Functions and Symmetry

Even and Odd Functions

Understanding whether a function is even, odd, or neither is fundamental in algebra. These classifications help recognize symmetries in graphs.

• Even Function: A function f(x) is even if f(-x) = f(x) for all x in its domain. Its graph is symmetric about 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 about the origin.

• Symmetry: Even functions show y-axis symmetry; odd functions show origin symmetry.

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

Piecewise Functions

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

• Evaluation: Substitute the input value into the appropriate piece.

• Graphing: Plot each piece over its specified interval.

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

Linear Equations and Graphs

Point-Slope and Slope-Intercept Forms

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

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

• Slope-Intercept Form: $y = mx + b$

• Example: If a line passes through (2, 3) with slope 4, its point-slope form is $y - 3 = 4(x - 2)$.

Parallel and Perpendicular Lines

Lines are parallel if they have the same slope, and perpendicular if their slopes are negative reciprocals.

• Parallel Lines: $m_1 = m_2$

• Perpendicular Lines: $m_1 \cdot m_2 = -1$

• Example: A line with slope 2 is perpendicular to a line with slope -1/2.

Transformations of Functions

Translations and Transformations

Transformations shift or change the shape of function graphs. Common transformations include translations, reflections, and stretches.

• Translation: Shifting the graph horizontally or vertically.

• Quadratic Function: $f(x) = a(x - h)^2 + k$ shifts the graph by h units horizontally and k units vertically.

• Square Root Function: $f(x) = \sqrt{x - h} + k$

• Absolute Value Function: $f(x) = |x - h| + k$

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

Distance and Circles

Distance Between Two Points

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

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

• Example: Between (1, 2) and (4, 6): $d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Equation of a Circle

The standard form of a circle's equation is:

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

• Center: $(h, k)$

• Radius: $r$

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

Graphing Circles

To graph a circle, plot its center and use the radius to draw the circle around it.

• Domain: $[h - r, h + r]$

• Range: $[k - r, k + r]$

Quadratic Functions

Graphing Quadratic Functions

Quadratic functions have the form $f(x) = ax^2 + bx + c$ or $f(x) = a(x - h)^2 + k$.

• Axis of Symmetry: $x = h$ or $x = -\frac{b}{2a}$

• Vertex: $(h, k)$ or $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$

• Domain: All real numbers

• Range: Depends on the direction of opening

• Example: $f(x) = (x - 1)^2 + 2$ has vertex (1, 2), axis of symmetry $x = 1$

Determining Quadratic Equations

Given a vertex and a point, you can determine the equation of a quadratic function.

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

• Find $a$: Substitute a known point and solve for $a$.

Polynomial Functions

Zeros of Polynomial Functions

Zeros are values of $x$ where $f(x) = 0$. They are found by factoring or using the quadratic formula.

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

• Example: $x^2 - 5x + 6 = 0$ has zeros $x = 2$ 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 $f(c) = 0$.

• Application: Used to show 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 $x = 3$.

Summary Table: Key Properties