## Introduction to the culminating assessment - Math journal

#### During the course:

Throughout the course you will be keeping a math journal based on the suggestions made at the end of each learning activity.

Take a moment to review the “Math Journal Assessment Break-down” (Opens in new window).

This assessment is worth 10% of your final mark.

#### End of the course:

You will select 2 entries from each unit (8 entries in total) to submit as part of your culminating assessment.

Here is a reminder of the 4 units in the course:

• Unit 1: Polynomial Functions
• Unit 2: Exponential Functions
• Unit 3: Trigonometric Functions
• Unit 4: Applications of Geometry

### How to format your journal

Your math journal entries can be presented in a variety of ways. They do not all need to be in the same format. Here are some suggestions:

• handwritten journal
• online journal
• video recordings
• pictures
• audio recordings

The content should be organized in a way that makes the most sense to you and will likely vary depending on the topic. Some ideas include:

• tables
• T charts to compare/contrast
• mind maps
• flow charts

### What you should include in your journal

Each journal entry should include the following:

• Unit and learning activity (1)
• Description of the task, as written in the learning activity (2)
• The required content, written by you with evidence of your learning (3)

How you present part (3) is up to you!

Here is an example:

#### Unit 1 Learning activity 1: Polynomial and power functions(1)

In your math journal, summarize how to determine the domain of a function. (2)

The domain of a function is all possible x values of that function. For example, consider the following function:

The domain of this function exists for all values of x. We would write this as $\left\{xϵ\mathbb{R}\right\}$(3)

Each journal entry you submit for your culminating assessment should include evidence of learning from that learning activity. Some ideas may be:

• Think about what would be most helpful as review if you returned to your journal at a later date:
• Examples
• Diagrams
• Written explanations
• Summary sheet
• Did you devise a memory trick to serve as a reminder for the steps of a process you found challenging?
• Were there particular types of questions where you frequently made errors?
• You may include personal reflections:
• Do you think this content is useful in your daily life, or might it be in the future? Why or why not?
• You could even use emojis to rate each aspect of the content throughout this assessment (😃,☹️,😠)!

### How the journal will be assessed

You will have the opportunity to submit two journal entries for feedback based on the rubric for this assessment; one in unit 2 and one in unit 3. You will want to review the feedback and make sure you understand the expectations for this assessment. You may decide to include these journals as part of your culminating assessment; make sure to update them based on the feedback.

Please read through the “Math Journal Assessment Rubric” to ensure full understanding of the assessment guidelines.

#### Polynomial and power functions

Throughout the course, you will have to answer problem solving questions. The following website has great tips on how to approach these types of questions:

Problem Solving and Mathematic Discovery

The aforementioned link has been provided as a suggestion only. You are encouraged to search for additional resources to broaden your range of problem solving strategies .

In mathematics, the word function refers to a relationship where one variable, or quantity, depends on another. More specifically, where there is only one value of the dependent variable (y) for each independent value (x).

Think of an example of a function in the real-world. Look at the suggested answers to check your ideas.

In previous math courses you learned about two main types of functions:

Linear Quadratic
$y=mx+b$ $y=a{x}^{2}+bx+c$
The graph of a linear function is a straight line. The graph of a quadratic function is a U-shaped curve, opening either up or down.

For each image that follows, identify whether it is a representation of a linear function, a quadratic function, or neither.

### Topic 1 – Polynomial functions

Here are examples of two types of polynomial functions you are familiar with:

Linear: $f\left(x\right)=x+3$
Quadratic:

Can you see a pattern? Try to think of another type of polynomial function. For example:

Cubic:
Quartic:

### The parts of a polynomial function

A polynomial function has the form

#### $n$: natural number

• any positive integer (1, 2, 3, ….)
• the highest value of any $n$ exponent is called the degree of the function

#### $x$: variable

• a letter that represents a quantity

#### ${a}_{2}$: coefficient

• any of are numbers in front of the variable, $x$

#### ${a}_{0}$: constant

• a number by itself

#### ${a}_{n}{x}^{n}$: leading (or dominant) term

• the term with the highest exponent or degree of the function

#### ${a}_{n}$: leading coefficient

The terms in a polynomial function are to be written in descending order of exponents. Some terms may appear to be ‘missing’ in the order because their coefficient is 0.

