Unit 1 · Largest assessed unit

Polynomial and Rational Functions

The unit that establishes the language for the rest of the course. It introduces rate of change as the thing that distinguishes function families, then works through polynomial behavior and the more delicate business of rational functions, where the graph depends on structure that is invisible until you factor.

What a strong answer looks like

A strong Unit 1 answer names the structure before the computation: here is the degree, here is the multiplicity, here is what cancels, and therefore here is what the graph does.

Practice this unit Course overview

Topics in this unit

Average Rate of Change

Know

The average rate of change of a function over an interval is the slope of the line joining the two endpoints on its graph. It measures net change per unit of input across the interval, and says nothing about what happened in between.

Apply

Compute it as the change in output divided by the change in input, keeping the order of subtraction consistent in numerator and denominator. Read it from a table as readily as from a formula.

Watch out

Reversing the subtraction in only one of the two differences flips the sign. If a rate comes out negative for a function you can see increasing, check the order before checking the arithmetic.

Study move

Take one quadratic and compute its average rate of change over three different intervals. Notice that the value changes, which is exactly what makes it non-linear.

Successive Differences and Degree

Know

Over equally spaced inputs, a linear function has constant first differences, a quadratic has constant second differences, and in general a polynomial of degree n has constant nth differences. This is the fastest way to identify a polynomial from a table.

Apply

Given a table with equal input spacing, difference the outputs repeatedly until the differences settle. The number of rounds is the degree.

Watch out

The rule requires equally spaced inputs. Applying it to an irregular table produces a confident wrong answer.

Study move

Build a table for a cubic, difference it three times, and confirm the third differences are constant.

Zeros and Multiplicity

Know

A zero of even multiplicity touches the x-axis and turns back; a zero of odd multiplicity crosses it. Multiplicity also controls how flat the graph is as it meets the axis, with higher multiplicity flattening the approach.

Apply

Factor completely, read each zero with its exponent, and sketch the crossing behavior before plotting any points.

Watch out

Counting distinct zeros rather than zeros with multiplicity gives the wrong degree. A degree-4 polynomial with three distinct zeros must have one of them repeated.

Study move

Sketch y = (x - 1)²(x + 2) from the factors alone, then check it against a graph.

End Behavior

Know

End behavior is determined entirely by the leading term. An even degree sends both ends the same way; an odd degree sends them opposite ways; and the sign of the leading coefficient decides which way.

Apply

Identify degree and leading coefficient, then state both ends. Ignore every other term, however large its constant.

Watch out

A large constant term tempts students into thinking it matters at the extremes. It does not: growth rate, not size at zero, governs the ends.

Study move

For four polynomials mixing odd and even degree with positive and negative leading coefficients, state the end behavior without graphing.

Vertical Asymptotes and Holes

Know

A zero of the denominator produces a vertical asymptote if the factor survives cancellation, and a hole if it cancels against the numerator. The two look completely different on a graph and come from the same starting equation.

Apply

Factor numerator and denominator, cancel common factors, and classify each denominator zero by whether it cancelled. Record the hole location by evaluating the reduced function there.

Watch out

Declaring every denominator zero an asymptote without factoring is the defining Unit 1 error, and it is deliberately baited on the exam.

Study move

Work through a rational function whose numerator and denominator share exactly one factor, and state both the hole and the asymptote.

End Behavior of Rational Functions

Know

Comparing degrees settles the non-vertical asymptote. Denominator degree larger gives a horizontal asymptote at zero; equal degrees give the ratio of leading coefficients; numerator larger by exactly one gives a slant asymptote; larger by two or more gives neither.

Apply

Compare degrees first and only then divide, using polynomial division when a slant asymptote is indicated.

Watch out

Reading the ratio of the constant terms rather than the leading coefficients gives a plausible wrong horizontal asymptote.

Study move

Classify four rational functions by degree comparison alone before computing anything.

Transformations of Functions

Know

Changes inside the function act horizontally and in the opposite direction to their sign; changes outside act vertically and as written. A coefficient inside compresses horizontally by its reciprocal.

Apply

Decompose a transformed function into its steps and apply them to a known point, working the inside changes backwards.

Watch out

The horizontal factor is the most-missed piece. In f(3(x - 1)), the input is compressed by a factor of three and then shifted right by one, not shifted by three.

Study move

Take a single point on y = f(x) and track it through a transformation with a reflection, a horizontal compression, a shift, and a vertical stretch.

Emphasized in this unit

Connections and techniques that receive extra attention in this AP unit.

• Reading degree from a table of equally spaced values rather than from an equation
• Describing a polynomial graph completely from its factored form before plotting a point
• Treating a hole and a vertical asymptote as different consequences of the same starting equation
• Applying horizontal transformations in the reverse of their apparent direction

Varies by course

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

Synthetic division. Some courses drill it as the route to slant asymptotes; others use long division throughout.
Complex zeros. The Fundamental Theorem is sometimes introduced here and sometimes left to a later course.
Polynomial inequalities. Sign charts appear in some Algebra 2 courses and are revisited here only briefly.

Mastery checklist

• Compute an average rate of change from a table and from a graph, with the correct sign.
• State the degree of a polynomial from constant nth differences.
• Predict crossing or touching at every zero from its multiplicity.
• State end behavior from the leading term alone.
• Classify each denominator zero as a hole or a vertical asymptote after factoring.
• Determine a horizontal or slant asymptote by comparing degrees.
• Apply a composite transformation to a known point, inside changes first and reversed.

Check yourself

• Why does a large constant term have no effect on end behavior?
• What has to be true of a factor for it to produce a hole rather than an asymptote?
• How do you know a table describes a quadratic rather than an exponential?
• In f(3(x - 1)), what happens to the input, and in what order?

Modeling drill

A company profit is modelled by a cubic with zeros at three production levels, one of them repeated. Sketch the profit curve from the factored form alone, then say which production levels the model claims break even and which merely touch zero.

Average rate of changeSuccessive differencesMultiplicityEnd behaviorVertical asymptoteHoleSlant asymptoteTransformation

Unit 2: Exponential and Logarithmic Functions