# Ch 12: Graphing Functions in Precalculus

### About This Chapter

## Graphing Functions in Precalculus - Chapter Summary and Learning Objectives

The lessons in this chapter were developed to help you fully comprehend the process of graphing functions in precalculus. Presented by top instructors, the lessons take an in-depth look at the mathematical operations associated with transformations, zeroes of functions, maximum and minimum values on a graph and more. Learn the definitions and get full explanations and examples of reflections in math, roots, x-intercepts, slopes and tangents. All of these concepts can help you complete homework assignments, or prepare you for quizzes and tests.

There are a variety of ways to use the resources in this chapter to boost your knowledge of graphing functions in precalculus. Watch fun video lessons with entertaining visual effects, or access written lessons that you can print to take notes and highlight important information. Submit questions to our experts if you need additional details about lesson topics. And gauge your comprehension of the lessons by taking self-assessment quizzes and a chapter exam. In no time, you will understand:

- How to graph basic functions and shift graphs on a plane
- The meaning of reflections, zeroes, roots and x-intercepts in math
- Steps involved in finding zeroes of functions
- How to identify where a function is linear, decreasing or increasing, positive or negative
- The process of determining maximum and minimum values of a graph and calculating slopes and tangents on a graph
- Horizontal and vertical asymptotes and the relationship between inverse and implicit functions

Video | Objective |
---|---|

Graphing Basic Functions | Explain the process of graphing functions and data. |

Transformations: How to Shift Graphs on a Plane | Discover how to shift graphs on the coordinate plane. |

Reflections in Math: Definition & Overview | Define a reflection, and explain its basics and how to recognize and create one. |

Zeroes, Roots & X-Intercepts: Definitions & Properties | Understand the differences between zeroes, roots and x-intercepts. |

Finding Zeroes of Functions | Review the process of finding zeroes of functions. |

Identify Where a Function is Linear, Increasing or Decreasing, Positive or Negative | Study the process of identifying traits of functions, including linear and nonlinear, increasing and decreasing and positive and negative. |

How to Determine Maximum and Minimum Values of a Graph | Examine through graphs how to determine maximum and minimum points and values. |

Slopes and Tangents on a Graph | Provide the definition and calculate tangents and slopes on a graph. |

Horizontal and Vertical Asymptotes | Discuss how to identify horizontal and vertical asymptotes. |

Implicit Functions | Describe the connection between implicit and inverse functions. |

### 1. Graphing Basic Functions

Graphs are just like maps - when you know the language! Review how locations have x and y coordinates similar to latitude and longitude, and how to plot points in the Cartesian plane.

### 2. Transformations: How to Shift Graphs on a Plane

What is a transformation? Well, it's something that transforms one function into another! To see what I mean and how that looks, check out this lesson!

### 3. Reflections in Math: Definition & Overview

A reflection of an object is a 'flip' of that object across a line. This lesson will describe the basics of reflection, how to recognize one and how to create one.

### 4. Zeroes, Roots & X-Intercepts: Definitions & Properties

We will learn the definition of zeros, roots, and x-intercepts, and we will see that these are all the same concept. We will look at algebraic and geometric properties of these concepts and how to use them to analyze functions.

### 5. Finding Zeroes of Functions

The zero of a function is the point (x,y) on which the graph of the function intersects with the x-axis. The y value of these points will always be equal to zero. There can be 0, 1, or more than one zero for a function.

### 6. Identify Where a Function is Linear, Increasing or Decreasing, Positive or Negative

Functions do all kinds of fun things. In this lesson, learn how to identify traits of functions such as linear or nonlinear, increasing or decreasing and positive or negative.

### 7. How to Determine Maximum and Minimum Values of a Graph

What is the highest point on a roller coaster? Most roller coasters have a lot of peaks, but only one is really the highest. In this lesson, learn the difference between the little bumps and the mother of all peaks on your favorite ride.

### 8. Slopes and Tangents on a Graph

Hit the slopes and learn how the steepness of a line is calculated. Calculate the slopes between points and draw the tangents of curves on graphs in this lesson.

### 9. Horizontal and Vertical Asymptotes

No matter how hard you try to get to them, asymptotes remain out of reach. Learn about these invisible lines on graphs that show you places your equations just can't go.

### 10. Implicit Functions

Sometimes inputting a variable into a function 'black box' doesn't yield a simple output. Find out what happens when you can't isolate the dependent variable on one side of the equal sign.

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### Other Chapters

Other chapters within the Honors Precalculus Textbook course

- Working with Linear Equations
- Working With Inequalities
- Absolute Value Equations
- Working with Complex Numbers
- Systems of Linear Equations
- Mathematical Modeling for Precalculus
- Introduction to Quadratics
- Working with Quadratic Functions
- Geometry Basics for Precalculus
- Types of Functions in Precalculus
- Understanding Function Operations
- Graph Symmetry
- Rate of Change
- Polynomial Function Basics in Precalculus
- Higher-Degree Polynomial Functions
- Rational Functions & Difference Quotients
- Rational Expressions & Function Graphs in Precalculus
- Exponential Functions & Logarithms in Precalculus
- Using Trigonometric Functions
- Trigonometric Graphs
- Solving Trigonometric Equations
- Trigonometric Identities
- Trigonometric Applications
- Graphing Piecewise Functions in Precalculus
- Vectors in Precalculus
- Matrices and Determinants in Algebra
- Understanding Arithmetic Sequences & Series
- Understanding Geometric Sequences & Series
- Analytic Geometry and Conic Sections
- Polar Coordinates and Parameterizations
- Working with Sequences & Series
- Sets & Probability
- Continuity
- Limits in Precalculus