Graphs, Functions, and Models

Distance and Midpoint

Understanding the distance and midpoint between two points is fundamental in coordinate geometry and is often used to analyze graphs and geometric figures.

• Distance Formula: The distance between two points and in the plane is given by:

• Midpoint Formula: The midpoint of the segment connecting and is:

• Example: For points (2, 3) and (6, 7):

  • Distance:

  • Midpoint:

Circles: Center, Radius, and Equation

The equation of a circle is derived from the distance formula and describes all points equidistant from a fixed center.

• Standard Form: The equation of a circle with center and radius is:

• Finding Center and Radius: Given an equation in standard form, the center is and the radius is .

• From Graph: Identify the center and a point on the circle to determine the radius.

• Example: has center and radius $4$.

Functions: Domain, Range, and Evaluation

Domain and Range

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

• Simple Functions: For , domain is all real numbers; range is .

• From Equations: Exclude values that make denominators zero or result in negative values under even roots.

• From Graphs: Estimate the lowest and highest x-values (domain) and y-values (range) shown on the graph.

• Example: has domain ; range .

Evaluating Functions

To evaluate a function, substitute the given input value into the function's formula.

• Example: If , then .

Graphing Functions and the Vertical Line Test

Graphing helps visualize the behavior of functions. The vertical line test determines if a graph represents a function: if any vertical line crosses the graph more than once, it is not a function.

• Example: The graph of passes the vertical line test; the graph of a circle does not.

Linear Functions and Equations of Lines

Linear Functions from Tables

Linear functions have a constant rate of change. Given a table of values, check if the difference in y-values divided by the difference in x-values is constant.

• Formula: , where is the slope and is the y-intercept.

• Example: If increases by 1 and increases by 2 each time, .

Average Rate of Change

The average rate of change of a function between and is the change in divided by the change in .

• Formula:

• Example: For , from to :

Finding Equations of Lines

To find the equation of a line, use the slope-intercept form or point-slope form, depending on the information given.

• Slope-Intercept Form:

• Point-Slope Form:

• Parallel Lines: Have the same slope.

• Perpendicular Lines: Slopes are negative reciprocals: .

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

Linear Word Problems and Finding Zeroes

Linear word problems often involve translating real-world situations into equations. The zero of a linear function is the x-value where .

• Finding Zero: Set and solve for .

• Example: ; .

More on Functions

Increasing, Decreasing, and Constant Intervals

From a graph, identify where the function is rising (increasing), falling (decreasing), or flat (constant).

• Increasing: As increases, increases.

• Decreasing: As increases, decreases.

• Constant: remains the same as increases.

Relative Maximum and Minimum

A relative maximum is a point where the function reaches a local highest value; a relative minimum is a local lowest value.

• Identified from Graph: Look for peaks (maxima) and valleys (minima).

Piecewise Defined Functions

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

• Example:

• To evaluate, determine which interval falls into and use the corresponding formula.

Operations with Functions

Functions can be added, subtracted, multiplied, or divided (except where the denominator is zero).

• Sum:

• Difference:

• Product:

• Quotient: ,

The Difference Quotient

The difference quotient is a formula that measures the average rate of change of a function and is foundational in calculus.

• Formula:

• Example: For :

Composition of Functions

Composition involves applying one function to the result of another: .

• Evaluating: Substitute into .

• Domain: The domain of consists of all in the domain of such that is in the domain of .

• Example: If and , then .

Transformations of Functions

Transformations change the position or shape of a function's graph. Common transformations include translations, reflections, stretches, and compressions.

• Vertical Shift: shifts up by units.

• Horizontal Shift: shifts right by units.

• Reflection: reflects over the x-axis; reflects over the y-axis.

• Vertical Stretch/Compression: stretches if , compresses if .

• Example: is shifted right 2 units and up 3 units.