AP® Precalculus Unit 1 Review: Polynomial and Rational Functions

In this AP® Precalculus Unit 1 Review, we will focus on polynomial and rational functions. Particularly, two essential types of functions in algebra. These functions help us understand how values change together and model real-world situations. For example, you can use them to calculate population growth or track changes in medicine’s effectiveness. As you can see, polynomial and rational functions are powerful tools for solving complex problems.

Throughout this unit, we will cover key concepts. Firstly, we’ll discuss how input and output values change together. Then, we’ll explore rates of change. We’ll also talk about important features like zeros, end behavior, and transformations. Learning these functions will help you with your algebra skills and prepare you for the AP® Precalculus exam. Ready to test your AP® Precalculus Unit 1 review skills? Try the Unit 1 Assessment on albert.io now!

AP® Precalculus Unit 1 Review Part 1: Understanding Change in Functions

1.1 Change in Tandem

In this part of the AP® Precalculus Unit 1 Review, we focus on how input and output values change together in functions. A function describes how each input value (the independent variable) relates to an output value (the dependent variable). For example, in the function f(x) = 2x + 1 , the output depends on the value of x .

When input and output values change together, we can describe a function as either increasing or decreasing. If a function is increasing, the output gets larger as the input increases. In contrast, if the function is decreasing, the output gets smaller as the input grows.

Additionally, the graph of a function can be concave up or concave down. A graph is concave up when the rate of change increases, and concave down when the rate of change decreases. A graph crosses the x-axis at points called zeros. They show where the output value is zero.

Understanding how input and output values change in tandem helps you recognize key behaviors in different types of functions, which we will explore throughout this unit.

1.2 Rates of Change

The average rate of change of a function over an interval tells us how the output values change as the input values change. We calculate this by comparing the difference in output values to the difference in input values over that interval. For example, if you know the distance a car has traveled over time, you can calculate the average speed, which represents the car’s average rate of change in distance.

We can calculate the average rate of change by the formula:

\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}

Here, f(b) and f(a) represent the output values at two points, and b and a are the corresponding input values. This formula gives us a constant rate that describes how fast one variable is changing compared to the other.

On the other hand, the rate of change at a specific point shows how quickly the output is changing at that exact point. We can estimate this by calculating the average rate of change over smaller and smaller intervals around that point. If this value exists, it gives us an idea of how the function behaves locally.

A positive rate of change means both quantities are increasing or decreasing together. In contrast, a negative rate of change means that when one quantity increases, the other decreases.

Understanding rates of change is key to analyzing functions, as it helps us see how two quantities vary together, whether increasing or decreasing.

1.3 Rates of Change in Linear and Quadratic Functions

In this part of the AP® Precalculus Unit 1 Review, we’ll explore how rates of change behave differently in linear and quadratic functions.

For linear functions, the average rate of change is always constant.

In contrast, the average rate of change for a quadratic function is not constant. However, when you measure the average rate of change over equal-length intervals, it behaves like a linear function. The slope of the secant line between two points (a, f(a)) and (b, f(b)) represents the average rate of change over the interval [a, b] .

Let’s compare these rates of change using a table:

Function Type	Input Interval	Change in Output	Average Rate of Change
Linear (e.g., f(x) = 2x + 3 )	[1, 2]	f(2) - f(1) = 2	\frac{2}{1} = 2
	[2, 3]	f(3) - f(2) = 2	\frac{2}{1} = 2
	[3, 4]	f(4) - f(3) = 2	\frac{2}{1} = 2
Quadratic (e.g., f(x) = x^2 )	[1, 2]	f(2) - f(1) = 3	\frac{3}{1} = 3
	[2, 3]	f(3) - f(2) = 5	\frac{5}{1} = 5
	[3, 4]	f(4) - f(3) = 7	\frac{7}{1} = 7

As you can see, the rate of change for the linear function remains constant at 2 for each interval, while the rate of change for the quadratic function increases as the intervals progress.

Additionally, for a quadratic function, the rate of change of the average rate of change is constant, +2 each time in this case.

Finally, a function is concave up when the average rate of change increases over equal-length intervals, creating a “cup-like” shape on the graph. It is concave down when the average rate of change decreases, forming a “hill-like” shape.

AP® Precalculus Unit 1 Review Part 2: Exploring Polynomial Functions

1.4 Polynomial Functions and Rates of Change

Up next in this AP® Precalculus Unit 1 Review, we explore the key characteristics of polynomial functions and how their rates of change reveal important features such as maxima, minima, and points of inflection.

