← Precalculus syllabus

United StatesPrecalculus

Unit 1: Polynomial and Rational Functions

14 dot points across 14 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do two quantities change together, and how do we describe whether one increases or decreases as the other does?

Topic 1.1 Change in Tandem: describe how the output values of a function change as the input values change, using increasing or decreasing behavior, concavity, and the relationship shown in a graph, table or context.
A focused answer to AP Precalculus Topic 1.1, covering how output values change as input values change, increasing and decreasing behavior, concavity, and reading change in tandem from graphs, tables and contexts.
9 min answer →

How can the same polynomial or rational expression be rewritten so that each form reveals a different feature?

Topic 1.11 Equivalent Representations of Polynomial and Rational Expressions: rewrite polynomial and rational expressions in equivalent forms using factoring, the binomial theorem and polynomial long division to reveal zeros, asymptotes and end behavior.
A focused answer to AP Precalculus Topic 1.11, covering standard, factored and divided forms, the binomial theorem, polynomial long division, and how each equivalent form reveals zeros, asymptotes or end behavior.
10 min answer →

How do you build an explicit polynomial or rational function from a context or data, and use it to answer questions?

Topic 1.14 Function Model Construction and Application: construct a polynomial or rational function model from a context, restricted domain or data set, and apply it to make predictions and solve problems.
A focused answer to AP Precalculus Topic 1.14, covering how to build linear, quadratic, polynomial and rational models from context or data, restrict domains appropriately, and apply the model to predictions.
10 min answer →

How do you choose an appropriate function model for a situation, and what assumptions does that choice make?

Topic 1.13 Function Model Selection and Assumption Articulation: select an appropriate type of function to model a data set or context, and articulate the assumptions and limitations of the chosen model.
A focused answer to AP Precalculus Topic 1.13, covering how to choose linear, quadratic, polynomial or rational models from data and context, and how to state the assumptions and limitations of a model.
9 min answer →

How do the real and complex zeros of a polynomial, and their multiplicities, determine its factored form and graph?

Topic 1.5 Polynomial Functions and Complex Zeros: relate the real and non-real complex zeros of a polynomial to its factored form, degree and graph, including the effect of multiplicity.
A focused answer to AP Precalculus Topic 1.5, covering real and complex zeros, multiplicity and its effect on the graph, the conjugate pairs of complex zeros, and writing a polynomial in factored form.
10 min answer →

How do the degree and leading coefficient of a polynomial determine what its graph does at the far left and far right?

Topic 1.6 Polynomial Functions and End Behavior: determine the end behavior of a polynomial from its degree and leading term, and express that behavior using limit notation.
A focused answer to AP Precalculus Topic 1.6, covering how the degree and leading coefficient control the end behavior of a polynomial, the four end-behavior cases, and writing end behavior in limit notation.
9 min answer →

How do the degree, local extrema and points of inflection of a polynomial describe the way it changes?

Topic 1.4 Polynomial Functions and Rates of Change: relate the degree of a polynomial to its number of local extrema and points of inflection, and analyze where its rate of change is positive, negative or zero.
A focused answer to AP Precalculus Topic 1.4, covering local maxima and minima, points of inflection, the relationship between degree and the number of extrema, and where a polynomial increases or decreases.
9 min answer →

What distinguishes the rate-of-change behavior of linear functions from that of quadratic functions?

Topic 1.3 Rates of Change in Linear and Quadratic Functions: characterize linear functions by their constant rate of change and quadratic functions by their constant rate of change of the rate of change.
A focused answer to AP Precalculus Topic 1.3, covering the constant rate of change of linear functions, the constant second difference of quadratic functions, and how to tell the two apart from tables and contexts.
9 min answer →

How do we measure how fast a function changes, both over an interval and at a single point?

Topic 1.2 Rates of Change: compute and interpret the average rate of change of a function over an interval, and estimate the rate of change at a point.
A focused answer to AP Precalculus Topic 1.2, covering average rate of change over an interval, the rate of change at a point, and how to compute and interpret both from graphs, tables and formulas.
9 min answer →

What does the graph of a rational function do at its far ends, and when is there a horizontal or slant asymptote?

Topic 1.7 Rational Functions and End Behavior: determine the end behavior of a rational function by comparing the degrees of its numerator and denominator, identifying horizontal or slant asymptotes.
A focused answer to AP Precalculus Topic 1.7, covering how comparing numerator and denominator degrees gives horizontal asymptotes, slant asymptotes or unbounded end behavior, with limit notation.
9 min answer →

What creates a hole in the graph of a rational function, and how do you find its location?

Topic 1.10 Rational Functions and Holes: identify removable discontinuities (holes) of a rational function from factors common to numerator and denominator, and find the coordinates of each hole.
A focused answer to AP Precalculus Topic 1.10, covering how common factors create removable discontinuities (holes), how to find a hole's coordinates by cancelling and substituting, and how holes differ from asymptotes.
8 min answer →

Where does a rational function shoot off to infinity, and how do you describe that behavior with limits?

Topic 1.9 Rational Functions and Vertical Asymptotes: locate the vertical asymptotes of a rational function from the zeros of the denominator that do not cancel, and describe the behavior with one-sided limits.
A focused answer to AP Precalculus Topic 1.9, covering how denominator zeros that do not cancel give vertical asymptotes, how to do sign analysis for one-sided behavior, and limit notation.
9 min answer →

Where does a rational function equal zero, and how do the numerator's zeros relate to the graph?

Topic 1.8 Rational Functions and Zeros: determine the real zeros of a rational function from the zeros of its numerator, accounting for values excluded by the denominator.
A focused answer to AP Precalculus Topic 1.8, covering how the zeros of a rational function come from the numerator, why denominator zeros are excluded, and how multiplicity shapes the graph at each x-intercept.
8 min answer →

How do shifts, stretches and reflections change the equation and graph of a function?

Topic 1.12 Transformations of Functions: construct and analyze additive and multiplicative transformations (translations, dilations and reflections) of a function and their effect on the graph, domain and range.
A focused answer to AP Precalculus Topic 1.12, covering vertical and horizontal translations, dilations and reflections, how each changes the equation, and the effect on domain and range.
10 min answer →