IB MATH ANALYSIS I SUMMER ASSIGNMENT

We are so happy that you have decided to take IB Math Analysis. We will begin the year with a unit on

functions so it will be helpful for you to refresh some of your Algebra II skills. Therefore, complete the

following problems to get a jump start on our first unit of study.

Each topic will start with a notes and video section, followed by a few IB questions on the topic. The

expectation is that you will complete this packet on your own, without assistance.

If you have any questions regarding the assignment, please email Mr. Devine at crdevine@fcps.edu .

Topic 1: Rational Functions

Definition: A rational function is a function of the form:

( )( )

( )

p xR x

q x=

where p and q are polynomial functions and q is not the zero polynomial.

Examples:

1( )f x

x=

3( )

( 5)( 6)

xf x

x x

−=

− −

2( )

3 9

xf x

x x=

− −

Domain: The domain consists of all real numbers except those for which the denominator q is 0

Examples: Find the domains of the following rational functions.

( ) 2

1f x

x= ( )

2

2

3 1

6

x xg x

x x

+ +=

− −

Solution:

1. Set the denominator equal to zero and solve. 1. Set the denominator equal to 0. Factor.

2. Remove all zeroes from the domain. 2. Solve.

3. Remove all zeroes from the domain.

Vertical Asymptotes:

Vertical Asymptotes occur at every x-value that is a zero of the ______________________, but not the

numerator.

Examples: Identify the vertical asymptotes of each rational function.

( ) 3

1

2f x

x= ( )

2

3 2

2 2

6

xf x

x x x

−=

+ −

1. Set the denominator equal to zero. 1. Set the denominator equal to 0. Factor.

2. Solve 2. Solve.

Holes:

A hole occurs in a rational function for every value of x that is a ______________ of the

numerator and the _______________________.

Examples: Identify any vertical asymptotes and holes of the rational functions below.

( )2

2

4

6

xf x

x x

−=

− − ( )

2

2

4

4 4

xf x

x x

−=

+ +

1. Factor the numerator and denominator. 1. Factor the numerator and denominator.

2. Notice that _________ is a factor of both 2. Notice that _____________ is a factor of

both the numerator and denominator. This the numerator and denominator; however,

forms a hole in the graph of the function at there is no hole in the graph at 2x = − .

2x = − . The ( )3x − factor forms a vertical Since the remaining factor in the denominator

asymptote. is ( )2x + , this factor forms a vertical asymptote.

Horizontal Asymptotes:

Horizontal asymptotes of rational functions are determined by comparing the degree numerator to the

degree of the denominator.

1. If the degree of the numerator is less than the degree of the denominator, then the

horizontal asymptote is always____________________.

2. If the degree of the numerator is the same as the degree of the denominator, then

The horizontal asymptote is ________________________________.

3. If the degree of the numerator is greater than the degree of the denominator, then

______________________________________________________.

Slant Asymptotes:

A slant asymptote is formed when the degree of the numerator is ________________________ degree

higher than the degree of the denominator.

The equation of the slant asymptote is found by using long division.

( )2 2

5

x xf x

x

− −=

− .

The equation of the asymptote is the quotient without the remainder.

Put it all together.

Identify any asymptotes and /or holes of the rational function.

( ) 2

3

3

xf x

x x

−=

−

First: Determine the equation the horizontal asymptote, if one exists. __________________

Compare the degree of the numerator to the degree of the denominator.

Second: Factor the numerator and denominator completely.

Third: Look for a common factor in the numerator and denominator that could form a hole.

Fourth: Any leftover factors in the denominator form vertical asymptotes. Set each factor equal to 0 and

solve.

The graphs of rational functions can have many different looks.

