Functions and Their Properties

Even and Odd Functions; Symmetry

Understanding the classification of functions as even, odd, or neither is essential for analyzing their graphs and symmetries.

• Even Function: A function f(x) is even if f(-x) = f(x) for all x in its domain. Its graph is symmetric with respect to the y-axis.

• Odd Function: A function f(x) is odd if f(-x) = -f(x) for all x in its domain. Its graph is symmetric with respect to the origin.

• Neither: If a function does not satisfy either condition, it is neither even nor odd.

• Example: f(x) = x^2 is even; f(x) = x^3 is odd.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• Evaluation: To evaluate, determine which interval the input belongs to and use the corresponding expression.

• Graphing: Graph each piece on its specified interval, paying attention to open or closed endpoints.

• Example: f(x) = \begin{cases} x+2 & x < 0 \\ x^2 & x \geq 0 \end{cases}

Linear Equations and Their Graphs

Forms of Linear Equations

Linear equations can be written in several forms, each useful for different purposes.

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Standard Form: $Ax + By = C$

• Example: The line through (2, 3) with slope 4: $y - 3 = 4(x - 2)$

Parallel and Perpendicular Lines

Understanding the relationship between slopes helps in writing equations for parallel and perpendicular lines.

• Parallel Lines: Have the same slope, $m_1 = m_2$.

• Perpendicular Lines: Slopes are negative reciprocals, $m_1 \cdot m_2 = -1$.

• Example: A line perpendicular to $y = 2x + 1$ has slope $-\frac{1}{2}$.

Transformations of Functions

Translations and Transformations

Functions can be shifted, reflected, stretched, or compressed. These transformations apply to quadratic, square root, and absolute value functions.

• Vertical Translation: $f(x) + k$ shifts up/down by $k$ units.

• Horizontal Translation: $f(x - h)$ shifts right by $h$ units.

• Reflection: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis.

• Stretch/Compression: $af(x)$ stretches if $|a| > 1$, compresses if $0 < |a| < 1$.

• Example: $f(x) = (x-2)^2 + 3$ is $y = x^2$ shifted right 2 units and up 3 units.

Distance and Circles

Distance Between Two Points

The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula:

• $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Example: Between (1,2) and (4,6): $d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = 5$

Equations and Graphs of Circles

The standard form of a circle's equation and its graph are fundamental in coordinate geometry.

• Standard Form: $(x - h)^2 + (y - k)^2 = r^2$

• Center: $(h, k)$; Radius: $r$

• Graphing: Plot the center, then all points at distance $r$ from the center.

• Example: $(x - 3)^2 + (y + 2)^2 = 16$ has center (3, -2) and radius 4.

Quadratic and Polynomial Functions

Graphing Quadratic Functions

Quadratic functions have the form $f(x) = ax^2 + bx + c$ and their graphs are parabolas.

• Vertex Form: $f(x) = a(x-h)^2 + k$; vertex at $(h, k)$

• Axis of Symmetry: $x = h$

• Domain: All real numbers; Range: $y \geq k$ if $a > 0$, $y \leq k$ if $a < 0$

• Example: $f(x) = 2(x-1)^2 - 3$ has vertex (1, -3), opens upward.

Finding Equations of Quadratic Functions

Given points or features (vertex, intercepts), you can determine the equation of a quadratic function.

• Use vertex form or standard form and substitute known values to solve for coefficients.

• Example: Vertex at (2, 5), passes through (0, 1): $f(x) = a(x-2)^2 + 5$; plug in (0,1) to solve for $a$.

Zeros of Polynomial Functions

Zeros (roots) are the x-values where the function equals zero.

• Set $f(x) = 0$ and solve for $x$.

• For quadratics: $ax^2 + bx + c = 0$

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

• Example: $x^2 - 5x + 6 = 0$ has zeros at $x = 2$ and $x = 3$.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on [a, b] and $f(a)$ and $f(b)$ have opposite signs, then there is at least one $c$ in (a, b) such that $f(c) = 0$.

• Used to show the existence of roots in an interval.

• Example: If $f(1) = -2$ and $f(3) = 4$, then $f(x)$ has a zero between $x = 1$ and $x = 3$.

Domain, Range, and Key Features

Domain and Range

The domain is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Quadratic Functions: Domain is all real numbers; range depends on the vertex and direction of opening.

• Circles: Domain and range are limited by the radius and center.

• Example: For $(x-1)^2 + (y+2)^2 = 9$, domain is $[1-3, 1+3] = [-2, 4]$; range is $[-2-3, -2+3] = [-5, 1]$.

Identifying Key Features

Be able to identify the vertex, intercepts, axis of symmetry for quadratics, and the center and radius for circles from their equations.

• Vertex: From vertex form or by completing the square.

• Axis of Symmetry: $x = -\frac{b}{2a}$ for standard form $ax^2 + bx + c$.

• Intercepts: Set $x = 0$ for y-intercept; $y = 0$ for x-intercepts.

• Center and Radius: From standard form of a circle.