DP Maths AI HL 2.5 Polynomials and Rational Functions

DP Maths AI HL 2.5 Polynomials and Rational Functions - Full notes

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DP Maths AI HL | 2.5 Polynomials and Rational Functions

DP MATHEMATICS AI · TOPIC 2: FUNCTIONS

2.5

Polynomials and Rational Functions

Full notes for Applications and Interpretation HL

Contents

1	Polynomial vocabulary
2	Quadratic functions and their forms
3	Solving quadratic equations
4	Quadratic models and regression
5	Cubic functions and their graphs
6	Cubic models and regression
7	Polynomials in factorized form
8	End behaviour and turning points
9	Polynomial equations and inequalities with technology
10	The reciprocal function
11	Rational functions of the form (ax + b)/(cx + d)
12	Inverse variation and rational models
13	Average cost and concentration models
14	Fitting polynomial models to data
15	Optimization with polynomial models
16	Choosing between polynomial and other models
17	Using technology
18	Mixed review: worked problems
19	Polynomial division and the factor theorem (HL)
20	Rational functions with quadratic denominators (HL)
21	Investigation: the open box
22	Exam-style questions with markscheme
23	Common misconceptions and links
24	Check your understanding
25	Summary and topic map
26	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the practice exercise.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
use the vocabulary of polynomials (degree, leading coefficient, roots)
work with quadratics in standard, factorized and vertex forms
solve quadratic and polynomial equations by algebra and technology
sketch cubic functions and describe end behaviour
fit quadratic and cubic models with regression
sketch rational functions and find their asymptotes
model inverse variation and average cost with rational functions
optimize polynomial models with technology

HL booklet: this booklet contains all the SL content plus polynomial division, the factor theorem and rational functions with quadratic denominators.

Guiding questions

How can polynomial and rational functions model real situations? What do roots, turning points and asymptotes mean in context?

How this topic is examined

Polynomial and rational models appear mainly on Paper 2: fit a quadratic or cubic model, find a maximum or minimum, solve an equation with the GDC, and interpret asymptotes of rational models (for example, the long-run average cost).

State the type of regression used, give the equation to 3 s.f., and interpret key features in context with units.

Mathematics: Applications and Interpretation places strong emphasis on technology, modelling and interpretation. Most papers allow a GDC, so you are expected to use it efficiently, record the method you used, and interpret every answer in context.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working, including what you entered into a GDC
Show that	obtain the given result; every step must be shown and the result must not be assumed
Estimate	obtain an approximate value
Sketch	a diagram showing the general shape and key features, labelled
Interpret / Comment	explain the meaning of a result in context, or give a judgement
Justify / Explain	give valid reasons or evidence
Hence	use the previous result

How to use these notes

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or explanation of a key result
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given), graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet

1 Polynomial vocabulary

Degree, coefficients and roots.

Think about it

The volume of a box made from a sheet of card depends on the size of the squares cut from the corners. What kind of function would you expect?

A polynomial is a sum of terms of the form axn with n a non-negative integer: P(x) = anxn + ... + a1x + a0. The degree is the highest power n; an is the leading coefficient and a0 the constant term (y-intercept). Degree 1 is linear, 2 quadratic, 3 cubic, 4 quartic.

A root (zero) of P is a value of x with P(x) = 0: an x-intercept of the graph. A polynomial of degree n has at most n real roots and at most n − 1 turning points.

Polynomial	Degree	Leading coefficient	Constant term
4x3 − 2x + 7	3	4	7
5 − x2	2	−1	5
0.5x4 + x3	4	0.5	0

Practice exercise 1

1. State the degree and leading coefficient of 3x2 − x5 + 1.

2. What is the maximum number of turning points of a quartic?

2 Quadratic functions and their forms

Three forms, three insights.

Form	Equation	Shows
Standard	y = ax2 + bx + c	y-intercept c; axis x = −b/(2a)
Factorized	y = a(x − p)(x − q)	x-intercepts p and q
Vertex	y = a(x − h)2 + k	vertex (h, k)

The graph is a parabola: opening up if a > 0 (minimum) and down if a < 0 (maximum), symmetric about the axis of symmetry x = −b/(2a), which passes through the vertex.

Figure 1. One parabola in factorized and vertex forms.

WORKED EXAMPLE 1 Converting forms (standard)

Write y = 2x2 − 12x + 10 in factorized and vertex forms.

