Modeling and Equation Solving 1.1 Modeling and Equation Solving

Quick Review

What you’ll learn about Numeric Models Algebraic Models Graphic Models The Zero Factor Property Problem Solving Grapher Failure and Hidden Behavior A Word About Proof … and why Numerical, algebraic, and graphical models provide different methods to visualize, analyze, and understand data.

A mathematical model is a mathematical structure that approximates phenomena for the purpose of studying or predicting their behavior. A numeric model is a kind of mathematical model in which numbers (or data) are analyzed to gain insights into phenomena. An algebraic model uses formulas to relate variable quantities associated with the phenomena being studied.

Example Comparing Pizzas

Graphical Model A graphical model is a visible representation of a numerical model or an algebraic model that gives insight into the relationships between variable quantities. A product of real numbers is zero if and only if at least one of the factors in the product is zero.

Example Solving an Equation

Fundamental Connection

Example Seeing Grapher Failure

Functions and Their Properties 1.2 Functions and Their Properties

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What you’ll learn about Function Definition and Notation Domain and Range Continuity Increasing and Decreasing Functions Boundedness Local and Absolute Extrema Symmetry Asymptotes End Behavior … and why Functions and graphs form the basis for understanding The mathematics and applications you will see both in your work place and in coursework in college.

Function, Domain, and Range A function from a set D to a set R is a rule that assigns to every element in D a unique element in R. The set D of all input values is the domain of the function, and the set R of all output values is the range of the function.

Mapping

Example Seeing a Function Graphically

Vertical Line Test A graph (set of points (x,y)) in the xy-plane defines y as a function of x if and only if no vertical line intersects the graph in more than one point.

Example Finding the Domain of a Function

Example Finding the Range of a Function

Continuity

Example Identifying Points of Discontinuity Which of the following figures shows functions that are discontinuous at x = 2?

Increasing and Decreasing Functions

Increasing, Decreasing, and Constant Function on an Interval A function f is increasing on an interval if, for any two points in the interval, a positive change in x results in a positive change in f(x). A function f is decreasing on an interval if, for any two negative change in f(x). A function f is constant on an interval if, for any two points in the interval, a positive change in x results in a zero change in f(x).

Example Analyzing a Function for Increasing-Decreasing Behavior

Lower Bound, Upper Bound and Bounded A function f is bounded below of there is some number b that is less than or equal to every number in the range of f. Any such number b is called a lower bound of f. A function f is bounded above of there is some number B that is greater than or equal to every number in the range of f. Any such number B is called a upper bound of f. A function f is bounded if it is bounded both above and below.

Local and Absolute Extrema A local maximum of a function f is a value f(c) that is greater than or equal to all range values of f on some open interval containing c. If f(c) is greater than or equal to all range values of f, then f(c) is the maximum (or absolute maximum) value of f. A local minimum of a function f is a value f(c) that is less than or equal to all range values of f on some open interval containing c. If f(c) is less than or equal to all range values of f, then f(c) is the minimum (or absolute minimum) value of f. Local extrema are also called relative extrema.

Example Identifying Local Extrema

Symmetry with respect to the y-axis

Symmetry with respect to the x-axis

Symmetry with respect to the origin

Example Checking Functions for Symmetry

Horizontal and Vertical Asymptotes

Twelve Basic Functions 1.3 Twelve Basic Functions

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What you’ll learn about What Graphs Can Tell You Twelve Basic Functions Analyzing Functions Graphically … and why As you continue to study mathematics, you will find that the twelve basic functions presented here will come up again and again. By knowing their basic properties, you will recognize them when you see them.

The Identity Function

The Squaring Function

The Cubing Function

The Reciprocal Function

The Square Root Function

The Exponential Function

The Natural Logarithm Function

The Sine Function

The Cosine Function

The Absolute Value Function

The Greatest Integer Function

The Logistic Function

Example Looking for Domains One of the functions has domain the set of all reals except 0. Which function is it?

