Precalculus

About this book

David H. Collingwood, K. David Prince, and Matthew M. Conroy develop precalculus through modeling and multistep problem solving. The linked University of Washington PDF is dated September 2, 2011, despite the imported catalog year. Its twenty chapters connect words, diagrams, coordinates, functions, and symbolic calculations before developing circular and trigonometric models. This guide follows that PDF and treats complete reasoning, units, and domain restrictions as part of an answer.

Chapter and section guide

1. Warming Up

Units, rates, and accumulated change introduce the modeling process. A formula becomes useful when its quantities are defined and its assumptions fit the situation. The relationship between rate, time, and total change applies directly under appropriate constant-rate conditions; it should not be extended without thought to a changing rate. The chapter encourages translating a verbal problem into a picture and then into symbols. Checking units can reveal an error before the final number is calculated.

Study application: Create an original constant-rate problem and solve it with units at every step. Then change the rate during the interval and explain why the original one-step calculation needs revision.

2. Imposing Coordinates

Coordinates connect a physical or geometric situation with numerical pairs. Choosing an origin, axis directions, units, and scale is part of building the model. A point’s coordinates depend on that choice, while the underlying object or event does not. Distance calculations become meaningful only when the coordinates represent compatible quantities. Students should label the diagram before manipulating equations and explain why the chosen system makes a relationship easier to express.

Study application: Draw the same arrangement with two different origins. Compute corresponding coordinates and show that a physical distance remains the same even though the coordinate values change.

3. Three Simple Curves

Horizontal and vertical lines and circles provide a small vocabulary for translating geometry into equations. Intersections represent conditions satisfied simultaneously. A diagram can suggest how many solutions are plausible, while algebra determines candidates to check. Students should distinguish the equation of a full curve from the portion relevant to a physical question. A circle’s center and radius should be interpreted geometrically rather than inferred from the appearance of an unlabeled sketch.

Study application: Construct a line and circle with two intersections, solve for the points, and verify both equations. Explain how a contextual restriction could make only one point relevant.

4. Linear Modeling

Linear relationships describe constant change and connect slope, intercept, equations, and graphs. The chapter applies these ideas to situations such as earnings and uniform motion. A slope is a rate with units, and an intercept may or may not have a meaningful interpretation within the application’s domain. Parallel and perpendicular relationships add geometric constraints. Students should identify what information determines a model and avoid assuming a straight line remains valid indefinitely.

Study application: Build two cost models with different fixed charges and rates. Find their intersection and interpret which is lower on either side, stating the meaningful input range.

5. Functions and Graphs

A function assigns one output to each input in its domain. Tables, equations, plots, and verbal descriptions emphasize different features of that relationship. The vertical-line test concerns uniqueness of output and should be connected with the definition. Applied constraints can restrict a mathematically available domain. Profit analysis illustrates why evaluating a function differs from solving for an input that achieves a target. Students should state variables and units before reading a graph.

Study application: Give an example of a relation that fails to be a function of one chosen variable but becomes a function after an appropriate restriction. Explain the restriction explicitly.

6. Graphical Analysis

The graph is used to read domain, range, intercepts, increasing and decreasing behavior, and values. Circles, semicircles, and multipart functions show why a familiar shape may need a restricted formula. A point on a graph connects an input with an output; it is not merely a visual marker. Students should distinguish exact information given by a formula from estimates read from a scale. Piecewise descriptions require attention to boundaries and included endpoints.

Study application: Write a two-part function and test its value at the joining point. Explain whether the two formulas agree there and how endpoint notation affects the definition.

7. Quadratic Modeling

Quadratic functions describe relationships with constant second differences and characteristic curvature. Vertex form makes shifts, reflection, scale, and an extremum visible. Completing the square connects this form with an expanded expression. In applications, a mathematically calculated vertex may fall outside the permitted input interval. Students should identify what data or conditions determine a parabola and check both the algebra and the contextual interpretation.

Study application: Construct a quadratic with a specified maximum and one additional point. Solve for its parameters and explain whether the maximum is reachable in a chosen application domain.

