for horizontal translation of k units to the right.
$f(x-k)$
Formula for vertical dilation by a factor of a.
$a*f(x)$
Formula for horizontal dilation by a factor of c.
$f(cx)$
Formula for reflection about the y-axis.
$f(-x)$
Formula for reflection about the x-axis.
$-f(x)$
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.