A polynomial function can be written in the form:

p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0

Here, n is the degree of the polynomial, a_n is the leading coefficient, and a_0 is the constant term. The degree of the polynomial determines its overall behavior, including how fast the output changes as the input increases.

Key Features of Polynomial Functions:

• Maxima and Minima: A polynomial function can switch between increasing and decreasing at certain points. These are called local maxima and local minima. For example, between two distinct zeros of a polynomial, there must be at least one local maximum or minimum. The largest of all local maxima is the global maximum, and the smallest of all local minima is the global minimum.
• Even-Degree Polynomials: Polynomials with an even degree will always have either a global maximum or a global minimum because their graphs will extend either upwards or downwards at both ends.
• Points of Inflection: A point of inflection occurs when the rate of change of a polynomial shifts from increasing to decreasing, or vice versa. This happens when the graph changes from concave up (a “cup-like” shape) to concave down (a “hill-like” shape), or the other way around.

Understanding these key features, particularly how rates of change affect the graph’s shape, helps us predict the behavior of polynomial functions. This knowledge becomes essential when analyzing real-world problems and interpreting how outputs change based on inputs.

1.5 Polynomial Functions and Complex Zeros

Next, we examine the zeros (or roots) of polynomial functions, focusing on both real and complex roots, their multiplicities, and how they impact the function’s graph.

Real and Complex Zeros

A zero of a polynomial function p(x) is any value of x that satisfies p(a) = 0 . If a is a real number, then (x - a) is a linear factor of the polynomial. For example, if a = 3 , then (x - 3) is a factor, and the graph of the function will have an x-intercept at (3, 0) .

In contrast, complex zeros appear in conjugate pairs. If a + bi is a non-real zero of the polynomial p(x) , then its conjugate a - bi is also a zero of the polynomial.

Multiplicity of Zeros

If a zero is repeated, the zero has multiplicity. For instance, if (x - a) is repeated n times, the corresponding zero has a multiplicity of n . A polynomial of degree n will have exactly n complex zeros, counting multiplicities.

The graph of a polynomial behaves differently depending on the multiplicity of the zero. If a real zero has an even multiplicity, the graph will be tangent to the x-axis at that point. This means the graph touches the axis but does not cross it. If a real zero has an odd multiplicity, the graph will cross the x-axis at that point.

Even and Odd Polynomials

A polynomial is considered even if it is symmetric over the y-axis (line x = 0 ). Analytically, an even function satisfies f(-x) = f(x) . For example, a polynomial of the form p(x) = a_n x^n , where n is even, is an even function.

In contrast, a polynomial is odd if it is symmetric about the origin (0,0) . An odd function satisfies f(-x) = -f(x) . For example, a polynomial like p(x) = a_n x^n with an odd degree exhibits this property.

1.6 Polynomial Functions and End Behavior

In this part of the AP® Precalculus Unit 1 Review, we focus on the end behavior of polynomial functions. Overall, end behavior allows us to predict what happens to the output values at extreme values.

End Behavior and the Leading Term

The end behavior of a polynomial function is determined by its leading term—the term with the highest degree. As input values increase or decrease without bound, the leading term dominates the function’s behavior. For example, in the polynomial p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 , the term a_nx^n determines how the function behaves at extreme values of x .

Describing End Behavior with Limits

We use limits to describe the behavior of a polynomial as input values become very large or very small. For instance, as x \to \infty , if the leading term is a_nx^n and a_n is positive, then:

\lim \limits_{x \to \infty} p(x) = \infty .

This means that as x increases, the output values of the function will also increase without bound.

Degree and Leading Coefficient Effects

The degree and leading coefficient determine the graph’s end behavior:

• If the degree n is even and a_n is positive, both ends of the graph rise.
• On the other hand, If the degree n is odd, the end behavior differs:
  • If a_n is positive, the graph falls as x \to -\infty and rises as x \to \infty .

AP® Precalculus Unit 1 Review Part 3: Understanding Rational Functions

1.7 Rational Functions and End Behavior

In this part of the AP® Precalculus Unit 1 Review, we focus on the end behavior of rational functions. A rational function is represented as a quotient of two polynomial functions, and the degree of the numerator and denominator plays a key role in determining how the function behaves as input values increase or decrease without bound.

End Behavior Based on Degree of Numerator and Denominator

The end behavior of a rational function is largely determined by the polynomial with the higher degree, as it dominates the function for very large or very small input values. To understand this, we can compare the degrees of the numerator and denominator.