Solution

2(x2 − 6x + 5) = 2(x − 1)(x − 5)A1

Axis x = 3, y(3) = −8: 2(x − 3)2 − 8A1

Practice exercise 2

1. Find the vertex of y = −x2 + 6x − 5.

2. Write y = (x + 2)(x − 4) in standard form and find its axis.

3. Find the equation of the parabola with vertex (2, 5) through (0, 1).

3 Solving quadratic equations

Factorizing, formula, technology.

Solve by factorizing when possible, the quadratic formula otherwise, or the GDC polynomial solver on Paper 2. The discriminant Δ tells how many real roots: Δ > 0 two, Δ = 0 one (repeated), Δ < 0 none.

WORKED EXAMPLE 2 Formula (standard)

Solve 3x2 − 7x + 1 = 0.

Solution

x = (7 ± √37)/6M1

x = 2.18 or 0.153A1

WORKED EXAMPLE 3 Discriminant (standard)

For which k does x2 + kx + 4 = 0 have no real roots?

Solution

k2 − 16 < 0M1

−4 < k < 4A1

Practice exercise 3

1. Solve x2 − 5x − 14 = 0.

2. Solve 2x2 + x = 6.

3. How many roots has 4x2 − 12x + 9 = 0?

4 Quadratic models and regression

Curves with one turning point.

Quadratic models describe projectiles, areas, and quantities that rise to a maximum then fall (or fall then rise). Given three points, solve simultaneous equations for a, b, c; given data, use quadratic regression on the GDC and report R2.

Time t (s)	0	0.5	1.0	1.5	2.0
Height h (m)	1.2	6.3	9.0	9.4	7.4

WORKED EXAMPLE 4 Quadratic regression (exam level)

Fit a quadratic model to the table and estimate the maximum height and when the object lands.

Solution

GDC: h = −4.71t2 + 12.5t + 1.20A2

Vertex t = 1.33 s: maximum height 9.53 mA1

h = 0: t = 2.75 sA1

Practice exercise 4

1. Find the quadratic through (0, 3), (1, 6) and (3, 0).

2. Explain what R2 = 0.998 tells you about a quadratic regression.

5 Cubic functions and their graphs

Up to two turning points.

A cubic y = ax3 + bx2 + cx + d has up to three real roots and up to two turning points. If a > 0, it falls on the left and rises on the right; if a < 0, the reverse. Every cubic crosses the x-axis at least once.

Figure 2. Cubics with positive and negative leading coefficients.

WORKED EXAMPLE 5 Features of a cubic (standard)

For y = x3 − 3x2 − 9x + 5, find the turning points with technology.

Solution

Local max (−1, 10)A1

Local min (3, −22)A1

Practice exercise 5

1. How many roots does y = x3 − 3x2 − 9x + 5 have? Find them.

2. Sketch y = −x3 + 4x.

6 Cubic models and regression

Volumes and more.

Cubic models arise naturally for volumes (length × width × height when all depend on one variable) and are also used to fit data with two bends. Cubic regression on the GDC fits y = ax3 + bx2 + cx + d.

WORKED EXAMPLE 6 Volume model (standard)

Squares of side x are cut from the corners of a 20 cm × 30 cm card, and the sides folded up to make an open box. Write V(x) and state the domain.

Solution

V = x(20 − 2x)(30 − 2x)A1

Domain 0 < x < 10A1

Practice exercise 6

1. Use technology to find the maximum volume of the box above.

2. Fit a cubic through (0, 1), (1, 0), (2, 3), (3, 16).

7 Polynomials in factorized form

Roots and sketches.

If P(x) = a(x − p)(x − q)(x − r), its roots are p, q, r. A squared factor (x − p)2 gives a root where the graph touches the axis. Given roots and one more point, the equation is determined.

WORKED EXAMPLE 7 Equation from a graph (standard)

A cubic has roots −1, 2 and 4, and passes through (0, 16). Find its equation.

Solution

y = a(x + 1)(x − 2)(x − 4)M1

16 = a(1)(−2)(−4), a = 2A1

y = 2(x + 1)(x − 2)(x − 4)A1

Practice exercise 7

1. Find the cubic touching the x-axis at x = 1 and crossing at x = −2, through (0, 4).

2. Write down the roots of y = 3x(x − 5)(x + 2).

8 End behaviour and turning points

Big-picture shape.

