Mathematics Final Review: Exponential, Logarithmic, Polynomial, and Absolute Value Functions

Flashcards covering the core concepts of Exponential and Logarithmic functions, Polynomial and Rational functions, and Absolute Value and Quadratic inequalities.

What is the equivalent exponential form of the logarithmic equation $y = \log_b(x)$?

$$b^y = x$$

According to the Product Property of Logs, what is the expansion of $\log_b(MN)$?

$$\log_b(M) + \log_b(N)$$

What is the Quotient Property of Logs for the expression $\log_b(\frac{M}{N})$?

$$\log_b(M) - \log_b(N)$$

How does the Power Property of Logs rewrite the expression $\log_b(M^p)$?

$$p \cdot \log_b(M)$$

How is the natural log $\ln(x)$ defined?

$\ln(x)$ is $\log_e(x)$

When solving exponential equations with the same base, if $a^x = a^y$, what is the relationship between $x$ and $y$?

$$x = y$$

What must be checked when solving Log Equations to avoid extraneous solutions?

The argument of a log must be > 0.

What is the formula for possible rational zeros according to the Rational Zero Theorem?

$\pm \frac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

What does the Fundamental Theorem of Algebra state about a polynomial of degree $n$?

A polynomial of degree $n$ has exactly $n$ zeros, counting multiplicity and complex zeros.

If a polynomial with real coefficients has a complex zero $a + bi$, what must another zero be?

Its conjugate, $a - bi$.

For rational inequalities, what are the two types of critical values found to create a Sign Chart?

Zeros of the numerator (where the graph hits the axis) and zeros of the denominator (where the graph has a vertical asymptote).

How is the absolute value inequality $|x| \leq c$ rewritten as an \"and\" statement?

$$-c \leq x \leq c$$

How is the absolute value inequality $|x| \geq c$ rewritten as an \"or\" statement?

$x \geq c$ or $x \leq -c$

What is the quadratic formula used to solve $ax^2 + bx + c = 0$?

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

In solving quadratic inequalities, what does a positive leading coefficient (a > 0) indicate about the parabola structure?

The parabola opens up.