Action

Notebook

You will benefit from keeping an organized notebook, as this notebook will inform your Learning Log. The Learning Log is an assessment for feedback, but not marks. Choose a notebook format that you prefer, it can be digital or analog (paper notebook.) There will be times where the course will refer to technological options for solving questions and there will be times when you have to complete them on paper, by hand. You will be prompted to reflect on your learning and document evidence of your growth throughout the course.

As you complete the learning activities, you will make use of a notebook of your choice to:

- - complete solutions to sample questions and problems;
  - define mathematical terms;
  - track webpage URLs for practice and information to support your learning;
  - reflect on your progress as a self-directed, independent learner.

You will be provided with further instructions at the end of Unit 1 in Learning activity 1.3 on:

- - how to prepare your Unit 1: Rates of Change Learning Log Entry
  - how to submit this assessment for feedback

Review

In this course, you will be asked to recall concepts you learned in prior mathematics courses. In order to ensure you are prepared with all the necessary skills required for success, it is a good idea to take the time to review.

This first Learning activity will review material from the grade nine mathematics through to the grade twelve functions course.

Expanding Polynomials

Here are some expanding polynomial questions for you to try.

Answers and solutions are available for comparison when you are ready. Please review the solutions to ensure you are following the correct format.

Expand and simplify.

a. $(x+3)(x-4)$

b. $(2x-1)(3x+5)$

Suggested Answer

(x+3(x−4)

=x²−4x+3x−12

=x²−x−12

Suggested Answer

(2x−1)(3x+5)

=6x²+10x−3x−5

=6x²+7x−5

c. $(3 x - 7) (3 x + 7)$

d. (5x – 2)²

Suggested Answer

(3x−7)(3x+7)

=9x²+21x−21x−49

=9x²−49

Suggested Answer

(5x – 2)²

=(5x−2)(5x−2)

=25x²−10x−10x+4

=25x²−20x+4

Linear Functions

Linear functions are polynomials with a degree of 1. They form straight lines.

The slope/intercept form of a linear function is $y=mx+b$ , where m is the slope and b is the y-intercept.

To determine the equation of a line given the slope and a point on the line use the formula $y=m(x-$ x₁ $)+$ y₁, where m is the slope and $($ x₁ $,$ y₁ $)$ is the point on the line.

The slope of a straight line going through $A($ x₁ $,$ y₁ $)$ and $B($ x₂ $,$ y₂ $)$ is $m=$ (y₂-y₁)/(x₂-x₁) $.$

When the slope of a line is positive, we say the line is increasing, which means it goes up from left to right.

Use the view icon to see the increasing movement.

When the slope of a line is negative, we say the line is decreasing, which means it goes down from left to right.

Use the view icon to see the decreasing movement.

A horizontal line has a slope of 0.

The slope of a vertical line is undefined. To understand this better, let’s consider riding a bike. Riding the bike on a flat road there is no slope. When riding the bike up a hill the steepness of the hill is the slope. Now consider a vertical wall. It is impossible to ride the bike straight up the wall. Therefore, the slope is said to be undefined.

Quadratic Functions

Quadratic functions are polynomials with a degree of 2. They form parabolas that open upwards or downwards.

A parabola that opens upwards changes from decreasing to increasing. The change occurs at the vertex.

Use the view icon to see the movement.

A parabola that opens downwards changes from increasing to decreasing. Once again, the change occurs at the vertex.

Use the view icon to see the movement.

Factoring

Factoring can be used to find the roots (also known as the $x -$ intercepts) of a function.

These are the five most usual types of factoring:

Common factoring

Factor by inspection

Factor by decomposition

If you have difficulty finding the two numbers used in factoring by decomposition, there is an efficient method that uses the average of the two numbers.