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.
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.
$-f(x)$
Explain the product property.
When multiplying exponential terms with the same base, add the exponents: $b^m * b^n = b^{m+n}$.
Explain the power property.
When raising an exponential term to a power, multiply the exponents: $(b^m)^n = b^{mn}$.
Explain the negative exponent property.
A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent: $b^{-n} = \frac{1}{b^n}$.
Explain exponential unit fractions.
An exponential unit fraction, like $b^{1/k}$, represents the *k*th root of *b*. The value of $b^{1/k}$ is the *k*th root of *b*, when it exists.
Explain the relationship between horizontal translation and vertical dilation.
A horizontal translation of an exponential function, $f(x) = b^{x+k}$, is equivalent to a vertical dilation, $f(x) = ab^x$, where $a = b^k$.
Explain the relationship between horizontal dilation and changing the base.
A horizontal dilation of an exponential function, $f(x) = b^{cx}$, is equivalent to changing the base, $f(x) = (b^c)^x$.
Explain the effect of reflecting an exponential function over the y-axis.
Reflecting the graph of $f(x) = b^x$ over the y-axis gives the graph of $f(x) = b^{-x} = \frac{1}{b^x}$.
Why is the base of an exponential function restricted to positive numbers not equal to 1?
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.
Explain the importance of understanding transformations of exponential functions.
Understanding transformations helps analyze and predict the behavior of exponential functions under various conditions, which is crucial for modeling real-world scenarios.
Explain why $(-4)^{1/2}$ does not have a real value.
The square root of a negative number is not a real number because no real number multiplied by itself yields a negative result.