Example Analyzing a Function Graphically

Building Functions from Functions 1.4 Building Functions from Functions

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What you’ll learn about Combining Functions Algebraically Composition of Functions Relations and Implicitly Defined Functions … and why Most of the functions that you will encounter in calculus and in real life can be created by combining or modifying other functions.

Sum, Difference, Product, and Quotient

Example Defining New Functions Algebraically

Composition of Functions

Composition of Functions

Example Composing Functions

Example Decomposing Functions

Example Using Implicitly Defined Functions

Parametric Relations and Inverses 1.5 Parametric Relations and Inverses

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What you’ll learn about Defining Relations Parametrically Inverse Relations Inverse Functions … and why Some functions and graphs can best be defined parametrically, while some others can be best understood as inverses of functions we already know.

Example Defining a Function Parametrically

Example Defining a Function Parametrically

The inverse of a relation is a function if and only The ordered pair (a,b) is in a relation if and only if the pair (b,a) is in the inverse relation. The inverse of a relation is a function if and only if each horizontal line intersects the graph of the original relation in at most one point.

Example Finding an Inverse Function Algebraically

The Inverse Reflection Principle The points (a,b) and (b,a) in the coordinate plane are symmetric with respect to the line y=x. The points (a,b) and (b,a) are reflections of each other across the line y=x.

The Inverse Composition Rule

Example Verifying Inverse Functions

How to Find an Inverse Function Algebraically

Graphical Transformations 1.6 Graphical Transformations

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What you’ll learn about Transformations Vertical and Horizontal Translations Reflections Across Axes Vertical and Horizontal Stretches and Shrinks Combining Transformations … and why Studying transformations will help you to understand the relationships between graphs that have similarities but are not the same.

Translations Let c be a positive real number. Then the following transformations result in translations of the graph of y=f(x). Horizontal Translations y=f(x-c) a translation to the right by c units y=f(x+c) a translation to the left by c units Vertical Translations y=f(x)+c a translation up by c units y=f(x)-c a translation down by c units

Example Vertical Translations

Example Finding Equations for Translations

Reflections The following transformations result in reflections of the graph of y = f(x): Across the x-axis y = -f(x) Across the y-axis y = f(-x)

Graphing Absolute Value Compositions Given the graph of y = f(x), the graph y = |f(x)| can be obtained by reflecting the portion of the graph below the x-axis across the x-axis, leaving the portion above the x-axis unchanged; the graph of y = f(|x|) can be obtained by replacing the portion of the graph to the left of the y-axis by a reflection of the portion to the right of the y-axis across the y-axis, leaving the portion to the right of the y-axis unchanged. (The result will show even symmetry.)

Stretches and Shrinks

Example Finding Equations for Stretches and Shrinks

Example Combining Transformations in Order

Modeling with Functions 1.7 Modeling with Functions

Quick Review

What you’ll learn about Functions from Formulas Functions from Graphs Functions from Verbal Descriptions Functions from Data … and why Using a function to model a variable under observation in terms of another variable often allows one to make predictions in practical situations, such as predicting the future growth of a business based on data.

Example A Maximum Value Problem

Example A Maximum Value Problem

Example A Maximum Value Problem

Example A Maximum Value Problem

Example A Maximum Value Problem

Example Finding the Model and Solving Grain is leaking through a hole in a storage bin at a constant rate of 5 cubic inches per minute. The grain forms a cone-shaped pile on the ground below. As it grows, the height of the cone always remains equal to its radius. If the cone is one foot tall now, how tall will it be in one hour?

Constructing a Function from Data Given a set of data points of the form (x,y), to construct a formula that approximates y as a function of x: Make a scatter plot of the data points. The points do not need to pass the vertical line test. Determine from the shape of the plot whether the points seem to follow the graph of a familiar type of function (line, parabola, cubic, sine curve, etc.). Transform a basic function of that type to fit the points as closely as possible.

Functions

Functions (cont’d)

Chapter Test

Chapter Test