Exponential Functions, Sequences, Polynomials and Quadratics Review

Comprehensive practice flashcards covering exponential properties, sequences, polynomial operations, factoring patterns, and quadratic formula applications based on lecture units 6 through 10.

According to the key concept for any nonzero number $a$, what does $a^0$ equal?

$$1$$

What is the status of the power $0^0$?

Undefined

What is the Power of a Product Property formula?

$$(ab)^n = a^n b^n$$

How is an integer defined in these notes?

A whole number that can be positive, negative, or zero

Based on the jellyfish example, if $1.25 \times 10^8$ particles of light are emitted in $6.25 \times 10^{-4}$ second, how many photons are emitted each second?

$$2 \times 10^{11}$$

What is the mathematical definition of an $n\text{th}$ root of $a$?

For an integer $n > 1$, if $b^n = a$, then $b$ is an $n\text{th}$ root of $a$

If $n$ is an even integer and $a < 0$, how many real $n\text{th}$ roots does $a$ have?

No real $n\text{th}$ roots

What is the formula representing rational exponents for $a^{m/n}$?

$$a^{m/n} = (\sqrt[n]{a})^m$$

What is the standard nonlinear form of an exponential function?

$y = ab^x$, where $a \neq 0$, $b > 0$, and $b \neq 1$

What is the formula for an exponential growth function?

$y = a(1 + r)^t$, where $a > 0$ and $r > 0$

What formula is used to calculate the balance $y$ of an account earning compound interest?

$$y = P\left(1 + \frac{r}{n}\right)^{nt}$$

What is the Property of Equality for Exponential Equations?

Two powers with the same positive base $b$ (where $b \neq 1$) are equal if and only if their exponents are equal

What is the definition of a geometric sequence?

A sequence where the ratio between each pair of consecutive terms is the same ($common\,ratio$)

How is a monomial defined?

A number, a variable, or the product of a number and one or more variables with whole number exponents

What is the degree of a nonzero constant term?

$$0$$

What does it mean for a polynomial to be in standard form?

The exponents of the terms decrease from left to right

What polynomial represents the distance between a penny thrown from $200\,ft$ and a paintbrush dropped from $100\,ft$ after $t$ seconds?

$$-40t + 100$$

What is the FOIL method used for when multiplying binomials?

Multiplying First terms, Outer terms, Inner terms, and Last terms

What is the pattern for the square of a binomial for $(a + b)^2$?

$$a^2 + 2ab + b^2$$

What is the Zero-Product Property?

If $a$ and $b$ are real numbers and $ab = 0$, then $a = 0$ or $b = 0$

What is the rule for factoring $x^2 + bx + c$ when $c$ is positive and $b$ is negative?

$p$ and $q$ are both negative

What is the formula for the difference of two squares pattern?

$a^2 - b^2 = (a + b)(a - b)$, such as $x^2 - 9 = (x + 3)(x - 3)$

What is the vertex of the parent quadratic function $f(x) = x^2$?

$$(0, 0)$$

What is the vertex form of a quadratic function?

$$f(x) = a(x - h)^2 + k$$

What is the Product Property of Square Roots?

$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ where $a \geq 0$ and $b \geq 0$

What is the Quadratic Formula?

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

How many real solutions exist if the discriminant $b^2 - 4ac > 0$?

Two real solutions