06/06/24

The first big set of exercises

Hello all, hope you are enjoying this fabulous evening.

Today, begins the first of many tests of my skills and understandings of mathematics; to see where I am and where I need to be.

Starting with Questions 1 to 4, which just asked to convert an interval into set notation and then graph it. Fairly straightforward stuff, just had to go over my notes on how closed and open intervals are expressed, including both finite and infinite sets.

Closed brackets, represented by “[“, are expressed as less than or equal and means the endpoints are included. On the other hand, “(“ is open and expressed as less than (with the end point excluded - to represent the value could be anywhere along the real line).

Question 5 gave me trouble because I wasn’t sure what they meant by increasing or decreasing units of x and their effect on y. The equation was y = 5x - 12.

I believe they are asking how the changes in x affect y in terms of changes to the slope and how it will make y’s value (when y does not equal 0) different for the same value of x.

The red is the starting equation, the blue line is y = 8x - 12 and the green line is y = 3x - 12. It’s an excellent illustration of the y-intercept never changing regardless of x’s coefficient and how the coefficient pivots around a central point.

The image shows me why I believe I am meant to state something like “the coefficient of x affects the “steepness” of the slope” or it could be something else. Not sure. Will come back to it later.

Same kind of question for 6, so I moved onto 7. Another straightforward exercise where you are asked to find the slope from two different points. Below is the formula.

\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Up to Question 14, we will be using this formula. However, we have to be wary of when the x variables from each point equal each other. This will make the answer to the equation undefined. This is because of the rule:

\(x_2 \ne x_1\)

However, that is all for the session tonight, I will pick up from Question 8 tomorrow.

Thank you for reading this blog tonight, hope it means something to you.

Steve Frampton.