For large |x|, the leading term dominates: y ≈ anxn. Even degree: both ends point the same way; odd degree: opposite ways. Turning points are found with the GDC (or calculus, 5.3). In models, end behaviour usually lies outside the valid domain.

Practice exercise 8

1. Describe the end behaviour of y = −2x4 + x3.

2. Why do cubic regression models often give unrealistic predictions far from the data?

9 Polynomial equations and inequalities with technology

Graph and read.

Solve P(x) = k by intersecting y = P(x) with y = k. Solve P(x) > k by finding where the graph is above the line. Record the intersection values and state the solution as intervals.

WORKED EXAMPLE 8 Inequality (standard)

Solve x3 − 4x > 1.

Solution

Intersections of y = x3 − 4x and y = 1: x = −1.86, −0.254, 2.11M1 A1

−1.86 < x < −0.254 or x > 2.11A1

Practice exercise 9

1. Solve x2(3 − x) = 2.

2. For the box V(x) = x(20 − 2x)(30 − 2x), find x such that V > 800.

10 The reciprocal function

y = 1/x.

y = 1/x has a vertical asymptote x = 0 and a horizontal asymptote y = 0. It is self-inverse and models inverse proportion. y = k/x is a vertical stretch.

Figure 3. Reciprocal functions.

11 Rational functions of the form (ax + b)/(cx + d)

Asymptotes from the equation.

For f(x) = (ax + b)/(cx + d): vertical asymptote x = −d/c (denominator zero); horizontal asymptote y = a/c (ratio of x coefficients); x-intercept −b/a; y-intercept b/d. The graph is a hyperbola with these asymptotes.

Figure 4. Asymptotes x = 2 and y = 3.

WORKED EXAMPLE 9 Asymptotes (standard)

Find the asymptotes and intercepts of y = (3x + 1)/(x − 2).

Solution

x = 2 and y = 3A1 A1

Intercepts (−1/3, 0) and (0, −0.5)A1

Practice exercise 10

1. Find the asymptotes of y = (2x − 5)/(x + 4).

2. Find the range of y = (x + 1)/(x − 3).

12 Inverse variation and rational models

y = k/x and variations.

Inverse variation y = k/x is the simplest rational model: time against speed for a fixed distance, pressure against volume for a gas (Boyle's law), and the number of days to complete a job against the number of workers.

WORKED EXAMPLE 10 Boyle's law (standard)

A gas has pressure 150 kPa at volume 2.4 L. Find the pressure at 3.6 L.

Solution

PV = 360M1

P = 100 kPaA1

Practice exercise 11

1. 8 workers complete a job in 15 days. How long would 12 workers take?

2. Explain the asymptotes of P = 360/V in context.

13 Average cost and concentration models

Asymptotes as limits.

The average cost per item when fixed costs F and variable cost v per item are involved is A(x) = (F + vx)/x = v + F/x: a rational function with horizontal asymptote y = v. As production increases, the average cost approaches the variable cost. Concentration models such as C(t) = (5t)/(t + 2) approach a limiting value.

WORKED EXAMPLE 11 Average cost (exam level)

A company's cost for x items is 4000 + 12x. Find the average cost function, its asymptote, and the production level for an average cost of $15.

Solution

A(x) = 12 + 4000/xA1

Asymptote y = 12: the average cost approaches $12A1 R1

12 + 4000/x = 15: x = 1334 items (1333.3)M1 A1

Practice exercise 12

1. Salt solution: C(t) = 30t/(t + 5) g/L. Find C(10) and the long-term concentration.

2. Explain why A(x) = 12 + 4000/x is not valid for x = 0.

14 Fitting polynomial models to data

Regression and R2.

Choose the lowest degree polynomial that fits well. Compare R2 values and examine the shape in context: higher-degree polynomials always fit the data points better but may behave unrealistically between and beyond them (overfitting).

Practice exercise 13

1. Why might a quartic with R2 = 1 be a poor model for 5 data points?

15 Optimization with polynomial models

Maximum and minimum values.

WORKED EXAMPLE 12 Maximum profit (exam level)

Profit P(x) = −0.02x3 + 1.2x2 + 10x − 150 (x hundred items, 0 ≤ x ≤ 60). Find the maximum profit.

