Precalculus Study Notes: Quadratic and Trigonometric Function Attributes

Quadratic Functions

Domain, Range, and Vertex of a Quadratic Function

Quadratic functions are polynomial functions of degree two and are commonly written in the form $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

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

• 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 $(- , -1]$.

• Vertex: The point $(h, k)$ where the parabola 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: $(- , )$

• Range: $(- , -1]$

• Vertex: $(4, -1)$

Even, Odd, or Neither Functions

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

• Example: $f(x) = 2x^4 - x^2 + 5$

• Calculate $f(-x)$:

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

• Conclusion: $f(x)$ 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 ($b$): Number of cycles per $2\pi$ units.

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

• Phase Shift ($c$): Horizontal shift; the graph moves left or right by $c$ units.

• Vertical Shift ($d$): Moves the graph up or down by $d$ units; the midline is $y = d$.

Attributes of Sine and Cosine Functions

The amplitude of a sinusoidal function is half the distance between its maximum and minimum values. The midline is the horizontal axis that cuts the graph vertically in half.

• For $y = a \sin(bx + c) + d$ and $y = a \cos(bx + c) + d$, amplitude $= |a|$.

• The midline is $y = d$.

Example:

• For $y = 2\sin(x)$, amplitude $= 2$, range $= [-2, 2]$.

• For $y = 2\cos(x)$, amplitude $= 2$, range $= [-2, 2]$.

Period and Frequency

The period of a sine or cosine function is the distance required for the function to complete one full cycle. The frequency is the number of cycles per $2\pi$ units.

• Period: $T = \frac{2\pi}{b}$

• Frequency: $b$

Example:

• For $f(x) = \sin(x)$, $b = 1$, period $= 2\pi$.

• For $g(x) = \sin(3x)$, $b = 3$, period $= \frac{2\pi}{3}$.

Summary Table: Sine and Cosine Function Attributes

Function	Amplitude	Period	Range	Midline
$y = \sin(x)$	1	$2\pi$	[-1, 1]	$y = 0$
$y = 2\sin(x)$	2	$2\pi$	[-2, 2]	$y = 0$
$y = \sin(3x)$	1	$\frac{2\pi}{3}$	[-1, 1]	$y = 0$
$y = 2\cos(x)$	2	$2\pi$	[-2, 2]	$y = 0$

Graphical Transformations

Vertical dilations, phase shifts, and vertical shifts affect the shape and position of sine and cosine graphs:

• Vertical Dilation: Changes amplitude.

• Phase Shift: Moves the graph horizontally.

• Vertical Shift: Moves the graph up or down.

Example: Comparing $y = \cos(x)$ and $y = 3\cos(x)$, the amplitude increases from 1 to 3, so the graph stretches vertically.

Key Formulas

• General Sine/Cosine Function:

  • $y = a \sin(b(x - c)) + d$

  • $y = a \cos(b(x - c)) + d$

• Amplitude: $|a|$

• Period: $T = \frac{2\pi}{b}$

• Frequency: $b$

• Phase Shift: $c$

• Vertical Shift: $d$

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 higher mathematics.