8. Composition

Composition describes one process feeding into another. The output of the inner function must be a permitted input of the outer function. The order of operations matters, and composing two functions in reverse order generally gives a different relationship. Students should track units and domains rather than perform symbolic substitution alone. A meaningful composite model explains what the intermediate quantity represents and why both stages are appropriate.

Study application: Use one function to convert time to distance and another to convert distance to cost. Interpret their composition and explain why reversing them would not describe the same process.

9. Inverse Functions

An inverse reverses an input-output relationship when the original function is one-to-one on the relevant domain. The inverse is not a reciprocal. Graphical reflection and symbolic solution offer complementary views, while domain and range exchange roles. A function that is not globally one-to-one can sometimes be restricted so an inverse exists. Students should connect that restriction with the real question rather than select a branch without explanation.

Study application: Find an inverse for a restricted quadratic and verify both compositions on their valid domains. Explain why removing the restriction would prevent a single-valued inverse.

10. Exponential Functions

Exponential change involves a constant multiplicative factor over equal input steps. The initial amount and base determine different features of the model. Growth and decay can be distinguished by the factor, while exponent rules support calculation. Students should compare exponential and linear patterns using ratios and differences rather than only the curve’s appearance. A zero or negative input may be mathematically valid while requiring a contextual interpretation.

Study application: Create equal-step data for one linear and one exponential model. Show how differences and ratios distinguish them, and interpret the parameters with units.

11. Exponential Modeling

Compound interest and the number e provide settings for exponential models with different time conventions. A nominal rate, compounding frequency, and elapsed time must be used consistently. The chapter’s broader lesson is that similar verbal statements can define different mathematical processes. Students should separate a model’s formula from assumptions about constant rates and unchanged conditions. A prediction should state those conditions rather than imply that a mathematical growth law guarantees an actual outcome.

Study application: Compare two fictional compounding conventions using the same stated principal and nominal rate. Explain the source of the difference and avoid treating the example as an investment forecast.

12. Logarithmic Functions

Logarithms invert exponential relationships and allow multiplicative patterns to be expressed through additive comparisons. Their domain and base matter. Applications such as sound level show that a logarithmic scale is not an ordinary linear measure of the underlying quantity. Students should translate a logarithmic equation into its equivalent exponential statement and verify candidate values. Rules for products and powers should not be misapplied to sums.

Study application: Solve an original exponential equation using a logarithm, then substitute the result back. Explain why the logarithm of a sum cannot generally be split into a sum of logarithms.

13. Construction Tools

Reflection, shifting, and dilation generate new graphs from known ones. Changes inside and outside a function affect inputs and outputs differently, and the order of transformations can matter. The chapter encourages interpreting transformations as operations on points, not memorized verbal rules alone. Students should identify a reference graph and track a few anchor points through the construction. This also provides a way to check a symbolic expression against the intended graph.

Study application: Apply a shift and a reflection in two orders to a simple function. Track an anchor point and explain why the resulting graphs can differ.

14. Rational Functions

A ratio of expressions introduces restrictions where the denominator vanishes and behavior that can include asymptotes. Linear-to-linear models provide applications in which a quantity approaches a limiting value. Cancellation can simplify an expression without restoring a point excluded by the original formula. Students should identify domain restrictions before solving and distinguish a hole from an asymptote. Graphical features need to be checked against the original model.

Study application: Build a rational expression with a removable factor and explain the missing point after simplification. Contrast it with a genuine vertical asymptote in another example.

15. Angle Measurement

Degrees and radians measure the same rotation through different conventions. Radians connect an angle with arc length and radius, making them especially useful in later formulas. Sector area and navigation applications connect geometry with physical scales. Students should distinguish an angle from the length of the associated arc and state the unit used. A formula expecting radians cannot be applied unchanged to a degree value.

Study application: Choose a radius and angle, calculate arc length and sector area using radians, and verify that a full rotation gives the familiar circumference and area.

