class 051005

Give the two basic points of the function y = f(x) = x.  Answer the following, including a sketch for each:

The two basic points are (0, 0) and (1, 1).

• If the function is vertically stretched by factor -2, to what coordinates will the two basic points move?
• A vertical stretch by factor -2 will leave (0, 0) unchanged and will take (1, 1) to (1, -2).

• If the function is then vertically shifted +5 units, to what coordinates will these points then move?
• A vertical shift will then move (0, 0) to (0, 5) and (1, -2) to (1, 3).

• What is the expression for the stretched and shifted function?
• The function is now y = -2 f(x) + 5 = -2 x + 5.

Give the three basic points of y = f(x) = x^2.   Answer the same questions as above for this function.

The three basic points are (-1, 1), (0, 0) and (1, 1).

A vertical stretch by factor -2 will take these points to (-1, -2),  (0, 0) and (1, -2).

A vertical shift of +5 will take these points to (-1, 3), (0, 5) and (1, 3).

The function will be y = -2 f(x) + 5 = -2 x^2 + 5.

If f(x) = -.3 x + 2, then give expressions for each of the following, simplifying where possible:

• f(a+b)

f(a+b) = -.3 ( a + b) + 2, which might be expressed as -.3 a - .3 b + 2.

• f(x2) - f(x1)

f(x2) - f(x1) = [ -.3 x2 + 2 ] - [-.3 x1 + 2 ] = -.3 x2 + 2 + .3 x1 - 2 = -.3 x2 + .3 x1 = -.3(x2 - x1).

• ( f(x2) - f(x1) ) / (x2 - x1)

f(x2) - f(x1) = -.3(x2 - x1), by the steps shown above.

So

( f(x2) - f(x1) ) / (x2 - x1) = -.3 (x2 - x1) / (x2 - x1) = -.3.