For example,

has a coefficient of 0 for the ${x}^{2}$, but the terms are still in descending exponential order.

It may be helpful to search the internet for tutorials explaining the “parts of a polynomial”.

### Example:

Let’s go through an example of identifying parts of a polynomial equation. Remember to check your solutions with the suggested answers.

We’ll use the polynomial function , to identify the following:

#### The leading, or dominant, term

The leading, or dominant, term is the one with the highest exponent, it is ${4x}^{5}$.

#### The degree of the polynomial

The degree is the highest exponent, or the exponent with the dominant term, it is 5.

#### The leading coefficient

The leading coefficient is the number in front of the leading term, it is 4.

#### The constant term

The constant term is the number with no variable. The constant term is 15.

### Types of functions:

#### Rational function:

Created when one polynomial function, $f\left(x\right)$, is divided by another, $g\left(x\right)$

Note: The polynomial function in the denominator, g(x), must have a degree of at least 1.

Think of an example of a rational function and record it in your notebook. Look at the suggested answers to check your ideas.

#### Square Root Function:

Created by taking the square root of a polynomial function, $f\left(x\right)$

Think of an example of a square root function and record it in your notebook. Look at the suggested answers to check your ideas.

### Examples:

Let’s go through some examples of identifying polynomial functions.

Identify the degree and leading coefficient for the polynomial functions.

Now, see if you can match each function included on the left with its correct classification on the right.

### Notebook ### Polynomial functions

Complete these “Polynomial Function Practice Questions” (Opens in new window). You may want to use your notebook to record the solutions. When finished, use the “Polynomial Function Practice Suggested Answers” (Opens in new window) to check your work.

### Power function:

A power function is the simplest form of a polynomial function:

$a:$ any real number

$x:$ a variable

$n:$ any natural number

Some power functions are given special names based on their degree. Review each of the following power functions, identifying the degree and type of power function. Look at the suggested answer to check your ideas.

1. Identify the degree and the type of power function $y=a$.

2. Identify the degree and the type of power function $y=ax$.

3. Identify the degree and the type of power function $y={ax}^{2}$.

4. Identify the degree and the type of power function $y={ax}^{3}.$

5. Identify the degree and the type of power function $y={ax}^{4}$.

### Properties of a function

The end behaviour of a function tells us how the y-values are changing at both horizontal ends of the graph or as the $x$-values get very large or very small:

• $x$-values getting larger in the negative direction
• $x$-values getting larger in the positive direction

Next, you’ll have an opportunity to review each in more detail.

### Example: Large Negative and Positive Value of $x$

For each example included in the, “Large Negative and Positive Value(s) of x Practice Exercise” (Opens in new window), identify the end behaviour of each function. Then, check your solutions with the suggested answers that follow.

The following tabs include the end behaviours for large negative values of $x$ (the left, or beginning, of the function). Select each tab to review the suggested answers for all three examples.

#### Example 1 - End Behaviour Large Negative Values

Starts low

The $y-$values are getting larger in the negative direction

#### Example 2: End Behaviour Large Negative Values

Starts low

The $y$-values are getting larger in the negative direction

#### Example 3 – End Behaviour Large Negative Values

Starts high

The $y$-values are getting larger in the positive direction

Suggested Answers for Large Positive Value of $x$ Practice Exercise
Now, review the suggested answers for large positive value of immediately.

The following tabs include the end behaviours for large positive values of $x$ (the right, or ending, of the function). Select each tab to review the suggested answers for all three examples.

#### Example 1 – End Behaviour Large Positive Values

Ends high

The $y$-values are getting larger in the positive direction.

#### Example 2 – End Behaviour Large Positive Values

Ends low

The $y$-values are getting larger in the negative direction.

#### Example 3 – End Behaviour Large Positive Values

Ends low

The $y$-values are getting larger in the negative direction.

The domain of a function tells us for which $x-$values the function is possible.