• Numerator’s Degree is Greater:
  If the degree of the numerator is greater than that of the denominator, the rational function behaves like a polynomial. For example, if the numerator is a linear polynomial after simplifying, the function will have a slant asymptote. This asymptote is a line parallel to the graph of the leading term in the numerator.
• Degrees are Equal:
  When the degrees of the numerator and denominator are equal, the end behavior approaches a horizontal asymptote. The quotient of the leading terms determines the location of this horizontal asymptote. For example, if the leading coefficients of the numerator and denominator are a_n and b_n , then the horizontal asymptote will be located at y = \frac{a_n}{b_n} .
• Denominator’s Degree is Greater:
  If the degree of the denominator is greater than the numerator, the rational function has a horizontal asymptote at y = 0 . This means the output values approach zero as the input values grow larger in magnitude.

Describing End Behavior with Limits

When the graph of a rational function has a horizontal asymptote at y = b , the output values approach b as the input values increase or decrease without bound. In mathematical terms, this is expressed as:

\lim \limits_{x \to \infty} r(x) = b or \lim \limits_{x \to -\infty} r(x) = b .

1.8 Rational Functions and Zeros

In this part of the AP® Precalculus Unit 1 Review, we focus on determining the zeros of rational functions. Zeros are the points where the output of the function equals zero, which correspond to the real zeros of the function’s numerator.

Finding Zeros of Rational Functions

A rational function is defined as the quotient of two polynomials. To find the real zeros of a rational function r(x) = \frac{p(x)}{q(x)} , you only need to focus on the numerator p(x) . The real zeros of p(x) are the values of x that make the function equal to zero. Therefore, the zeros of the rational function are the same as the zeros of the numerator, provided these values are in the domain of the function.

For example, if p(x) = (x - 2)(x + 3) , then the real zeros of the numerator are x = 2 and x = -3 . These values will be the zeros of the rational function unless they also cause the denominator to be zero, which would make the function undefined.

Role of Zeros in Inequalities

The real zeros of both the numerator and denominator play a role in determining intervals that satisfy rational inequalities. For example, if you are solving the inequality r(x) \geq 0 or r(x) \leq 0 , the real zeros of the numerator serve as endpoints for the intervals where the function is positive or negative. The real zeros of the denominator create vertical asymptotes, which also divide the graph into different regions for solving inequalities.

By understanding how to find the zeros of rational functions and their role in the function’s graph, you can better analyze the behavior of the function across its domain.

1.9 Rational Functions and Vertical Asymptotes

In this part of the AP® Precalculus Unit 1 Review, we focus on how to determine the vertical asymptotes of rational functions. Vertical asymptotes occur when the denominator of a rational function equals zero, causing the function to be undefined at certain points.

Identifying Vertical Asymptotes

To find vertical asymptotes, we look for the real zeros of the denominator. If a is a real zero of the denominator, and it is not a zero of the numerator, then the graph of the rational function has a vertical asymptote at x = a . For example, if the denominator of a rational function is q(x) = (x - 2)(x + 1) , the function will have vertical asymptotes at x = 2 and x = -1 .

In some cases, the zero of the denominator may also be a zero of the numerator. If the multiplicity of the zero in the denominator is greater than its multiplicity in the numerator, a vertical asymptote will still occur at that point.

Behavior Near Vertical Asymptotes

Near a vertical asymptote, the values of the denominator become arbitrarily close to zero, causing the values of the rational function to increase or decrease without bound. This is expressed using limits:

• As x approaches the asymptote x = a from the right ( x \to a^+ ), the function can approach \infty or -\infty .
• Similarly, as x approaches the asymptote from the left ( x \to a^- ), the function can also approach \infty or -\infty .

In mathematical terms, this behavior is written as:

\lim \limits_{x \to a^+} r(x)=\infty or \lim \limits_{x \to a^+} r(x)=-\infty ,
and
\lim \limits_{x \to a^-} r(x)=\infty or \lim \limits_{x \to a^-} r(x)=-\infty .

This shows that the function’s values grow without bound as the input values get closer to the vertical asymptote from either direction.

1.10 Rational Functions and Holes

In this part of the AP® Precalculus Unit 1 Review, we focus on identifying holes in the graphs of rational functions. A hole occurs when a real zero is shared by both the numerator and denominator, but the function is undefined at that point.

