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.

• 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 $(4, -1)$.

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 the domain, and odd if $f(-x) = -f(x)$. If neither condition is met, the function is neither even nor odd.

• Example: $f(x) = 2x^4 - x^2 + 5$

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

• Conclusion: $f(x)$ is even.

Trigonometric Functions

Attributes of Sine and Cosine Functions

Sine and cosine functions are periodic and can be transformed using amplitude, period, 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 ($A$): $|a|$; half the distance between the maximum and minimum values.

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

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

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

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

Amplitude of Sinusoidal Functions

The amplitude of a sinusoidal function is half the distance between its maximum and minimum values, or $|a|$ in the general form.

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

• The midline is the horizontal line $y = d$ that cuts the graph vertically in half.

Examples: Amplitude and Range

Transformations affect the amplitude and range of sine and cosine functions.

• Example 1: $y = 2\sin(x)$

• Amplitude: $2$

• Range: $[-2, 2]$

• Example 2: $y = 2\cos(x)$

• Amplitude: $2$

• Range: $[-2, 2]$

Period and Frequency of Sine and Cosine Functions

The period is the distance required for the function to complete one full cycle. The frequency is the number of cycles per unit interval.

• For $y = a \sin(bx)$ or $y = a \cos(bx)$:

• Period: $T = \frac{2\pi}{b}$

• Frequency: $b$

Example: Compare $f(x) = \sin(x)$ and $g(x) = \sin(3x)$:

• Frequency of $f(x)$: $1$

• Period of $f(x)$: $2\pi$

• Frequency of $g(x)$: $3$

• Period of $g(x)$: $\frac{2\pi}{3}$

Summary Table: Attributes of Sine and Cosine Functions

Attribute	Symbol	Formula	Description
Amplitude	$A$	$|a|$	Half the distance between max and min values
Period	$T$	$\frac{2\pi}{b}$	Length of one cycle
Frequency	$b$	$b$	Number of cycles per $2\pi$ units
Vertical Shift	$d$	$d$	Moves graph up or down
Phase Shift	$c$	$c$	Moves graph left or right

Key Points for Graphing Sinusoidal Functions

• Identify amplitude, period, phase shift, and vertical shift from the equation.

• Plot the midline and mark maximum and minimum points using amplitude.

• Count increments to find x-intercepts, maxima, and minima.

Additional info: The notes also reference graphing techniques and transformations, which are essential for understanding how changes in parameters affect the shape and position of trigonometric graphs.