Quadratic Functions

Domain, Range, and Vertex

Quadratic functions are polynomial functions of degree 2, typically written in the form $f(x) = a(x-h)^2 + k$. The graph of a quadratic function is a parabola.

• 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 highest or lowest point on the graph. 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 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 $f(-x)$: $f(-x) = 2(-x)^4 - (-x)^2 + 5 = 2x^4 - x^2 + 5 = f(x)$

• Therefore, $f(x)$ is an even function.

Trigonometric Functions

General Form and Transformations

Trigonometric functions such as sine and cosine can be transformed using the general forms:

• $y = a \sin(b(x - c)) + d$

• $y = a \cos(b(x - c)) + d$

Each parameter affects the graph in a specific way:

• Amplitude (a): The vertical stretch or compression. $\text{Amplitude} = |a|$

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

• Phase Shift (c): The horizontal shift. $x$ is replaced by $x - c$.

• Vertical Shift (d): The midline of the graph moves up or down by $d$ units.

Key Attributes of Sine and Cosine Functions

• Amplitude: Half the distance between the maximum and minimum values of the function.

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

• Period: The distance (along the x-axis) required for the function to complete one full cycle.

• Frequency: The number of cycles the function completes in a given interval, typically $2\pi$ for sine and cosine. Frequency is $b$ in $y = a\sin(bx)$ or $y = a\cos(bx)$.

• Phase Shift: The horizontal translation of the graph, determined by $c$ in $y = a\sin(b(x-c))$.

Formulas

• Amplitude: $|a|$

• Period: $\frac{2\pi}{b}$

• Frequency: $b$

• Phase Shift: $c$

• Vertical Shift: $d$

Examples: Amplitude and Range

• Example 1: $y = 2\sin(x)$ Amplitude: $2$ Range: $[-2, 2]$

• Example 2: $y = 2\cos(x)$ Amplitude: $2$ Range: $[-2, 2]$

Comparing Parent and Transformed Functions

• Parent Function: $y = \cos(x)$

• Transformed Function: $y = 3\cos(x)$

• The amplitude increases from $1$ to $3$, so the graph stretches vertically.

Period and Frequency Examples

• Example: $y = \sin(5x)$ Frequency: $5$ Period: $\frac{2\pi}{5}$

• Example: $y = \cos(\frac{x}{2})$ Frequency: $\frac{1}{2}$ Period: $2\pi \div \frac{1}{2} = 4\pi$

Summary Table: Attributes of Sine and Cosine Functions

Graphical Interpretation

• The amplitude is the height from the midline to a peak (maximum) or trough (minimum).

• The midline is the horizontal axis about which the function oscillates.

• The period is the horizontal length of one complete cycle.

• Vertical and horizontal shifts move the graph up/down or left/right, respectively.

Applications

• Trigonometric functions model periodic phenomena such as sound waves, tides, and seasonal temperatures.

• Understanding transformations allows for the analysis and prediction of real-world cycles.