Question 176248
Well, from the data you provided, you would have to conclude that the change in the temperature over the give time span (3 days) is a linear function.
How would you come to that conclusion? By simply plotting the given data as points on a graph. Choose the temperature (T) as the dependent variable and the day number as the independent variable, you know, like y is commonly the dependent variable and x is commonly the independent variable.
So you are given three points which you can express as: (d, T) for ( day No., Temp).
The three points are:
(1, 27), (2, 26.5), (3, 26)
So let's find the slope of the line represented by these points using the standard slope formula (m) and two of the given points:
{{{m = (y[2]-y[1])/(x[2]-x[1])}}} but, of course, we will use d (day) instead of x and T (temp) instead of y, so...
{{{m = (T[2]-T[1])/(d[2]-d[1])}}} Let's use the two points (1, 27) and (3, 26), so...
{{{m = (26-27)/(3-1)}}} Simplifying this, we get:
{{{m = -(1/2)}}} so we have a negative slope which means the temperature (T) is decreasing with the passage of days, where T = Temp. and d = day number.
Now we can start the formula using the "Point-slope" form of a linear equation: {{{y = mx+b}}} or, in this problem, {{{T = md+b}}}, so we can write:
{{{T = (-1/2)d+b}}} But we need to find the value of b, the T-intercept. We can do this by substituting the T- and d-values from any one of the three given points. Let's choose (2, 26.5), so...
{{{26.5 = (-1/2)(2)+b}}} Simplify this and solve for b.
{{{26.5 = -1+b}}} Add 1 to both sides.
{{{27.5 = b}}} Now we can write the final equation.
{{{highlight(T = (-1/2)d+27.5)}}}
Let's see what this would look like on a graph:
{{{graph(400,400,-3,60,-2,30,(-1/2)x+27.5)}}}