• For example, the function $y=\sqrt{x}$, only positive values of $x$ and 0 are possible, so the domain would be
• $\left\{xϵ\mathbb{R}|x\ge 0\right\}$, ‘$x$ is an element of real numbers, $R$, such that, |, $x$ is greater than or equal to, $\ge$, 0’

The range of a function tells us for which $y$-values the function is possible

• For example, the function , is a parabola that opens upward with vertex (0,-2), so only $y$-values greater than or equal to $-2$ are possible for this function
• $\left\{yϵ\mathbb{R}|y\ge -2\right\}$, ‘$y$ is an element of real numbers, $R$, such that, |, $y$ is greater than or equal to, $\ge$, -2’

### Examples:

Let’s go through some examples of identifying the domain and range. Remember to check your solutions with the suggested answers.

### Notebook Study the graph and record the domain and range in your notebook. Then, continue to the next tab to check your answers.

$\left\{x\in R\right\}$

All values of $x$ are possible for this linear function

$\left\{y\in R\right\}$

All values of $y$ are possible for this linear function

#### Example 2:

Here is another example to try.

### Notebook Study the graph and record the domain and range in your notebook. Then, continue to the next tab to check your answers.

$\left\{x\in R\right\}$

All values of $x$ are possible for this quadratic function

$\left\{y\in R|y\le -2\right\}$

Only $y$-values less than or equal to -2 are possible for this quadratic function

#### Example 3:

Great work! Here is one more example to try.

### Notebook Study the graph and record the domain and range in your notebook. Then, continue to the next tab to check your answers.

$\left\{x\in R|x\ge 0\right\}$

Only $x$-values greater than or equal to 0 are possible for this function

$\left\{y\in R|y\le 0\right\}$

Only $y$-values less than or equal to 0 are possible for this function

The $x$-intercept(s) of a function are the point(s) where the function crosses the $x$-axis or when $y=0$

For example, this function has $x$-intercepts at $\left(-1,0\right)$ and $\left(4,0\right)$

The $y$-intercept of a function is the point where the function crosses the $y$-axis or when $x=0$

For example, this function has the $y$-intercept at $\left(0,2\right)$

### Examples:

Let’s go through another example of identifying the $x$- and $y$-intercept(s). Select each tab to learn more.

$\left(2,0\right)$

$\left(0,1\right)$

Now, you give it a try. The answers are provided so that you can check your work.

### Notebook Study the graph and record the $x$-intercept(s) and $y$-intercept in your notebook. Then, continue to the next tab to check your answers.

$\left(-5,0\right)$ and $\left(3,0\right)$

$\left(0,-6\right)$

### Notebook How did you make out? Here’s one more practice opportunity. Study the following graph and record the $x$-intercept(s) and $y$-intercept in your notebook. Then, review the "x-intercept(s) and y-intercept exercise suggested answers” (Opens in new window).

#### Domain, Range, and End Behaviours Practice Exercise

Using Geogebra or another free online graphing tool, investigate how the domain and end behaviour changes for positive and negative leading coefficients. Then use the, “Domain, Range, and End Behaviours Suggested Answers” (Opens in new window) to check your ideas.

### Notebook Complete these practice questions on power functions. You may want to use a notebook either online or on paper.

### Self Check As a self-directed learner, you will be reflecting on your learning process and checking your understanding in order to plan for success.

Rate your understanding on a scale of 1 to 5, where 1 = “I am confused” and 5 =”I have a thorough understanding.

Agree or Disagree statements
Statement 1 2 3 4 5
Recognize the equation of a polynomial function and give reasons why it is a function and identify linear and quadratic functions as examples of polynomial functions
Describe the type and degree (linear, quadratic, cubic, quartic) for graphical and algebraic representations of a polynomial function
Describe the domain and range for graphical and algebraic representations of a power function
Describe the end behaviours for graphical and algebraic representations of a power function

If you rated your understanding below 3, take some time now to think about how you will work to improve your rating where it is needed.

### Math Journal In your math journal, summarize how to determine the domain, range, and end behaviour of different power functions. Remember to include evidence of your learning with examples, pictures, and/or explanations.

### Notebook Now, work through the “Lesson 1: Final Practice Questions”, making sure to check your answers with the suggested solutions. You can use your notebook, a free online graphing application such as Desmos, or any graphing tool that you find helpful.