Study Guide: Graphical Behavior, Maxima/Minima, Average Rate of Change, and Polynomial Functions

Graphical Behavior of Functions

Increasing and Decreasing Functions

Understanding whether a function is increasing or decreasing is fundamental in precalculus. These concepts describe how the output values of a function change as the input values change.

• Increasing Function: A function is increasing on an interval if, for any two values a and b in the interval with a < b, then $f(a) < f(b)$. In Words: As inputs increase, outputs increase.

• Decreasing Function: A function is decreasing on an interval if, for any two values a and b in the interval with a < b, then $f(a) > f(b)$. In Words: As inputs increase, outputs decrease.

Intervals of Increase and Decrease

Intervals are often written in interval notation to specify where a function is increasing or decreasing.

• Example: The function is increasing on $(-1, 1) \cup (3, \infty)$ and decreasing on $(-\infty, -1) \cup (1, 3)$.

Local and Absolute Extrema

Local Maximum and Minimum

Local extrema are points where a function reaches a highest or lowest value within a certain neighborhood.

• Local Maximum: A point where the y-value is greater than the immediately surrounding y-values.

• Local Minimum: A point where the y-value is less than the immediately surrounding y-values.

Absolute Maximum and Minimum

Absolute extrema are the greatest or smallest y-values for the entire function.

• Absolute Maximum: The greatest y-value for the entire function.

• Absolute Minimum: The smallest y-value for the entire function.

Finding Extrema from Graphs

To find local and absolute maxima and minima, examine the graph for peaks (maxima) and valleys (minima).

• Example: At $x = 1$, the maximum value is ... (context inferred: likely a y-value at x=1).

• At $x = -1$ or $x = 3$, there may be local minimum values.

Average Rate of Change

Definition and Formula

The average rate of change of a function between two points describes how the output changes per unit change in input. It is equivalent to the slope of the secant line connecting the two points.

• Formula: $m_{sec} = \frac{f(b) - f(a)}{b - a}$

• Notation: $\frac{\Delta y}{\Delta x}$

Examples of Average Rate of Change

• Example 1: For points (1, 3) and (3, 27): $\frac{27-3}{3-1} = \frac{24}{2} = 12$

• Example 2: For points (5, 75) and (1, 3): $\frac{75-3}{5-1} = \frac{72}{4} = 18$

Secant Line

A secant line is a straight line connecting two points on the graph of a function. Its slope represents the average rate of change between those points.

• Definition: The secant line through points $(a, f(a))$ and $(b, f(b))$ has slope $m_{sec}$.

Polynomial Functions and Their Degree

Identifying Polynomials

Polynomials are algebraic expressions consisting of terms with non-negative integer exponents.

• Terms: Each part of a polynomial separated by plus or minus signs.

• Coefficients: The numerical factors of each term.

• Degree: The highest exponent of the variable in the polynomial.

Polynomial Functions

A polynomial function is a function that can be written in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$, where $n$ is a non-negative integer and $a_n, ..., a_0$ are constants.

• Example: $f(x) = 5x^3 + \frac{1}{4}x^2 - 9$

End Behavior of Polynomial Functions

The end behavior of a polynomial function describes how the function behaves as $x$ approaches positive or negative infinity. It is determined by the leading term (the term with the highest degree).

• Constant: The function does not change as $x$ increases or decreases.

• Leading Coefficient (L.C.): The coefficient of the highest degree term.

Summary Table: Graphical Behavior of Functions

The following table summarizes the graphical behavior of functions:

Behavior	Description
Increasing	As input values increase, output values increase ($f(a) < f(b)$ for $a < b$)
Decreasing	As input values increase, output values decrease ($f(a) > f(b)$ for $a < b$)
Concave Up	The rate of change is increasing
Concave Down	The rate of change is decreasing

Practice and Assignments

• Practice identifying intervals of increase/decrease, local/absolute extrema, and calculating average rate of change.

• Assignment: pg 99 #87, 89. Quiz CED 2.1 to 2.3 upcoming.

Additional info:

• Some context and examples were inferred for completeness, especially regarding the calculation of average rate of change and secant lines.

• Graphs and interval notation were expanded for clarity.