16. Circular Motion

Circular motion can be described by angular change and by linear travel along a path. Radius connects angular speed with tangential speed, while period and frequency describe timing in different ways. Belt and wheel examples make the relationship concrete. Students should identify which motion is shared in a coupled system and which depends on size. Equal angular speeds do not imply equal tangential speeds at different radii.

Study application: Compare two points on a rotating wheel at different radii. State their angular speeds, tangential speeds, and periods, explaining which quantities agree.

17. Circular Functions

Right-triangle ratios connect with coordinates on a circle, extending familiar trigonometry beyond acute angles. Reference angles, signs, and radius allow the same definitions to describe a full rotation. Students should relate each ratio to a diagram and explain why changing the quadrant changes signs. This geometric foundation is more transferable than a list of isolated numerical values. Undefined ratios should be recognized from their denominators and geometry.

Study application: Place an angle in a nonfirst quadrant and determine the signs of its circular functions. Explain the result from coordinates rather than recalling a sign mnemonic alone.

18. Trigonometric Functions

The chapter develops identities, periodicity, and graphs from the circular definitions. Sine, cosine, and tangent have different ranges and points of definition. An identity holds throughout a shared valid domain, while an equation may be true only at selected inputs. Scaling the argument changes the period and requires attention to the angle unit. Students should test a claimed identity algebraically and with domain awareness rather than trust a few calculator values.

Study application: Compare an identity with a trigonometric equation that has isolated solutions. Explain why checking several values cannot prove the identity.

19. Sinusoidal Models

Amplitude, period, phase, and vertical shift allow a sinusoid to model recurring behavior. Each parameter has a distinct interpretation, and a graph can help estimate the values before algebra refines them. Periodic appearance alone does not prove that a sine wave is an adequate model. Students should compare the model with the data and identify departures from symmetry or constant period. A fitted curve remains an approximation to the observed process.

Study application: Construct a sinusoid from a stated maximum, minimum, and period. Choose a phase convention and verify that the formula reaches a specified peak at the intended time.

20. Inverse Circular Functions

Inverse circular functions require restricted ranges, so a calculator’s principal value may not be the complete solution of a trigonometric equation. The chapter connects these functions with geometric and modeling problems. Students should identify all solutions in the requested interval and check the original equation. Domain restrictions and units remain essential. A principal value is a convention for making an inverse function, not a declaration that other geometrically valid angles do not exist.

Study application: Solve an original sine equation over one full rotation. Show how the inverse function gives a starting value and how symmetry supplies another solution when appropriate.

Appendices: Formulas and Answers

The reference formulas and answer list support checking, while the source explicitly distinguishes answers from worked solutions. A correct final value does not show that a multistep model was justified. Students should use an answer to locate a discrepancy and then reconstruct the reasoning. The licensing appendix concerns reuse of the source and serves a different purpose from the mathematical references.

Study application: Select a problem with a listed answer and write a complete solution including diagram, variables, equations, domain checks, and interpretation.

Study instructions

1. Translate words into a labeled picture and then into equations before computing.
2. State units and contextual domain restrictions at the beginning of each model.
3. Practice function families with graphs, tables, and formulas, explaining how the representations correspond.
4. Attempt multistep exercises before consulting answer lists, recording the first obstacle encountered.
5. Verify results by substitution, scale, limiting behavior, and interpretation in the original question.

Practice and review task

Create an original modeling portfolio with a linear cost comparison, a quadratic optimization, exponential change, and a sinusoidal process. For each, include variables, assumptions, a graph, a calculation, and a reason the model has limits.

Questions for reflection

How can an algebraically correct value fail to answer the original problem? Why does an inverse trigonometric value sometimes provide only part of an equation’s solution set?

Sources and coverage

This original study guide follows University of Washington PDF dated September 2, 2011. It draws on the source contents, chapter introductions, selected explanations and examples, and review material. It condenses the resource rather than replacing the full text; study applications are original exercises. The imported record’s 2016 year is not the date printed on the reviewed PDF. The guide and edition field explicitly identify the linked September 2011 source.

Reviewed source text; Book and cover catalog record.