Solution

GDC maximumM1

x = 43.8: P = 910 (hundred dollars)A1 A1

Practice exercise 14

1. A garden of perimeter 40 m is enclosed against a wall (three sides fenced). Find the maximum area.

16 Choosing between polynomial and other models

Shape and context.

Data pattern	Model
single maximum or minimum	quadratic
two bends	cubic
levels off	exponential with asymptote or rational
inverse proportion	rational y = k/x
constant percentage change	exponential

17 Using technology

Solver, regression, graph.

GDC tip

Use the polynomial root finder for equations, the graph tools (zero, max, min, intersect) for features, and QuadReg/CubicReg for data. For rational functions, use a window that avoids the vertical asymptote misleading the display (some GDCs draw a false vertical line).

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 13 Mixed problem 1 (exam level)

A ball's height is h(t) = at2 + bt + c. It is at 2 m at t = 0, 12 m at t = 1 and 12.4 m at t = 2. Find h(t).

Solution

c = 2; a + b = 10; 4a + 2b = 10.4M1 A1

a = −4.8, b = 14.8: h = −4.8t2 + 14.8t + 2A1

Practice exercise 15

1. Find the intersections of y = x2 − 4 and y = 2/x.

2. The average cost per item is A(x) = 8 + 600/x. Find the number of items for which the average cost is within 10% of $8.

19 Polynomial division and the factor theorem (HL)

Remainders and factors.

When P(x) is divided by (x − a), the remainder is P(a) (remainder theorem). So (x − a) is a factor if and only if P(a) = 0 (factor theorem). Test integer factors of the constant term to find roots, then divide to factorize fully.

WORKED EXAMPLE 14 Factorizing (standard)

Factorize x3 − 2x2 − 5x + 6.

Solution

P(1) = 0, so (x − 1) is a factorM1

Divide: x2 − x − 6 = (x − 3)(x + 2)A1

(x − 1)(x − 3)(x + 2)A1

Practice exercise 16

1. Find the remainder when x3 + 4x − 1 is divided by (x − 2).

2. Find k if (x + 1) is a factor of x3 + kx2 + 3.

20 Rational functions with quadratic denominators (HL)

Two possible vertical asymptotes.

For f(x) = (ax + b)/(cx2 + dx + e), vertical asymptotes occur at the real zeros of the denominator and the horizontal asymptote is y = 0. If the numerator has the same degree as a quadratic denominator, the horizontal asymptote is the ratio of the leading coefficients.

Figure 5. Asymptotes x = ±2 and y = 0.

Practice exercise 17

1. Find the asymptotes of y = 3/(x2 − x − 6).

2. Find the horizontal asymptote of y = (2x2 + 1)/(x2 + 3).

21 Investigation: the open box

A Paper 3 style exploration.

A square sheet of side 30 cm has squares of side x cut from each corner and is folded into an open box. 1. Show that V = x(30 − 2x)2 and state the domain. 2. Use technology to find the maximum volume. 3. Repeat for sheets of side 20, 40 and 60 cm. What do you notice about the optimal x? 4. Conjecture a formula for the optimal x for a sheet of side s and test it. 5. For a rectangular sheet 20 cm × 32 cm, find the optimal x.

Answers. 1. 0 < x < 15. 2. x = 5, V = 2000 cm3. 3. Optimal x is s/6 each time (3.33, 6.67, 10). 4. x = s/6. 5. x = 4 cm, V = 1152 cm3.

22 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 10 minutes)

f(x) = (2x + 6)/(x − 1).

(a) Write down the equations of the asymptotes. [2]

(b) Find the intercepts. [2]

Markscheme

Part	Answer	Marks
(a)	x = 1	A1
	y = 2	A1
(b)	(−3, 0)	A1
	(0, −6)	A1
(c)	two branches with asymptotes and intercepts	A2

Examiner's comments. Asymptotes must be written as equations.

Question 2 (Paper 2 style) (about 15 minutes)

The table shows the number of visitors (thousands) to a park in month m: m = 1, 3, 5, 7, 9, 11; visitors 4.2, 9.8, 14.1, 14.6, 10.9, 5.1.

(a) Find a quadratic regression model. [2]

(b) Use the model to estimate the month with the most visitors. [2]

Markscheme

Part	Answer	Marks
(a)	V = −0.397m2 + 4.89m − 0.596	A2
(b)	vertex m ≈ 6.15	M1 A1
(c)	V(12) ≈ 0.83 thousand	A1
	reasonable as a seasonal low but the model eventually goes negative	R1

Examiner's comments. Reasonable coefficients from the GDC are accepted.

