Precalculus Study Guide: Quadratic, Polynomial, Exponential, and Logarithmic Functions

Quadratic Functions

Standard, Vertex, and Factored Forms

Quadratic functions are polynomial functions of degree 2 and are fundamental in precalculus. They can be expressed in several forms, each useful for different purposes.

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

• Vertex Form: $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola

• Factored Form: $f(x) = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the roots

Quadratic functions graph as parabolas. The sign of $a$ determines if the parabola opens upward ($a > 0$) or downward ($a < 0$).

• Vertex: The point $(h, k)$ is the maximum or minimum of the function.

• Axis of Symmetry: $x = h$ in vertex form, or $x = -\frac{b}{2a}$ in standard form.

• Roots/Zeros: Solutions to $f(x) = 0$ found by factoring, completing the square, or using the quadratic formula:

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

• Concavity: If $a > 0$, the parabola is concave up; if $a < 0$, concave down.

Example: For $h(x) = -0.2d^2 + 4d + 3$, find the vertex, axis of symmetry, and interpret the physical meaning (e.g., maximum height of a ball).

Polynomial Functions

General Properties and Graphs

Polynomial functions are sums of terms of the form $a_nx^n$, where $n$ is a non-negative integer. Their degree determines the number of roots and the end behavior.

• Degree: The highest power of $x$ in the polynomial.

• End Behavior: Determined by the leading term $a_nx^n$.

• Roots: The values of $x$ where the polynomial equals zero.

• Factored Form: Useful for finding roots and graphing.

Example: $p(x) = (x - 1)(x + 5)(x - 4)$ is a cubic polynomial with roots at $x = 1, -5, 4$.

Exponential Functions

Growth and Decay

Exponential functions model rapid growth or decay and are of the form $f(x) = ab^{x}$ or $f(x) = ae^{kx}$.

• Growth: If $b > 1$ or $k > 0$, the function increases as $x$ increases.

• Decay: If $0 < b < 1$ or $k < 0$, the function decreases as $x$ increases.

• Growth Factor: For $f(x) = ab^{x}$, $b$ is the growth factor.

• Continuous Growth Rate: For $f(x) = ae^{kx}$, $k$ is the continuous growth rate.

Example: Find the growth factor for a rate of change of 5%: $b = 1.05$.

Applications: Population growth, compound interest, radioactive decay.

Compound Interest

Compound interest is calculated using exponential functions. The formula for compound interest is:

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

• $A$ = final amount

• $P$ = principal (initial amount)

• $r$ = annual interest rate (decimal)

• $n$ = number of compounding periods per year

• $t$ = number of years

Effective Annual Yield: The actual interest earned in a year, accounting for compounding.

Logarithmic Functions

Definitions and Properties

Logarithms are the inverses of exponential functions. The logarithm base $b$ of $x$ is the exponent to which $b$ must be raised to get $x$.

$y = \log_b x \iff b^y = x$

• Common Logarithm: Base 10, $\log x$

• Natural Logarithm: Base $e$, $\ln x$

• Properties:

  • $\log_b(xy) = \log_b x + \log_b y$

  • $\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$

  • $\log_b(x^k) = k \log_b x$

Example: Rewrite $\log 8,000 = x$ as $10^x = 8,000$.

Graphing Exponential and Logarithmic Functions

Matching Equations to Graphs

Exponential functions have characteristic graphs:

• Growth: $y = ab^x$ with $b > 1$ rises rapidly.

• Decay: $y = ab^x$ with $0 < b < 1$ falls rapidly.

• Horizontal Asymptote: $y = 0$ for $y = ab^x$ as $x \to -\infty$ (if $a > 0$).

Logarithmic functions $y = \log_b x$ have a vertical asymptote at $x = 0$ and increase slowly for large $x$.

Applications and Problem Solving

Population Growth and Decay

Population models use exponential functions to describe growth or decline over time.

• General Model: $P(t) = P_0e^{kt}$

• Continuous Growth Rate: $k$ is positive for growth, negative for decay.

Example: A town starts with 10,000 residents and grows at a continuous rate of 4% per year. Formula: $P(t) = 10,000e^{0.04t}$.

Radioactive Decay and Half-Life

Radioactive decay is modeled by exponential functions with negative growth rates.

• Half-Life Formula: $N(t) = N_0e^{-kt}$, where $k = \frac{\ln 2}{\text{half-life}}$

Example: Determine the half-life of a substance decaying at a continuous rate of 7% per day.

Tables: Comparison of Exponential Growth Factors

Growth Rate (%)	Growth Factor (b)
5	1.05
10	1.10
4	1.04
40	1.40
Additional info: Table can be extended for other rates as needed.

Key Formulas and Conversions

• Exponential to Logarithmic: $b^y = x \iff y = \log_b x$

• Continuous Growth: $Q = Q_0e^{kt}$

• Discrete Growth: $Q = Q_0b^t$

• Logarithm Properties: $\log_b(xy) = \log_b x + \log_b y$

• Half-Life: $t_{1/2} = \frac{\ln 2}{k}$

Practice Problems Overview

• Convert between exponential and logarithmic forms.

• Find growth factors and rates from given data.

• Match functions to their graphs.

• Apply formulas to population and investment scenarios.

• Order numbers involving exponents and logarithms.

Additional info: These notes synthesize exam review and practice questions covering quadratic, polynomial, exponential, and logarithmic functions, as well as their applications to growth, decay, and financial mathematics. All content is relevant to Precalculus topics listed in the provided chapter titles.