Lay the Ground Work

When graphing a rational function, gather all of the information about the important characteristics first:

Horizontal Asymptote Slant Asymptote

Holes Vertical Asymptotes

x-intercepts y-intercept

Example 1: Graph the rational function ( )2

4

xf x

x=

−

x-intercept: _______

y-intercept: _______

holes: ___________

Vertical Asymptote: ________

Horizontal Asymptote: _______

Additional Points:_______ _______

Domain:_________

Range:__________

-

x

y

Example 2: Graph the rational function ( )−

=−2

2(x 3)f x

x 9

x-intercept: _______

y-intercept: _______

holes: ___________

Vertical Asymptote: ________

Horizontal Asymptote: _______

Additional Points:_______ _______

Domain:_________

Range:__________

Example 3: Graphing the rational function 1

( ) 52

f xx

= +−

using transformations:

What would the parent function be?

What transformations would happen from the parent?

What impacts do those transformations have?

Graph the parent and transformed function below:

-

x

y

-

x

y

Practice Problems- Rational Functions

1. The function f is given by

f (x) = , x , x 3.

(a) (i) Show that y = 2 is an asymptote of the graph of y = f (x).

(ii) Find the vertical asymptote of the graph.

(iii) Write down the coordinates of the point P at which the asymptotes intersect.

(b) Find the points of intersection of the graph and the axes.

(c) Hence sketch the graph of y = f (x), showing the asymptotes by dotted lines.

312

−+

xx

2. Consider the function f (x) = + 8, x 10.

(a) Write down the equation of

(i) the vertical asymptote;

(ii) the horizontal asymptote.

(b) Find the

(i) y-intercept;

(ii) x-intercept.

(c) Sketch the graph of f , clearly showing the above information.

(d) Let g (x) = , x 0.

The graph of g is transformed into the graph of f using two transformations.

Describe the two transformations from the graph of f.

10

16

−x

x

16

3. The function f (x) is defined as f (x) = 3 + , x .

(a) Sketch the curve of f for −5 x 5, showing the asymptotes.

(b) Using your sketch, write down

(i) the equation of each asymptote;

(ii) the value of the x-intercept;

(iii) the value of the y-intercept.

52

1

−x

2

5

4. Let f (x) = , where p, q +

(the positive rational numbers).

Part of the graph of f, including the asymptotes, is shown below.

(a) The equations of the asymptotes are x =1, x = −1, y = 2. Write down the value of

(i) p;

(ii) q.

22

3

qx

xp

−−

Quadratics

Please follow this link to find the tutorial video: https://youtu.be/a5fHHIKULOQ

The general equation for a quadratic equation is 2 , 0y ax bx ac+ + = .

Solutions of Quadratic Functions

What is the solution to a quadratic equation?

Graphically, it is the _________________ of the parabola.

Algebraically, it is the value(s) of x when y is set equal to _____.

How do you solve a quadratic equation?

1. Graph the equation and ______________________________________.

2. ______________________________. Then use the zero product property.

3. _______________________________________.

The graph of a quadratic equation is a ___________.

A parabola has vertical line symmetry about a line called

the ___________________.

It has either a maximum (highest point) or a minimum

(lowest point) called the _____________.

The axis of symmetry always goes through the vertex of

the parabola.

The equation for the axis of symmetry is _________.

Therefore, the x – coordinate of the vertex is ______.

The y – intercept of the graph is ___________.

Domain: _________________

Range: __________________

The parent graph has a minimum value of

_____ at x = _____.

Decreasing on the interval _________

Increasing on the interval __________

Example: Solve the equation 2 12y x x= − − .

Graphing Method: Factoring Method:

( )( )4 3 0x x− + =

4 0 3 0x x− = + =

The solution is: _____________________

The Quadratic Formula

Use the Quadratic Formula when the quadratic expression cannot be factored.

Set the equation equal to zero:

2 0ax bx c+ + =

2 4

2

b b acx

a

− −=

The _________________ is the _____________ part of the Quadratic Formula. This indicates the type

of solutions the equation will have.

If:

2 4 0b ac− (negative) then _______________________________

2 4 0b ac− = then ______________________________

2 4 0b ac− (positive) then ______________________________

Vertex form of a quadratic equation

The vertex form of a quadratic equation is ______________________________.

The vertex of the parabola is _________.

How do you change a quadratic equation from standard form to vertex form?

Example: Change the equation 2 6 3y x x= + − to vertex form.