Question 3 (Paper 2 style) (about 10 minutes)

A printer has fixed costs of $2500 and a variable cost of $0.08 per page.

(a) Write the average cost per page A(n) for n pages. [2]

(b) Find the horizontal asymptote and interpret it. [2]

Markscheme

Part	Answer	Marks
(a)	A(n) = 0.08 + 2500/n	A2
(b)	y = 0.08	A1
	the average cost approaches the variable cost	R1
(c)	2500/n = 0.02	M1
	125 000 pages	A1

Examiner's comments. Interpretation in context earns the R mark.

23 Common misconceptions and links

Students often think…	Actually…
the horizontal asymptote of (ax + b)/(cx + d) is y = 0	y = a/c
higher-degree regression is always better	risk of overfitting
every cubic has three roots	between one and three real roots
vertex of a(x − h)2 + k is (−h, k)	(h, k)

How 2.5 connects to the rest of the course

Topic	Connection
2.2 Functions	domain and range
2.9 Transformations	vertex form
5.3 Derivatives	turning points by calculus
4.4 Regression	fitting models

24 Check your understanding

Section A: multiple choice

1. The degree of 4x2 − x3 is

A. 3 B. 2 C. 4 D. 1

2. The vertical asymptote of (x + 2)/(x − 5) is

A. y = 1 B. x = −2 C. x = 5 D. y = 5

3. A cubic has at most

A. 3 B. 2 turning points C. 1 D. 4

4. Δ < 0 means

A. two roots B. one root C. infinitely many D. no real roots

5. The average cost 12 + 4000/x approaches

A. 4000 B. 12 C. 0 D. 4012

Section B: structured

6. Find the quadratic with roots 2 and 6 passing through (4, −8). [3]

7. Solve x3 = 3x + 1. [2]

Answers

1. A 2. C 3. B 4. D 5. B

6. y = 2(x − 2)(x − 6).

7. x = −1.53, −0.347, 1.88.

25 Summary and topic map

Idea	Remember
Polynomial	degree, leading coefficient, roots
Quadratic	standard, factorized, vertex forms
Discriminant	b2 − 4ac
Cubic	up to 3 roots, 2 turning points
Rational	asymptotes x = −d/c, y = a/c
Models	regression; average cost; inverse variation

Before the exam, make sure you…

•state the regression type
•interpret asymptotes
•restrict domains
•avoid overfitting

Polynomials and Rational Functions on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 6. Topic map for Polynomials and Rational Functions.

26 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. 5; −1

2. 3

Practice exercise 2

1. (3, 4)

2. x2 − 2x − 8; x = 1

3. y = −(x − 2)2 + 5

Practice exercise 3

1. 7, −2

2. 1.5, −2

3. one repeated root (Δ = 0)

Practice exercise 4

1. y = −2x2 + 5x + 3

2. the model explains 99.8% of the variation: an excellent fit

Practice exercise 5

1. 3: x = −2.18, 0.489, 4.69

2. roots 0, ±2; falls on the right

Practice exercise 6

1. x = 3.92 cm, V = 1056 cm3

2. y = x3 − x2 − x + 1

Practice exercise 7

1. y = 2(x − 1)2(x + 2)

2. 0, 5, −2

Practice exercise 8

1. falls at both ends

2. the dominant x3 term grows without limit

Practice exercise 9

1. x = −0.732, 1, 2.73

2. 1.88 < x < 6.36

Practice exercise 10

1. x = −4, y = 2

2. y ≠ 1

Practice exercise 11

1. 10 days

2. as V → 0, P → ∞; as V grows, P → 0

Practice exercise 12

1. 20 g/L; 30 g/L

2. no items made: average cost undefined

Practice exercise 13

1. it passes exactly through 5 points but may oscillate and extrapolate badly

Practice exercise 14

1. x(40 − 2x): max 200 m2 at x = 10

Practice exercise 15

1. x = 2.21, −1.68 (and −0.54)

2. 600/x < 0.8: x > 750

Practice exercise 16

1. 15

2. k = −2

Practice exercise 17

1. x = 3, x = −2, y = 0

2. y = 2

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