class 051003
Let y = f(x) = 3 x - 1.
What are the two basic points of this function?
Show what happens to the basic points if the function y = 2 f(x) is then vertically shifted +4 units.
Each point is raised 4 units in the y direction, increasing its y coordinate by 4. Our points (0, -2) and (1, 4) become (0, -2+4) = (0, 2) and (1, 4+4) = (1, 8).
If we wanted to horizontally shift the function 10 units?
The function y = 2 f(x-10) + 4 would accomplish this shift.
The function would be y = 2 f(x-10) + 4.
Since f(x) = 3x - 1,
f(x-10) = 3 ( x-10) + 1 and
2 f(x-10) + 4 would be
2 ( 3(x-10) - 1) + 4 =
2 (3x - 30 - 1) + 4 = 2 ( 3x - 31) + 4 = 6x - 62 + 4 = 6x - 58.
What is f(3)?
f(x) = 3x - 1 so f(3) = 3 * 3 - 1 = 8.
What is the expression for f(a + b)?
f(a+b) = 3(a+b) - 1 = 3a + 3b - 1.
What is the expression for f(x1)?
f(x) = 3x - 1 so f(x1) = 3 x1 - 1.
What is the expression for f(x2)?
f(x) = 3x - 1 so f(x2) = 3 x2 - 1.
What is the expression for f(x2) - f(x1)?
f(x2) = (3 x2 - 1) and f(x1) = (3 x1 - 1) so
f(x2) - f(x1) = (3 x2 - 1) - (3 x1 - 1)
= 3 x2 - 1 - 3 x1 + 1
= 3 x2 - 3 x1 - 1 + 1; since -1 + 1 = 0 this is the same as
= 3 x2 - 3 x1
= 3 ( x2 - x1).
What is the expression for ( f(x2)- f(x1) ) / ( x2 - x1)?
The numerator is just what we saw above. We would substitute and simplify to get f(x2) - f(x1) = 3 ( x2 - x1).
So the expression
(f(x2) - f(x1) ) / ( x2 - x1)
= 3 ( x2 - x1) / (x2 - x1)
= 3.
Note that since f(x2) is the y value corresponding to x = x2 (let's say that y2 = f(x2) ) and f(x1) is the y value corresponding to x = x1 (we'll say y1 = f(x1) ):
On a graph of y vs. x, (x1, f(x1) ) and (x2, f(x2) ) are points on the graph, and the slope of the line segment connecting these points is ( f(x2) - f(x1) ) / (x2 - x1).
For the linear function y = 3 x - 1, this expression simplifies to just 3, which proves what we thought we already knew--that between any two points of this graph the slope is 3.