Identifying Holes in Rational Functions

A hole in a rational function’s graph happens when a zero of the numerator is also a zero of the denominator, with the same or greater multiplicity in the numerator. In such cases, the factor causes the zero cancels out, leaving a hole in the graph at the corresponding input value.

For example, if r(x) = \frac{(x - 3)(x + 2)}{(x - 3)(x + 5)} , the factor (x - 3) cancels out, creating a hole at x = 3 .

Locating Holes in the Graph

To find the location of the hole, we can evaluate the rational function at points very close to x = c , where the hole occurs. If the output values approach a specific value L as the input approaches c , then the hole is located at the point (c, L) .

Mathematically, we describe this using limits. If:

\lim \limits_{x \to c} r(x) = L

Then the hole is located at (c, L) . Additionally, the limits from both the left and the right of c must approach the same value, meaning:

\lim \limits_{x \to c^-} r(x) = \lim \limits_{x \to c^+} r(x) = L .

This ensures that the function approaches the same output value from both sides of the hole.

By understanding how holes form and using limits to find their exact location, you can better analyze the graph of a rational function.

AP® Precalculus Unit 1 Review Part 4: Function Representations and Transformations

1.11 Equivalent Representations of Polynomial and Rational Expressions

In this part of the AP® Precalculus Unit 1 Review, we focus on how to rewrite polynomial and rational expressions in equivalent forms. Each form provides different insights into the function, helping us answer key questions about zeros, asymptotes, domain, range, and more.

Factored vs. Standard Form

Rewriting a polynomial or rational function in factored form reveals key details about the function’s real zeros and, for rational functions, any vertical asymptotes or holes. For example, in the polynomial p(x) = (x - 2)(x + 3) , the factored form tells us the zeros are x = 2 and x = -3 , which correspond to the x-intercepts of the graph.

On the other hand, the standard form of a polynomial, such as p(x) = x^2 + x - 6 , helps us understand the function’s end behavior. The leading term in the standard form determines how the graph behaves as x approaches infinity or negative infinity.

Using Polynomial Long Division

When dividing two polynomials, we use polynomial long division, which works much like numerical long division. If we divide a polynomial f(x) by another polynomial g(x) , we can rewrite f(x) in the form:

f(x) = g(x)q(x) + r(x)

So, q(x) is the quotient and r(x) is the remainder. The remainder r(x) will have a degree lower than g(x) . This process is especially useful for finding slant asymptotes in rational functions. As has been noted, this occurs when the degree of the numerator is greater than the degree of the denominator.

Binomial Theorem and Expanding Polynomials

We can also use the binomial theorem to expand polynomials of the form (x + c)^n . The binomial theorem simplifies the expansion of powers of binomials by utilizing Pascal’s Triangle. For instance, expanding (x + 2)^3 using the binomial theorem gives:

(x + 2)^3 = x^3 + 6x^2 + 12x + 8

This method is especially helpful when expanding products of binomials in polynomial functions.

By understanding how to rewrite polynomial and rational functions in different forms, you can extract valuable information about their behavior and use this knowledge to answer questions in context.

1.12 Transformations of Functions

In this part of the AP® Precalculus Unit 1 Review, we explore how functions can be transformed using additive and multiplicative changes. These transformations impact the graph of the original function, resulting in translations, dilations, and reflections.

Additive Transformations

Additive transformations shift the graph either vertically or horizontally:

• For vertical shifts, the function g(x) = f(x) + k moves the graph of f(x) up or down by k units. If k > 0 , the graph is translated upward, and if k < 0 , it shifts downward.
• In the case of horizontal shifts, g(x) = f(x + h) results in the graph moving horizontally. If h > 0 , the graph shifts left by h units, while a negative h moves the graph right.

Multiplicative Transformations

Multiplicative transformations involve stretching, compressing, or reflecting the graph:

• Vertical dilation occurs with g(x) = af(x) , where the graph of f(x) is stretched or compressed depending on the value of a . If a > 1 , the graph stretches vertically. Conversely, if 0 < a < 1 , it compresses. A negative value of a not only compresses or stretches but also reflects the graph across the x-axis.
• Horizontal dilation occurs with g(x) = f(bx) . Here, the graph stretches or compresses horizontally based on the value of b . If |b| > 1 , the graph compresses horizontally. However, if 0 < |b| < 1 , the graph stretches horizontally. A negative b introduces a reflection across the y-axis.

Combined Transformations

Both additive and multiplicative transformations can be combined to produce more complex changes to the graph. For instance, the function g(x) = af(x + h) + k introduces a vertical and horizontal shift, along with a vertical stretch or compression.