1. Add the 3 to both sides of the equation and reformat.

2. Complete the square.

3. Factor the quadratic trinomial.

4. Subtract 12 from each side of the equation.

5. The vertex is ___________.

Use the video to change 23 6 7y x x= − + into vertex form.

The vertex is : ________________

Practice Problems:

1. Find the values of m for which the quadratic x x m2 2 0 has

Hint: Consider the discriminant. (a) one real solution a) _____________

(b) two real solutions b) _____________

(c) no real solutions c) _____________

2. Let ( ) ( )2

f x a x h k= − + . The vertex of the graph of f is at ( )2,3 and the graph

passes through the point ( )1,7 .

a) Write down the value of h and k . a) ____________________

b) Find the value of a . b) ____________________

3. Let ( ) ( )2

1 5.g x x= − − +

a) Write down the coordinates of the vertex of the graph of .g a) _____________

Let ( ) 2.f x x= The following diagram shows part of the graph of .f

The graph of g intersects the graph of f at 1x = − and 2x = .

b) On the grid above, sketch the graph of g for 2 4.x−

4. The diagram below shows part of the graph of ( ) ( )( )1 3 .f x x x= − +

a) Write down the x – intercepts of the graph of f .

__________________

b) Find the coordinates of the vertex of the graph of .f

____________________

5. The following diagram shows part of the graph of a quadratic function f .

6. Let ( ) ( )2

3 1 12f x x= + − .

a) Show that ( ) 23 6 9f x x x= + − .

b) For the graph of f

i) Write down the coordinates of the vertex. i) _____________

ii) Write down the y – intercept. ii) _____________

iii) Find both x – intercepts. iii) _____________

c) Hence sketch the graph of .f

The vertex is at ( )1, 9−

, and the graph crosses the x – axis at the point

( )0,c.

The function can be written in the form ( ) ( )

2f x x h k= − +

.

a) Write down the value of h and k .

a) ______________________

b) Find the value of c.

b) ____________

Section 3: Polynomials

Definition: A polynomial is an equation in the form 𝑃(𝑥) = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + ⋯ + 𝑎1𝑥 + 𝑎0.

Key Terms: Coefficient: Each of the 𝑎𝑛 in the equation. The leading coefficient is the coefficient

of the first term.

Term: Each 𝑎𝑛𝑥𝑛

Degree: The n value.

End Behavior: What the function is doing as it approaches ±∞. The notation for

end behavior is “As 𝑥 → ±∞ , 𝑓(𝑥) → _____"

Even & Odd Functions

A function is even if its degree is even. A function is odd if its degree is odd. Depending on the

degree and whether the leading coefficient is positive or negative, we can determine the end

behavior of the polynomial.

When looking at the end behavior, it gives us insight into the domain and range of the

polynomials. Notice, for all 4 situations, the domain is all real numbers, but the range may be

restricted. For the odd functions, the range is also all real numbers, but since the even functions

approach the same infinity as x approaches both positive and negative infinity, the range will be

restricted. In order to find the range, we need to look at calculating maximums and minimums.

Maximums/Minimums

Key Terms: Increasing: A function is increasing if the y-values are increasing as the x-values

increase. When a function is increasing, it has a positive slope.

Decreasing: A function is decreasing if the y-values are decreasing as the x-values

increase. When a function is decreasing, it has a negative slope.

Maximum: A function has a maximum when it switches from increasing to

decreasing.

Minimum: A function has a minimum when it switches from decreasing to

increasing.

We can find the maximum and minimum on the calculator by graphing the function in “Y=” and

using “2nd TRACE (calc)” then 3 for a maximum or 4 for a minimum. When using the calculator

to find either of these, you will need to choose a point to the left and right of the max/min value,

and then finally choose a point close to the max/min value.

