Precalculus Study Notes: Quadratic and Trigonometric Function Attributes

Quadratic Functions

Domain, Range, and Vertex

Quadratic functions are polynomial functions of degree two, typically written in the form $f(x) = a(x-h)^2 + k$. Understanding their domain, range, and vertex is essential for graphing and analyzing their behavior.

• Domain: The set of all possible input values (x-values) for the function. For any quadratic function, the domain is $(-\infty, \infty)$.

• Range: The set of all possible output values (y-values). For $f(x) = -(x-4)^2 - 1$, the parabola opens downward, so the range is $(-\infty, -1]$.

• Vertex: The point $(h, k)$ where the function reaches its maximum or minimum. For $f(x) = -(x-4)^2 - 1$, the vertex is at $(4, -1)$.

Example: For $f(x) = -(x-4)^2 - 1$:

• Domain: $(-\infty, \infty)$

• Range: $(-\infty, -1]$

• Vertex: $(4, -1)$

Even and Odd Functions

A function is even if $f(-x) = f(x)$ for all $x$ in its domain, and odd if $f(-x) = -f(x)$. Otherwise, it is neither.

• Even Function: Symmetric about the y-axis.

• Odd Function: Symmetric about the origin.

Example: For $f(x) = 2x^4 - x^2 + 5$:

• Calculate $f(-x) = 2(-x)^4 - (-x)^2 + 5 = 2x^4 - x^2 + 5 = f(x)$

• Therefore, $f(x)$ is even.

Trigonometric Functions

General Form and Transformations

Trigonometric functions such as sine and cosine can be transformed using amplitude, frequency, phase shift, and vertical shift. The general forms are:

• $y = a \sin(b(x - c)) + d$

• $y = a \cos(b(x - c)) + d$

Where:

• Amplitude ($A$): $|a|$ — the height from the midline to the maximum or minimum.

• Frequency ($b$ or $\omega$): Number of cycles per unit interval.

• Period ($T$): The length of one complete cycle, $T = \frac{2\pi}{b}$ or $T = \frac{2\pi}{\omega}$.

• Vertical Shift ($d$): Moves the graph up or down.

• Phase Shift ($c$ or $\phi$): Moves the graph left or right.

Attributes of Sine and Cosine Functions

The amplitude, period, and midline are key features of sinusoidal functions.

• Amplitude: Half the distance between the maximum and minimum values. For $y = a \sin(bx + c) + d$ or $y = a \cos(bx + c) + d$, amplitude is $|a|$.

• Midline: The horizontal line $y = d$ that bisects the graph vertically.

• Period: The distance required for the function to complete one full cycle. $T = \frac{2\pi}{b}$.

Example Table: Amplitude and Range

Function	Amplitude	Range
$y = \sin(x)$	1	[-1, 1]
$y = 2\sin(x)$	2	[-2, 2]
$y = 2\cos(x)$	2	[-2, 2]

Graphing Sinusoidal Functions

Transformations affect the graph of sine and cosine functions:

• Vertical Dilation: Changes amplitude.

• Horizontal Dilation: Changes period.

• Phase Shift: Moves the graph horizontally.

• Vertical Shift: Moves the graph up or down.

Example: Comparing $y = \cos(x)$ and $y = 3\cos(x)$:

• $y = \cos(x)$ has amplitude 1, range $[-1, 1]$.

• $y = 3\cos(x)$ has amplitude 3, range $[-3, 3]$.

Frequency and Period

The frequency determines how many cycles occur in a given interval, while the period is the length of one cycle.

• Frequency ($b$): Number of cycles per $2\pi$ units.

• Period ($T$): $T = \frac{2\pi}{b}$

Example Table: Frequency and Period

Function	Frequency ($b$)	Period ($T$)
$y = \sin(x)$	1	$2\pi$
$y = \sin(3x)$	3	$\frac{2\pi}{3}$
$y = \cos(5x)$	5	$\frac{2\pi}{5}$

Summary of Sinusoidal Function Transformations

• Amplitude ($|a|$): Controls the height of the wave.

• Period ($\frac{2\pi}{b}$): Controls the length of one cycle.

• Frequency ($b$): Number of cycles per $2\pi$ units.

• Phase Shift ($c$): Horizontal translation.

• Vertical Shift ($d$): Up/down translation.

Example: For $y = -2\sin(5x)$:

• Amplitude: $|-2| = 2$

• Period: $\frac{2\pi}{5}$

• Vertical shift: None

• Phase shift: None

Additional info: These notes cover foundational concepts in Precalculus, including quadratic and trigonometric function properties, transformations, and graphing techniques. Understanding these attributes is essential for analyzing and graphing functions in advanced mathematics.