# Precalculus

The Precalculus exam tests students' knowledge of specific properties of many types of functions.

## Overview

The Precalculus exam assesses student mastery of skills and concepts required for success in a first-semester calculus course. A large portion of the exam is devoted to testing a student’s understanding of functions and their properties. Many of the questions test a student’s knowledge of specific properties of the following types of functions: linear, quadratic, absolute value, square root, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise defined. Questions on the exam will present these types of functions symbolically, graphically, verbally or in tabular form. A solid understanding of these types of functions is at the core of all precalculus courses, and it is a prerequisite for enrolling in calculus and other college-level mathematics courses.

The exam contains approximately 48 questions, in two sections, to be answered in approximately 90 minutes.

- Section 1: 25 questions, approximately 50 minutes. The use of an online graphing calculator (non-CAS) is allowed for this section. Only some of the questions will require the use of the calculator.
- Section 2: 23 questions, approximately 40 minutes. No calculator is allowed for this section.

Although most of the questions on the exam are multiple choice, there are some questions that require students to enter a numerical answer.

### Graphing Calculator

A graphing calculator, the TI-84 Plus CE, is integrated into the exam software and available to students during Section 1 of the exam. Only some of the questions actually require the graphing calculator.

Visit ETS to learn more and to practice using the graphing calculator.. In order to answer some of the questions in the calculator section of the exam, students may be required to use the online graphing calculator in the following ways:

- Perform calculations (e.g., exponents, roots, trigonometric values, and logarithms)
- Graph functions and analyze the graphs
- Find zeros of functions
- Find points of intersection of graphs of functions
- Find minima/maxima of functions
- Find numerical solutions to equations
- Generate a table of values for a function

## Knowledge and Skills Required

Questions on the exam require candidates to demonstrate the following abilities:

- Recalling factual knowledge and/or performing routine mathematical manipulation
- Solving problems that demonstrate comprehension of mathematical ideas and/or concepts
- Solving nonroutine problems or problems that require insight, ingenuity, or higher mental processes

The subject matter of the Precalculus exam is drawn from the following topics. The percentages next to the topics indicate the approximate percentage of exam questions on that topic.

### Algebraic Expressions, Equations, and Inequalities (20%)

- Ability to perform operations on algebraic expressions
- Ability to solve equations and inequalities, including linear, quadratic, absolute value, polynomial, rational, radical, exponential, logarithmic, and trigonometric
- Ability to solve systems of equations, including linear and nonlinear

### Functions: Concept, Properties, and Operations (15%)

- Ability to demonstrate an understanding of the concept of a function, the general properties of functions (e.g., domain and range), function notation, and to perform symbolic operations with functions (e.g., evaluation and inverse functions)

### Representations of Functions: Symbolic, Graphical, and Tabular (30%)

- Ability to recognize and perform operations and transformations on functions presented symbolically, graphically, or in tabular form
- Ability to demonstrate an understanding of basic properties of functions and to recognize elementary functions (linear, quadratic, absolute value, square root, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions) that are presented symbolically, graphically, or in tabular form

### Analytic Geometry (10%)

- Ability to demonstrate an understanding of the analytic geometry of lines, circles, parabolas, ellipses, and hyperbolas

### Trigonometry and its Applications* (15%)

- Ability to demonstrate an understanding of the basic trigonometric functions and their inverses and to apply the basic trigonometric ratios and identities (in right triangles and on the unit circle)
- Ability to apply trigonometry in various problem-solving contexts

### Functions as Models (10%)

- Ability to interpret and construct functions as models and to translate ideas among symbolic, graphical, tabular, and verbal representations of functions

*Note that trigonometry permeates most of the major topics and accounts for more than 15% of the exam. The actual proportion of exam questions that requires knowledge of either right triangle trigonometry or the properties of the trigonometric functions is approximately 30%–40%.

## Score Information

### ACE Recommendation for Introductory Precalculus

Credit-granting Score | 50 |

Semester Hours | 3 |

**Note:** Each institution reserves the right to set its own credit-granting policy, which may differ from the American Council on Education (ACE). Contact your college to find out the score required for credit and the number of credit hours granted.