Graph the function 𝑓(𝑥) = −2𝑥3 + 3𝑥

Graph the function 𝑓(𝑥) = −𝑥4 + 3𝑥2 + 3

Coordinates of max:

Coordinates of min:

Increasing Interval:

Decreasing Interval:

Coordinates of max:

Coordinates of min:

Increasing Interval:

Decreasing Interval:

-

x

y

-

x

y

Zeros/Solutions/Roots/x-intercepts

Key Terms: Factor: (𝑥 − 𝑏) is a factor of a polynomial if and only if b is a zero of the function

Solving a polynomial can be thought of as finding the zeros, roots, or x-intercepts of the

function. We can do this graphically on a calculator, or analytically by factoring or using

polynomial division.

When factoring polynomials of degree 3 or higher, we typically use either grouping or

u-substitution.

Solve the following polynomials:

1: 3𝑥3 + 9𝑥2 − 162𝑥 = 0 2: 𝑥3 + 40𝑥 = 13𝑥2

3: 𝑥3 + 𝑥2 = 9𝑥 + 9 (Grouping) 4: 𝑥4 − 9𝑥2 + 8 = 0 (U-substitution)

We can also using polynomial division (either long or synthetic division) to tell us about roots.

𝑥3+3𝑥2−4𝑥−12

𝑥−2 (Long division)

𝑥3+3𝑥2−4𝑥−12

𝑥−2 (Synthetic Division)

Sometimes the division leaves us with a remainder that we will put over the original

denominator in our solution. We may also have a polynomial missing a specific term where

we will need to take that term into account with its coefficient as 0.

𝑥3−2𝑥2−22𝑥+40

𝑥−4

𝑥3+5𝑥2−18

𝑥+3

In the one of the above expressions, we had a remainder. If the remainder is zero, we know that

it is a factor of the polynomial. If it is not a factor, then there will be a non-zero remainder. We

can find this using the remainder theorem:

Remainder Theorem: If a polynomial expression, 𝑓(𝑥) is divided by 𝑥 − 𝑎, then the remainder is

the number 𝑓(𝑎).

Ex: Find the remainder if the polynomial 𝑓(𝑥) = 2𝑥3 + 7𝑥2 + 2𝑥 + 1 is divided by 𝑥 − 5.

It is beneficial when we are looking for roots to know how many we are looking for. The

fundamental theorem of algebra tells us exactly how many we can find!

Fundamental Theorem of Algebra: Every polynomial of degree n has exactly n zeros, counting

multiplicities.

There are 2 clarifications we need when looking at the Fundamental Theorem of Algebra

1) Multiplicity: The number of times a single value is a factor of a polynomial. For

example, the same value may be a factor twice, giving it a multiplicity of 2.

2) The roots DO NOT have to be real, they may be imaginary!

Solve the following polynomials.

1: 4𝑥3 − 20𝑥2 − 3𝑥 + 15 = 0

2: 2𝑥3 − 𝑥2 + 𝑥 −1

2= 0

3: 𝑥4 − 14𝑥2 + 45 = 0

4: 𝑥6 − 2𝑥4 − 4𝑥2 + 8 = 0

Some practice IB Problems:

1) Let 𝑔(𝑥) = 𝑥4 − 2𝑥3 + 𝑥2 − 2

a. Solve 𝑔(𝑥) = 0

Let 𝑓(𝑥) =2𝑥3

𝑔(𝑥)+ 1. A part of 𝑓(𝑥) is shown below.

b. The graph has vertical asymptotes with equations 𝑥 = 𝑎 and 𝑥 = 𝑏 where 𝑎 < 𝑏. Write down the values of a and b.

c. The graph intersects the x-axis at the points A and B. Write down the exact value

of the x-coordinate at A and B

y

x0A B

C

2) The polynomial 𝑥4 + 𝑝𝑥3 + 𝑞𝑥2 + 𝑟𝑥 + 6 is exactly divisible by each of (𝑥 − 1), (𝑥 − 2), and (𝑥 − 3). Find the values of 𝑝, 𝑞, and 𝑟.

3) The same remainder is found when 2𝑥3 + 𝑘𝑥2 + 6𝑥 + 32 and 𝑥4 − 6𝑥2 − 𝑘2𝑥 + 9 are

divided by (𝑥 + 1). Find the possible values of k.