Graphs, Functions, and Models

Distance and Midpoint

Understanding the distance and midpoint between two points is fundamental in coordinate geometry. These concepts are often used to analyze geometric figures and solve real-world problems.

• 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: Find the distance and midpoint between and .

  • Distance:

  • Midpoint:

Circles: Center, Radius, and Equation

The equation of a circle is derived from the distance formula, representing 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 .

• Example: For , the center is and the radius is $4$.

Equations from Graphs

Given a graph, you can determine the equation of a circle or line by identifying key features such as the center, radius, or intercepts.

• Process: Identify the center and a point on the circle to find the radius, then write the equation in standard form.

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 , the domain is all real numbers, and the range is .

• From Graphs: Estimate the domain and range by observing the extent of the graph along the x- and y-axes.

• From Equations: Exclude values that make the denominator zero or result in an even root of a negative number.

• Example: For , the domain is .

Evaluating Functions

To evaluate a function, substitute the given value for the variable and simplify.

• 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

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: Slope

• Example: If increases by 1 and increases by 3 each time, the function is linear with slope 3.

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 , .

Equations of Lines

Lines can be described by their slope and a point or by their intercepts.

• Slope-Intercept Form:

• Point-Slope Form:

• Parallel Lines: Have the same slope.

• Perpendicular Lines: Slopes are negative reciprocals:

• Example: Find the equation of a line through with slope :

Linear Word Problems and Zeroes

Linear equations are used to model real-world situations. The zero of a linear function is the x-value where .

• Example: If , the zero is .

More on Functions

Increasing, Decreasing, and Constant Intervals

A function is increasing where its graph rises as you move left to right, decreasing where it falls, and constant where it remains flat.

• Identifying from Graph: Look for intervals where the y-values increase, decrease, or stay the same as x increases.

Relative Maximum and Minimum

A relative maximum is a point where the function reaches a peak locally; a relative minimum is a local trough.

• Identifying 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:

• Graphing: Plot each piece on its respective interval.

Operations on Functions

Functions can be added, subtracted, multiplied, or divided to create new functions.

• 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.