Unit 3 · Core unit

Functions and Graphs

The major function families revisited at the depth calculus assumes. The emphasis falls on graph behavior, inverse relationships, and reading one function algebraically, graphically, numerically, and in context.

What a strong answer looks like

Unit 3 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 unit Course overview

Topics in this unit

Polynomial Zeros and Factors

Know

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.

Apply

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.

Watch out

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.

Study move

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

Rational Functions and Their Discontinuities

Know

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.

Apply

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.

Watch out

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

Study move

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

End Behavior and Asymptotes

Know

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.

Apply

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

Watch out

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

Study move

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

Exponential and Logarithmic Functions

Know

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.

Apply

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

Watch out

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.

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.

Inverses, Piecewise, and Multiple Representations

Know

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.

Apply

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.

Watch out

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.

Study move

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

Emphasized in this unit

Connections and techniques that receive extra attention in this honors sequence.

• 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

Varies by course

Related topics some schools attach to this unit and others leave out. Covered on request rather than assumed.

Conic sections. Some schools place conics alongside functions; this sequence develops them in Unit 5.
Systems of equations and inequalities. Some schools revisit these inside the functions unit; this sequence develops systems and matrix methods in Unit 4.
Sequences, series, and sigma notation. Placement varies widely; this sequence groups them with systems and matrices in Unit 4.

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

Unit 2: Polar, Complex Numbers, and Vectors

Unit 4: Systems, Matrices, and Sequences