Answer to Question #269034 in Algebra for Paul Carpenter

Question

Lines can be used to approximate a wide variety of functions; often a function can be described using many lines.

If a stock price goes from $10 to $12 from January 1st to January 31, from $12 to $9 from February 1st to February 28th, and from $9 to $15 from March 1st to March 31th is the price change from $10 to $15 a straight line?

It is clear that in each of the three time intervals mentioned there was a complex daily variation of prices as in an electrocardiogram. But what would be a simplified solution for a first naive view of the situation? Would a simple function hold up? What is the simplest function to represent this situation? Does your naïve initial and simplified model allow you to predict the behavior of the stock in the next month?

How can I use three “pieces” of lines to describe the price movements from the beginning of January to the end of March? Show the graph for the price movement.

y = x + 2 {0 < x < 2}

y = –x + 6 {2 < x < 5}

y = 2x – 9 {5 < x < 8}

Answer 1

Assumption: End of previous month price is the starting price of the following month

Number of days from January to March=90days up to end if April =120 days

TABLE OF VALUES

$\begin{matrix} Month&JAN&FEB&MARCH \\ Days&0&31&59&90 \\ Stock Price (\$)&10&12&9&15 \end{matrix}$

DESCRIPTION OF STOCK VARIATION USING LINE GRAPH

Let $x=$ Number of Days and $y=$ stock price ($)

JANUARY

Line passes through points $(1,10)(31,12)$

Gradient of line $=\frac{12-10}{31-1}=\frac{1}{15}$

$\frac{y-10}{x-1}=\frac{1}{15}$

Simplify $y=\frac{1}{15}x+\frac{149}{15}$

FEBRUARY

Line passes through points $(32,12)$ and $(59,9)$

Gradient of line $=\frac{9-12}{59-32}=\frac{-3}{27}=\frac{-1}{9}$

$\frac{y-12}{x-32}=\frac{-1}{9}$

Simplifying $Y=\frac{-1}{9}x+\frac{140}{9}$

MARCH

Line passes through points $(60,9)$ and $(90,15)$

Gradient of line $=\frac{15-9}{90-60}=\frac{6}{30}=\frac{1}{5}$

$\frac{y-9}{x-60}=\frac{1}{5}$

Simplifying $y=\frac{1}{5}x-3$

The pattern that emerges after plotting the graphs suggests a linear relation between price variation with days

The simplest solution for a first naive view of the situation is a line of best fit between January and March.

The line passes through points $(0,10)$ and $(120,13)$

Gradient of the line $=\frac{13-10}{120-0}=\frac{1}{40}$

$\frac{y-10}{x-0}=\frac{1}{40}$

Simplifying $y=\frac{1}{40}x+10$

Although there is a complex daily variation of price and a simple function cannot hold up, but the line of best fit can be extrapolated to predict stock price in the month of April

The stock will go from $\$15$ to $\$13$ from March to April as per the coordinates read from the line of best fit on $120^{th}$day (End of April)

From January to April , though there are sharp variation , the price movement can simply be said to be gradually increasing.