Graphs of Equations

Even and Odd Functions; Symmetry

Understanding the properties of even and odd functions is essential for analyzing graphs and their 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.

• Symmetry: Recognizing symmetry helps in sketching and analyzing graphs efficiently.

• 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 and closed endpoints.

• Example: f(x) = \begin{cases} x^2 & x < 0 \\ x+1 & x \geq 0 \end{cases}

Equations & Inequalities

Equations of Lines

Lines can be represented in several forms, each useful for different purposes.

• Point-Slope Form:

• Slope-Intercept Form:

• Graphing: Use the slope and y-intercept to plot the line quickly.

• Example: A line through (2, 3) with slope 4:

Parallel and Perpendicular Lines

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

• Parallel Lines: Have the same slope, .

• Perpendicular Lines: Slopes are negative reciprocals, .

• Example: If a line has slope 2, a perpendicular line has slope .

Functions

Transformations of Functions

Transformations shift or change the shape of basic functions, such as quadratic, square root, and absolute value functions.

• Translations: Shift the graph horizontally or vertically.

• Reflections: Flip the graph over the x- or y-axis.

• Stretching/Compressing: Change the steepness or width of the graph.

• Example: is a translation of right by 2 units and up by 3 units.

Absolute Value and Square Root Functions

Transformations can also be applied to the absolute value and square root functions.

• Absolute Value: ; vertex at (0,0), V-shaped graph.

• Square Root: ; starts at (0,0), increases slowly.

• Example: shifts right by 1 and up by 2.

Graphs of Equations

Distance Between Two Points

The distance formula calculates the length between two points in the coordinate plane.

• Formula:

• Example: Between (1,2) and (4,6):

Circles: Standard Form and Graphing

The standard form of a circle's equation reveals its center and radius.

• Standard Form:

• Center: (h, k)

• Radius:

• Graphing: Plot the center, then use the radius to draw the circle.

• Example: has center (3, -2) and radius 4.

Polynomial Functions

Quadratic Functions: Graphs and Properties

Quadratic functions have the form and their graphs are parabolas.

• Vertex Form:

• Axis of Symmetry:

• Domain: All real numbers

• Range: if ; if

• Vertex: (h, k)

• Intercepts: Find by setting (y-intercept) and (x-intercepts/zeros)

• Example: has vertex (1, -3), opens upward.

Finding Equations of Quadratic Functions

Given points or properties, you can determine the equation of a quadratic function.

• Given Vertex and a Point: Use vertex form and substitute to solve for a.

• Given Three Points: Set up a system of equations to solve for a, b, and c in standard form.

Zeros of Polynomial Functions

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

• Finding Zeros: Set and solve for x.

• Example: ; zeros at and .

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental property of continuous functions.

• Statement: If is continuous on and is between and , then there exists in such that .

• Application: Used to show that a function has a root in an interval.

• Example: If and , then for some in (1, 3).

Domain and Range

Understanding domain and range is crucial for analyzing functions and their graphs.

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

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

• Example: For , domain: ; range: .