Quadratic, Polynomial, Rational, Exponential, and Logarithmic Functions: Key Concepts and Properties

Quadratic Functions

Vertex Form and Standard Form

Quadratic functions are polynomials of degree two and can be written in vertex form or standard form. Understanding their structure is essential for graphing and analyzing their properties.

• Vertex Form: $f(x) = a(x-h)^2 + k$

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex: In vertex form, the vertex is $(h, k)$. In standard form, the vertex is $\left( \frac{-b}{2a}, f\left(\frac{-b}{2a}\right) \right)$.

• Direction of Opening: If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

• x-intercepts: Solve $f(x) = 0$ to find where the graph crosses the x-axis.

• y-intercept: Set $x = 0$ to find where the graph crosses the y-axis.

Example: For $f(x) = 2(x-1)^2 + 3$, the vertex is $(1, 3)$ and the parabola opens upward.

Maximum and Minimum Values

The vertex of a quadratic function represents either the maximum or minimum value, depending on the sign of $a$.

• If $a > 0$, the function has a minimum at $x = \frac{-b}{2a}$, with value $f\left(\frac{-b}{2a}\right)$.

• If $a < 0$, the function has a maximum at $x = \frac{-b}{2a}$, with value $f\left(\frac{-b}{2a}\right)$.

Example: For $f(x) = -x^2 + 4x + 1$, the maximum occurs at $x = 2$.

Polynomial Functions and Their Graphs

Leading Coefficient Test

The end behavior of a polynomial function is determined by its degree and the sign of its leading coefficient.

• n odd, coefficient positive: Graph falls to the left and rises to the right.

• n odd, coefficient negative: Graph rises to the left and falls to the right.

• n even, coefficient positive: Graph rises to both the left and right.

• n even, coefficient negative: Graph falls to both the left and right.

Zeros, Multiplicity, and x-intercepts

• Zeros: The real solutions to $f(x) = 0$ are the x-intercepts of the graph.

• Multiplicity: If a zero $r$ has even multiplicity, the graph touches and turns at $r$. If $r$ has odd multiplicity, the graph crosses the x-axis at $r$. Higher multiplicity causes the graph to flatten near the intercept.

Example: $f(x) = (x-2)^2(x+1)$ has a zero at $x=2$ (multiplicity 2, touches) and $x=-1$ (multiplicity 1, crosses).

Rational Functions and Their Graphs

Definition and Domain

A rational function is a ratio of two polynomials: $f(x) = \frac{p(x)}{q(x)}$. The domain is all real numbers except where $q(x) = 0$.

Vertical and Horizontal Asymptotes

• Vertical Asymptote (V.A.): Set $q(x) = 0$ and solve. If a factor cancels with the numerator, it creates a hole, not an asymptote.

• Horizontal Asymptote (H.A.): Compare degrees of numerator ($n$) and denominator ($m$):

  • If $n < m$, $y = 0$ is the H.A.

  • If $n = m$, $y = \frac{a_n}{b_m}$, where $a_n$ and $b_m$ are leading coefficients.

  • If $n > m$, there is no horizontal asymptote.

Slant (Oblique) Asymptote

• If the degree of the numerator is exactly one more than the denominator, perform long or synthetic division to find the slant asymptote: $f(x) = mx + b + \frac{\text{remainder}}{q(x)}$.

Graphing Rational Functions: Steps

1. Check for symmetry: $f(-x) = f(x)$ (y-axis), $f(-x) = -f(x)$ (origin).

2. Find x-intercepts: Set $p(x) = 0$.

3. Find y-intercept: Set $x = 0$.

4. Find vertical asymptotes: Set $q(x) = 0$.

5. Find horizontal/slant asymptotes as above.

6. Plot points between and beyond intercepts and asymptotes.

7. Sketch the graph using all information.

Exponential Functions

Definition and Properties

An exponential function has the form $f(x) = b^x$, where $b > 0$ and $b \neq 1$. The variable $x$ is in the exponent.

• Domain: All real numbers.

• Range: $(0, \infty)$ for $b > 1$.

Compound Interest Formulas

• n Compounding Periods per Year: $A = P\left(1 + \frac{r}{n}\right)^{nt}$

• Continuous Compounding: $A = Pe^{rt}$

Where $A$ is the amount, $P$ is the principal, $r$ is the annual interest rate, $n$ is the number of compounding periods per year, and $t$ is time in years.

Logarithmic Functions

Definition and Forms

A logarithmic function is the inverse of an exponential function. The logarithm base $b$ of $x$ is written $\log_b x$ and is defined as the exponent $y$ such that $b^y = x$.

• Logarithmic Form: $\log_b x = y$

• Exponential Form: $b^y = x$

• Domain: The argument $x$ must be positive ($x > 0$).

• Common Logarithm: $\log x$ (base 10)

• Natural Logarithm: $\ln x$ (base $e$)

Properties of Logarithms

• $\log_b 1 = 0$

• $\log_b b = 1$

• $\log_b b^x = x$

• $b^{\log_b x} = x$

Logarithm Laws

• Product Rule: $\log_b (MN) = \log_b M + \log_b N$

• Quotient Rule: $\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N$

• Power Rule: $\log_b M^p = p \log_b M$

Order of Operations for Logs

• Expanding: First apply quotient/product rules, then power rule.

• Condensing: First apply power rule, then product/quotient rules.

Change-of-Base Formula

• $\log_b M = \frac{\log_a M}{\log_a b}$ (works for any base $a$, including 10 or $e$)

Concept	Key Formula	When to Use
Quadratic Vertex	$\left( \frac{-b}{2a}, f\left(\frac{-b}{2a}\right) \right)$	Finding maximum/minimum of $ax^2 + bx + c$
Compound Interest	$A = P\left(1 + \frac{r}{n}\right)^{nt}$	Interest compounded $n$ times per year
Continuous Interest	$A = Pe^{rt}$	Interest compounded continuously
Change of Base	$\log_b M = \frac{\log_a M}{\log_a b}$	Evaluating logs with calculators