## 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 $\left\{x\u03f5\mathbb{R}\right\}$(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.

There are many examples of functions that exist in the real-world, here are just some suggestions:

- The time it takes for a car to stop depends on the speed of the car
- The length of a tree’s shadow depends on the time of day
- The number of daylight hours depends on the time of year
- The dollar amount earned in interest depends on the interest rate

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: $f\left(x\right)={x}^{2}+4x+3$

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

Cubic: $f\left(x\right)={x}^{3}+{5x}^{3}+x-7$

Quartic: $f\left(x\right)={x}^{4}-{3x}^{3}+{x}^{2}-5$

### The parts of a polynomial function

A **polynomial function **has the form

$f\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+{a}_{n-2}{x}^{n-2}+...+{a}_{3}{x}^{3}+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$

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 $f\left(x\right)={4x}^{5}-{7x}^{4}+{2x}^{3}-{6x}^{2}+10x+15$ , to identify the following:

### Types of functions:

#### Rational function:

$y=\frac{f\left(x\right)}{g\left(x\right)}$

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.

There are many examples of rational functions, here are just some suggestions:

$y=\frac{{2x}^{2}-3x+2}{x-1}$

$y=\frac{4}{8{x}^{2}-3}$

$y=\frac{x+4}{3x-5}$

#### Square Root Function:

$y=\sqrt{f\left(x\right)}$

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.

There are many examples of square root functions, here are just some suggestions:

$y=\sqrt{{x}^{2}+1}$

$y=\sqrt{{4x}^{3}-x+8}$

$y=\sqrt{x}$

### 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.

### Topic 2 – Power functions

### Power function:

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

$f\left(x\right)={ax}^{n}$

$a:$ 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 $y=a$.

**Degree:** 0

**Type:** Linear

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

**Degree:** 1

**Type:** Linear

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

**Degree:** 2

**Type: **Quadratic

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

**Degree:** 3

**Type:** Cubic

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

**Degree:** 4

**Type:** Quartic

### 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.

**Suggested Answers for Large Positive Value of $x$ Practice Exercise**

Now, review the suggested answers for large positive value of $x$immediately.

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
- $\left\{x\u03f5\mathbb{R}|x\ge 0\right\}$, ‘$x$ 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 $y={x}^{2}-2$, 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

- $\left\{y\u03f5\mathbb{R}|y\ge -2\right\}$, ‘$y$ 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.

#### Example 1:

### Notebook

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

#### 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.

#### 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.

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 $(-\mathrm{1,0})$ and $\left(\mathrm{4,0}\right)$

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 $\left(\mathrm{0,2}\right)$

### Examples:

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

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.

### 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.

#### Power functions

### Notebook

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

#### Review the success criteria:

### 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.

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.

### Connections

### 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.