class 051028

1.  If y = f(x) = A x^p, and the points (2, 8) and (5, 3) lie on the graph of y vs. x, then what are the value of A and p?

Plugging in the coordinates of our points we get the equations

8 = A * 2^p and

3 = A * 5^p.

Dividing the first by the second we get

8/3 = (A * 2^p) / (A * 5^p) = A / A * ( 2^p / 5^p) = 1 * (2/5)^p so that

(2/5)^p = 8/3.

We solve by trial and error, finding that p = -1.1, approx..

Substituting this back into the first equation we get 8 = A * 2^   so that

A = 8 / 2^-1.1  = 3.8.

Substituting these values of A and p into the original form we have

y = 3.8  x ^-1.1.

2.  A team of 100 workers can scrub the face of a 300-foot-high pyramid in a week.  Assuming that the workers in both teams are equally productive, how many workers would it take to scrub the fact of a 400-foot-high pyramid in a week?

The workers are cleaning the surface of the pyramid.   The number of workers needed is therefore proportional to the surface area of the pyramid.

The surface area of the pyramid is proportional to the square of its altitude.

We have y = k x^2 where y is number of workers and x is the altitude of the pyramid.

Plugging in our information

100 = k * 300^2 so

k = 100 / 300^2 = .00111 so

y = .00111 x^2.

If x = 400 we get

y = .00111 * 400^2 = 177.6,

so we better hire 178 workers if we want to get done on time.

3.  What are the basic points and asymptotes of y = f(x) = x^-2?

The basic points are the x = -1, 0, 1/2, 1 and 2 points. 

At x = 0 the function is undefined, but by thinking about what happens when x is close to 0 conclude that there is a vertical asymptote at x = 0.

At x = 1 we have y = (-1)^(-2) = 1 / (-1)^2 = 1.

At x = 1 we have y = (1)^(-2) = 1 / (1)^2 = 1.

At x = 1/2 we have y = (1/2)^(-2) = 1 / (1/2)^2 = 1/(1/4) = 4/1 = 4.

At x = 2 we have y = (2)^(-2) = 1 / (2)^2 = 1/4.

What are the basic points and asymptotes of y = 2 f(x - 1) + 3?

Each basic point gets vertically stretched by factor 2, horizontally shifted 1 unit and vertically shifted 3 units. 

the point (-1, 1) becomes (-1, 2 * 1) = (-1, 2), then (-1 + 1, 2) = (0, 2), then (0, 2 + 3) = (0,5).

the point (1, 1) becomes (1, 2 * 1) = (1, 2), then (1 + 1, 2) = (2, 2), then (2, 2 + 3) = (2,5).

the point (1/2, 4) becomes (1/2, 4 * 1) = (1/2, 8), then (1/2 + 1, 8) = (3/2, 8), then (3/2, 8 + 3) = (3/2, 11).

What is the equation of this function?

Sketch the graph of this function.