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← Precalculus Honors Study Guide

Unit 1 · Calculus foundation

Functions and Graphs

The families AP Calculus draws on first. They are revisited here at the depth calculus assumes, with the emphasis on graph behavior, composition and inverse relationships, and reading one function algebraically, graphically, numerically, and in context.

What a strong answer looks like

Unit 1 answers should account for the whole graph, not one feature. Domain, zeros, holes, asymptotes, and end behavior together describe a function; naming one of them is a partial answer.

Practice this unitCourse overview

Chapter introduction

Understand the idea before memorizing the rule

The families AP Calculus draws on first. They are revisited here at the depth calculus assumes, with the emphasis on graph behavior, composition and inverse relationships, and reading one function algebraically, graphically, numerically, and in context.

In Precalculus Honors, success means moving among words, symbols, tables, and graphs while preserving the meaning of each quantity. Use the lessons below as connected notes: explain the concept, carry out the procedure, test the result, and interpret it in context.

Detailed study notes

Core lessons, reasoning, and common errors

1

Lesson 1

Polynomial Zeros and Factors

Concept explanation

The Remainder Theorem says dividing p(x) by x - a leaves p(a), so a zero remainder means x - a is a factor. Zeros and factors are the same information stated two ways.

How to apply it

Test a candidate by evaluating rather than dividing. Once a zero is found, divide it out and factor what remains, which is usually a quadratic you can handle directly.

Common mistake

A zero and a factor are not the same object. If x = 3 is a zero, the factor is x - 3, and the sign trips up more students than the theorem does.

Guided study move

Given a cubic and one zero, produce the complete factorization and confirm it by substituting each remaining zero.

2

Lesson 2

Rational Functions and Their Discontinuities

Concept explanation

A rational function is defined by what its denominator forbids. A denominator factor that cancels completely creates a hole; if any copy of that factor remains in the denominator, the corresponding zero is a vertical asymptote.

How to apply it

Factor the numerator and denominator completely before concluding anything. Record the original domain restrictions, simplify common factors, and inspect the remaining denominator to distinguish holes from vertical asymptotes.

Common mistake

Reading vertical asymptotes straight off the original denominator without factoring is the standard error, and it converts every hole into an asymptote.

Guided study move

For each rational function, list domain, holes, vertical asymptotes, horizontal or slant asymptote, and intercepts before sketching anything.

3

Lesson 3

End Behavior and Asymptotes

Concept explanation

Comparing the degrees of numerator and denominator determines end behavior. A smaller numerator degree gives y = 0, equal degrees give the ratio of the leading coefficients, and a numerator exactly one degree larger gives a slant asymptote.

How to apply it

Use long division to find a slant asymptote. The quotient is the asymptote and the remainder term vanishes as x grows without bound.

Common mistake

A graph may cross a horizontal asymptote. The asymptote describes behavior far from the origin, not a boundary the curve cannot touch.

Guided study move

Given a rational function, predict its end behavior from the degrees, then confirm it by evaluating at a very large input.

4

Lesson 4

Exponential and Logarithmic Functions

Concept explanation

These two families are inverses of each other, which is why their graphs mirror across y = x and why each undoes the other in an equation. The logarithm is defined only for positive arguments, and that restriction drives most of the difficulty.

How to apply it

Combine logarithms into a single term, convert to exponential form, then solve. Check every solution against the domain the original equation required.

Common mistake

Extraneous solutions are not arithmetic errors; they are algebraically valid values the original equation cannot accept. Discarding them is part of the method, not an afterthought.

Guided study move

Before solving any logarithmic equation, write the domain restriction at the top of the work so the final check is already set up.

5

Lesson 5

Inverses, Piecewise, and Multiple Representations

Concept explanation

A function has an inverse function only when no two inputs share an output. Piecewise definitions let one rule describe different behavior on different intervals, and absolute value is the most common example.

How to apply it

Find an inverse by exchanging the variables and solving. Write an absolute value as a piecewise function by locating where the inside expression changes sign.

Common mistake

The branch point of an absolute value sits where the inside expression is zero, not at x = 0. For |x - 3| that is x = 3.

Guided study move

Given a function as a formula and another as a table, compare their growth and say which representation made the comparison easier.

Formula and visual reference

Connect the symbols to the picture

Average rate of change = (f(b) − f(a))/(b − a)

Composition: (f ∘ g)(x) = f(g(x))

Inverse reflection line: y = x

Concept graph
-5-4-3-2-1123456-5-4-3-2-1123456xy

A function and its inverse exchange coordinates and reflect across y = x.

Complete solution

Follow the reasoning, not just the arithmetic

Cover each next step and predict it before reading on. Then redo the example with one number changed to confirm that the method—not the answer—stuck.

Worked example

Find and verify an inverse

Problem

Find the inverse of f(x) = (3x − 5)/2.

  1. 1Write y = (3x − 5)/2 and exchange x and y.
  2. 2Solve x = (3y − 5)/2: 2x + 5 = 3y, so y = (2x + 5)/3.
  3. 3Composition in either order simplifies to x.

Answer and interpretation

f⁻¹(x) = (2x + 5)/3.

Chapter synthesis

Ideas to connect

  • Linear and quadratic families revisited before the harder function types
  • Composition of functions alongside inverses
  • Rational expressions and fractional exponents as prerequisite fluency
  • Real-world exponential and logarithmic modeling, not only equation solving

Modeling lab

Transfer the chapter to a new setting

A population is 1200 at time zero and 1800 after four years, growing exponentially. Write the model, predict the population at year ten, and state one assumption built into the model that would make the prediction unreliable at year fifty.

Mastery checklist

  • • Can find all real zeros of a polynomial given one of them.
  • • Can distinguish a hole from a vertical asymptote and justify the difference.
  • • Can determine end behavior from degrees and find a slant asymptote by division.
  • • Can solve exponential and logarithmic equations and reject extraneous solutions.
  • • Can find an inverse and explain why a domain restriction is sometimes needed.
  • • Can rewrite an absolute value function as a piecewise function.

Check yourself

  • • Why can a rational function have a zero denominator at a point without having an asymptote there?
  • • Why should domain restrictions be recorded before solving a logarithmic equation and checked again after candidate solutions are found?
  • • What does the horizontal line test tell you that the vertical line test does not?

Modeling drill

A population is 1200 at time zero and 1800 after four years, growing exponentially. Write the model, predict the population at year ten, and state one assumption built into the model that would make the prediction unreliable at year fifty.

Remainder Theoremzerofactorholevertical asymptotehorizontal asymptoteslant asymptoteend behaviorextraneous solutioninverse functionone-to-onepiecewise function

Next

Unit 2: Trigonometry