These transformations can also affect the domain and range of the new function. Consequently, be sure to analyze the new inputs and outputs to determine these characteristics.

By mastering these transformations, you can adjust graphs to model different real-world situations or understand the changes to a function’s behavior.

AP® Precalculus Unit 1 Review Part 5: Function Models in Action

1.13 Function Model Selection and Assumption Articulation

In this part of the AP® Precalculus Unit 1 Review, we focus on how to choose the appropriate function model for different scenarios. Different types of functions are used to represent various patterns in real-world data, and understanding which model fits best is key to constructing accurate function models.

Selecting the Right Function Model

• Linear Functions: These are best suited for scenarios where data shows a roughly constant rate of change. For example, a steady increase in speed or a straight-line relationship between two quantities can be modeled with a linear function.
• Quadratic Functions: When the data displays a linear rate of change or forms a symmetric pattern with a single maximum or minimum value, quadratic functions are ideal. Common applications include modeling the area of a two-dimensional shape or analyzing the trajectory of a thrown object.
• Cubic Functions: Geometric contexts involving volume or three-dimensional measurements often call for cubic functions, which capture changes in three dimensions.
• Polynomial Functions: For more complex scenarios with multiple real zeros or several turning points (maxima and minima), polynomial functions of higher degrees are appropriate. A polynomial of degree n can model scenarios with n+1 distinct data points.
• Piecewise-Defined Functions: When a scenario involves different characteristics across intervals, a piecewise-defined function is useful. This type of function combines multiple functions over nonoverlapping domain intervals, ideal for modeling data with varying trends over time.

Articulating Assumptions and Restrictions

When constructing a function model, it’s important to state any assumptions clearly. For example, a model may assume that the rate of change remains constant throughout the scenario or that quantities change together in predictable ways.

Additionally, domain restrictions may apply, especially when certain input values don’t make sense within a real-world context (e.g., negative time values). Similarly, range restrictions might be needed when rounding or limiting extreme values, based on the data set or context.

By selecting the appropriate function model and articulating any assumptions or restrictions, you can accurately represent real-world scenarios and make reliable predictions.

1.14 Function Model Construction and Application

In this part of the AP® Precalculus Unit 1 Review, we focus on constructing function models—whether linear, polynomial, piecewise-defined, or rational—that represent real-world scenarios. Once constructed, these models can be used to answer important questions, make predictions, and analyze rates of change.

Constructing Function Models

• Linear and Polynomial Models: To construct a function model, begin by identifying restrictions from the context or data set. A linear model may be appropriate for data with a constant rate of change, while a polynomial model works for scenarios with multiple turning points (maxima or minima) or when modeling more complex behaviors.
• Transformations of Parent Functions: Often, you can build a model by applying transformations (translations, stretches, compressions) to a known parent function. For example, transforming a quadratic parent function f(x) = x^2 can help model scenarios involving projectile motion or area.
• Using Technology and Regressions: When working with large data sets, regression tools can help generate a function model. Technology such as graphing calculators or software can fit a linear or polynomial curve to data points, producing a model that accurately reflects the underlying pattern.
• Piecewise-Defined Models: Some real-world scenarios may require piecewise-defined functions, which combine multiple models across different intervals. For example, a piecewise function might be used to represent varying rates of growth in different time periods.
• Rational Function Models: Rational functions are useful for modeling situations involving inverse relationships, such as when two quantities are inversely proportional. For example, gravitational force between two objects is inversely proportional to the square of the distance between them, making rational functions ideal for these contexts.

Applying the Function Model

Once the model is constructed, it can be used to answer questions about the data set or scenario. Function models allow you to:

• Predict values based on the model.
• Calculate rates of change, average rates of change, and analyze how these rates change over time.
• Make inferences about the behavior of the system or scenario being modeled, ensuring the appropriate units of measure are used based on the given context.

By constructing and applying function models, you can better understand and analyze real-world situations, using mathematics to make informed decisions and predictions.

AP® Precalculus Unit 1 Review: Conclusion

As you finish this AP® Precalculus Unit 1 Review, you’ve gained a deeper understanding of polynomial and rational functions, rates of change, and function models. Mastering these concepts will be essential for success on the AP® Precalculus Exam. To strengthen your skills, revisit this review regularly, practice with different function models, and focus on applying these ideas in real-world contexts. For additional support, follow these AP® Precalculus tips in our other blog post.

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